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%
% Author:   Ruslan Abdulovich Sharipov
% Title:    Dynamical systems admitting the normal shift
% Comment:  Thesis for the degree of Doctor of Sciences in Russia
% Address:  Rabochaya street 5, Ufa, 450003, Russia
% E-mail:   R_Sharipov@ic.bashedu.ru
%           ruslan-sharipov@usa.net
% Web-page: http://www.geocities.com/CapeCanaveral/Lab/5341
%
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\def\SO{\operatorname{SO}}
\def\const{\operatorname{const}}
\def\compos{\,\raise 1pt\hbox{$\sssize\circ$} \,}
\pagewidth{12.8cm}
\pageheight{20cm}
\Monograph
\topmatter
\title\chapter{1}Introduction.
\endtitle
\endtopmatter
\loadbold
\startpage{6}
\document
\head
\S\,1. Classical constructions of surface transformation in 
three-dimensional space. 
\endhead
\rightheadtext{\S\,1. Classical constructions \dots}
     Interest to geometry of surfaces in the beginning of
\uppercase\expandafter {\romannumeral 19}-th century was stimulated 
by practical tasks of a geodesy. For the military purposes and for 
calculation of taxes the exact geographical maps were required. It 
is known, that Gauss since 1816 was engaged in organization of geodesic 
shooting of the Hanover kingdom, his own measurements were made in
1821-1825, while whole works lasted till 1841 (see \cite {1}, chapter 
\uppercase\expandafter {\romannumeral 1}). It was the very time when
Gauss obtained his basic results in the theory of surfaces \cite{2}
(including the theorem of invariance of Gaussian curvature under the 
bending of a surface without stretching).\par
     Here we consider later time, when differential geometry has
been developed into a separate mathematical discipline.
In papers of Bonnet (see \cite{3--5}), Bianchi \cite{6}, Sophus Lie
\cite{7}, B\"acklund \cite{8} and Darboux \cite{9} special 
transformations of surfaces in three-dimensional Euclidean space
were constructed. Totally in these papers four transformations
were determined: {\bf Bonnet} transformation, {\bf Bianchi and Lie}
transformation, {\bf B\"acklund} transformation and {\bf Darboux}
transformation. In each of four above constructions pairs of surfaces
$S$ and $\tilde S$ connected by a transformation $f\!:S\to \tilde S$
are considered. The appropriate points of two
surfaces are bound by a segment of straight line: point $A$ on $S$ 
is bound with a point $\tilde A=f(A)$ on $\tilde S$. Each transformation 
is defined by imposing certain limitations on mutual arrangement
\vadjust{\vskip 1.2cm \special{em:graph edis-01a.gif} \vskip -1.2cm}of
surfaces $S$ and $\tilde S$ and segments $[A\tilde A]$ that connect
appropriate points on $S$ and $\tilde S$.
\definition{Definition 1.1}\parshape 2 6cm 6.8cm 6cm 6.8cm Bonnet
transformation is defined by the following two conditions: 
\roster
\rosteritemwd=0pt 
\item "1)"\parshape 2 6cm 6.8cm 6cm 6.8cm segment $[A\tilde A]$ is
          orthogonal to the tangent plane $T_A(S)$ to $S$ at the
          point $A$;
\item "2)"\parshape 2 6cm 6.8cm 6cm 6.8cm length of the segment
          $[A\tilde A]$ is the same for all points $A\in S$ (it is a
          parameter of construction).
\endroster
\enddefinition
\parshape 6 6cm 6.8cm 6cm 6.8cm 6cm 6.8cm 6cm 6.8cm 6cm 6.8cm 0cm 12.8cm
Let $t=|A\tilde A|$. In Bonnet construction we can choose absolutely
arbitrary surface $S$ and arbitrary value of parameter $t$. Choosing
various values of $t$ for the fixed initial surface $S$, we obtain
a family of surfaces $S_t$ parallel to the surface $S$ (as shown
on Fig\. 1.1). If surface $S$ is compact, then for sufficiently small 
$t$ surfaces $S$ and $S_t$ do not intersect. And they and are at the
distance of $\rho (S,S_t)=t$ apart from each other. In this situation
Bonnet transformation yields a set of diffeomorphisms $f_t\!:S\to S_t$. 
It possess the following property.
\proclaim{Theorem 1.1} In Bonnet construction the segment $[A\tilde A]$, 
which connect appropriate points $A$ and $\tilde A$ on $S$ and $\tilde S$,
is orthogonal not only to initial surface of $S$, but to all intermediate 
surfaces $S_t$ including the surface $\tilde S$.
\endproclaim
Due to this property Bonnet transformation from definition~1.1 is often 
called {\it a shift along normal-vector} or {\it normal shift}. 
\definition{Definition 1.2} Bianchi-Lie transformation of $f\!:S\to\tilde S$ 
is defined by the following three conditions:
\roster
\rosteritemwd=0pt
\item "1)" segment $[A\tilde A]$ connecting appropriate points $A$ and 
           $\tilde A$ on $S$ and $\tilde S$ lays in tangent planes
           $T_A(S)$ and $T_{\tilde A}(\tilde S)$ simultaneously;
\item "2)" tangent planes of $T_ A(S)$ and $ T_{\tilde A}(\tilde S)$ 
           are orthogonal to each other;
\item "3)" length of the segment $[A\tilde A]$ is the same for all 
           points $A\in S$ (it is a parameter of construction).
\endroster
\enddefinition
\definition{Definition 1.3} B\"acklund transformation $f\!:S\to\tilde S$ 
is defined by the following three conditions:
\roster
\rosteritemwd=0pt
\item "1)" segment $[A\tilde A]$ connecting appropriate points $A$ 
           and $\tilde A$ on $S$ and $\tilde S$ lays in tangent 
           planes $T_A(S)$ and $T_{\tilde A}(\tilde S)$simultaneously;
\item "2)" the angle $\gamma$ between tangent planes of $T_A(S)$ and 
           $T_{\tilde A}(\tilde S)$ has the same value for all points 
           of $A\in S$ (it is a parameter of construction);
\item "3)" length of the segment $[A\tilde A]$ is the same for all 
           points $A\in S$ (it is a parameter of construction).
\endroster
\enddefinition
\definition {Definition 1.4} Darboux transformation $f\!:S\to\tilde S$ 
is defined by the following three conditions:
\roster
\rosteritemwd=0pt
\item"1)" segment $[A\tilde A]$ connecting appropriate points $A$ and 
          $\tilde A$ on $S$ and $\tilde S$ forms two angles $\alpha $ 
          and $\beta$ with tangent planes $T_A(S)$ and $T_{\tilde A}
          (\tilde S)$, the values of which are same for all points
          $A\in S$ (they are parameters of construction);
\item "2)" the angle $\gamma$ between tangent planes of $T_A(S)$ and 
           $T_{\tilde A}(\tilde S)$ has the same value for all points 
           of $A\in S$ (it is a parameter of construction);
\item "3)" length of the segment $[A\tilde A]$ is the same for all 
           points $A\in S$ (it is a parameter of construction).
\endroster
\enddefinition
     Despite of presence of one (or two parameters) the Bianchi-Lie 
transformation and B\"acklund transformation are discrete by their 
nature. As to Bianchi-Lie transformation this fact is explained by 
the following theorem.
\proclaim{Theorem 1.2} Bianchi-Lie transformation with parameter 
$\rho=|A\tilde A|$ can be realized on an initial surface of $S$ 
if and only if $S$ is a surface of constant negative Gaussian curvature 
$K=-\rho^{-2}$, the resulting surface $\tilde S$ being also a surface
of constant negative Gaussian curvature with the same value of curvature
$\tilde K=-\rho^{-2}$.
\endproclaim\pagebreak
\noindent Similar theorem holds for the B\"acklund transformation. 
\proclaim{Theorem 1.3} B\"acklund transformation with parameters 
$\rho=|A\tilde A|$ and $\gamma$ can be realized on an initial 
surface $S$ if and only if $S$ is a surface of constant negative 
Gaussian curvature $K=-\sin^2\!\gamma\cdot\rho^{-2}$. The resulting
surface $\tilde S$ in this construction is also a surface of constant
negative Gaussian curvature with the same value of curvature
$\tilde K=-\sin^2\!\gamma\cdot\rho^{-2}$.
\endproclaim
Suppose that surface $S$ of constant negative curvature $K$ is given.
By choosing it as an initial surface in Bianchi-Lie construction
we fix the value of parameter $\rho$:
$$
\rho=\frac{1}{\sqrt{-K}}.
$$
In B\"acklund construction we have additional parameter $\gamma$.
Therefore we can construct a family of transformations $f_\gamma\!:S\to 
S_\gamma$ with common initial surface, where
$$
\rho=\frac{\sin\gamma}{\sqrt{-K}}.
$$
However, in contrast to Bonnet transformation, there is no canonical
way to unite transformations $f_\gamma$ with various values of $\gamma$
into one construction. Therefore we treat $f_\gamma$ as separate
transformations with fixed values of $\gamma$.\par
     In Darboux construction, except for $\rho$ and $\gamma$, we have 
two additional parameters $\alpha$ and $\beta$. Let's consider two points 
$A$ and $\tilde A=f(A)$ on $S$ and $\tilde S$ \vadjust{\vskip 0.7cm
\special{em:graph edis-01b.gif}\vskip 6.2cm}related by Darboux
transformation with parameters $\rho$, $\gamma$, $\alpha$ and $\beta$. 
Tangent planes $T_A(S)$ and $T_{\tilde A}(\tilde S)$ intersect along
the straight line $B\tilde B$ (see Fig\. 1.2). Points $B$ and $\tilde B$
are obtained by dropping perpendiculars from $A$ and $\tilde A$ to this
line. For the vector of displacement beginning at the point $A$ and
ending at the point $\tilde A=f(A)$ we have the expansion
$$
\overrightarrow{A\tilde A\,\,}=\overrightarrow{AB\vphantom{\tilde B}}+
\overrightarrow{B\tilde B\,\,}+\overrightarrow{\tilde B\tilde A\,\,}.
\hskip -2em
\tag1.1
$$
Let's project tangent planes $T_A(S)$ and $T_{\tilde A}(\tilde S)$
shown on Fig\. 1.2 onto the plane perpendicular to the line of their
intersection. As a result we get Fig\. 1.3. Normal vectors $\bold n$
and $\tilde\bold n$ in this projection can be extended up to the
intersection at the point $C$. Now the expansion \thetag{1.1} can be
replaced by the following one:
$$
\overrightarrow{A\tilde A\,\,}=\overrightarrow{AC\vphantom{\tilde B}}+
\overrightarrow{B\tilde B\,\,}+\overrightarrow{C\tilde A\,\,}.
\hskip -2em
\tag1.2
$$
Lengths of the vectors in \thetag{1.2} do not depend on the choice of
point $A$ on $S$. On the base of elementary geometry we can calculate
them:
$$
\align
&|AC|=t_1=\frac{\sin\alpha-\cos\gamma\,\sin\beta}{\sin^2\gamma}
\,|A\tilde A|,\\
\vspace{2ex}
&|C\tilde A|=t_2=\frac{\sin\beta-\cos\gamma\,\sin\alpha}{\sin^2\gamma}
\,|A\tilde A|,
\tag1.3\\
\vspace{2ex}
&|B\tilde B|=r=\sqrt{|A\tilde A|^2-|AC|^2-|C\tilde A|^2\,}.
\endalign
$$
The vector $\overrightarrow{AC\vphantom{\tilde B}}$ is directed along
normal-vector to $S$, and vector $\overrightarrow{C\tilde A\,\,}$ is
directed along normal-vector to $\tilde S$. Therefore taking into
account constancy of lengths of segments \thetag{1.3} we derive the
following theorem.
\proclaim{Theorem 1.4} Darboux transformation $f\!:S\to\tilde S$ with
parameters $\rho=|A\tilde A|$, $\alpha$, $\beta$, and $\gamma$ can be
represented as a composition
$$
f=f_{t_2}\compos f_{r\,\gamma}\compos f_{t_1},\hskip -2em
\tag1.4
$$
which consists of two normal shifts (Bonnet transformations) with
parameters $t_1$ and $t_2$, and one B\"acklund transformation with
parameters $r$ and $\gamma$, where $t_1$, $t_2$, and $r$ are
defined by formulas \thetag{1.3}.
\endproclaim
The theorem~1.4 shows that all above classical constructions of surface 
transformation can be reduced to the following two basic types: 
\roster
\item to normal shifts (Bonnet transformation), 
\item to tangent shifts (B\"acklund transformation).
\endroster
\head
\S\,2. Association with nonlinear integrable equations and
further generalizations of classical constructions.
\endhead
\rightheadtext{\S\,2. Association with nonlinear equations \dots}
     In the beginning of \uppercase\expandafter{\romannumeral 20}-th
century we observe intensive development of the theory of surface
transformations and the theory of congruences\footnote{Theory of
line families and their envelopes.}\ (see references in \cite{10}).
The latter one was stimulated by its application to the optics in
describing light beams in non-homogeneous refracting media.
\adjustfootnotemark{-1}
Development of classical constructions from \cite{5--9} in this
period was determined by complication of geometry, where they are 
realized. Thus in paper \cite{11} Bianchi has replaced three-dimensional
Euclidean space by three-dimensional spaces of negative constant 
sectional curvature\footnote{See definition and detailed description
in \cite{12} and \cite{13}.}.\par
    Paper of Tzitzeika \cite{14} should also be mentioned here. In
this paper he has considered three-dimensional Euclidean space, but
with special marked point $O$, and has constructed the transformation
quite different from that of Bianchi-Lie, B\"acklund, and Darboux.
His transformation relates two surfaces of negative Gaussian curvature
$K$, which is not constant however. The value of $K$ at the point $A$
of Tzitzeika's surface is defined by formula
$$
K=-\frac{\const}{\rho^4},\hskip -2em
\tag2.1
$$
where $\rho$ is a distance from marked point $O$ to the plane $T_A(S)$
tangent to $S$ at the point $A$.\par
    Further development of differential geometry in 
\uppercase\expandafter{\romannumeral 20}-th century has displaced 
interest toward more abstract constructions: connections on 
bundles, gauge fields etc. A lot of results were forgotten for many
years. Nowadays the advent of {\bf inverse scattering method} 
(see \cite{15}) followed by 
intensive development of the theory of integrable differential equations 
and integrable models in classical and quantum physics has resumed
interest to the constructions of Bianchi, Lie, B\"acklund, and Darboux.
We explain this by taking B\"acklund's construction as an example.
This construction is realized on a surface $S$ of constant negative
curvature
$$
K=-\frac{\sin^2(\gamma)}{\rho^2}\hskip -2em
\tag2.2
$$
(see theorem~1.3 above). It is known (see \cite {16}) that on such 
surface one can choose asymptotic coordinates $u^1$ and $u^2$. In 
asymptotic coordinates the components of metric tensor and the 
components of second fundamental form are as follows: 
$$
\xalignat 2
&\quad g_{11}=g_{22}=\rho^2,&&g_{12}=g_{21}=\rho^2\,\cos(\varphi).
\hskip -2em
\tag2.3\\
&\quad b_{11}=b_{22}=0,&&b_{12}=b_{21}=\rho\,\sin(\gamma)
\,\sin(\varphi).
\hskip -2em
\tag2.4
\endxalignat
$$
Here function $\varphi=\varphi(u^1,u^2)$ determines the angle between
asymptotic lines\footnote{Lines, tangent vector of which make zero
the value of second fundamental form, see more details in \cite{16}.}.
The condition of constancy of Gaussian curvature $K$ in \thetag{2.2}
appears to lead to the following partial differential equation for
$\varphi$:
$$
\frac{\partial^2\varphi}{\partial u^1\,\partial u^2}=\sin(\varphi).
\hskip -2em
\tag2.5
$$
This is well-known Sin-Gordon equation integrable by means of inverse
scattering method (see \cite{17}, \cite{18}). If we designate by $\psi$
the angle between asymptotic lines on the second surface $\tilde S$
related to $S$ by B\"acklund transformation, then angles $\varphi$ and
$\psi$ will be bound to each other by the following two differential
equations:
$$
\aligned
&\frac{\partial}{\partial u^1}\left(\frac{\varphi-\psi}{2}\right)=
-C_1\cdot\sin\left(\frac{\varphi+\psi}{2}\right),\\
\vspace{1ex}
&\frac{\partial}{\partial u^2}\left(\frac{\varphi+\psi}{2}\right)=
-C_2\cdot\sin\left(\frac{\varphi-\psi}{2}\right),
\endaligned\hskip -2em
\tag2.6
$$
Here $C_1$ and $C_2$ are two constants defined by parameter $\gamma$:
$$
\xalignat 2
&C_1=\frac{1-\cos(\gamma)}{\sin(\gamma)},
&&C_2=\frac{1+\cos(\gamma)}{\sin(\gamma)}.
\endxalignat
$$
From \thetag{2.5} and \thetag{2.6} we can derive Sin-Gordon equation
for the angle $\psi$:
$$
\frac{\partial^2\psi}{\partial u^1\,\partial u^2}=\sin(\psi).
\hskip -2em
\tag2.7
$$
The relationships \thetag{2.6} are known as B\"acklund transformation
for the Sin-Gordon equation. They are used to construct new solutions
of this equation on a base of some already known ones. Group analytic
treatment of these relationships can be found in \cite{19}.\par
      Tzitzeika's surfaces with Gaussian curvature \thetag {2.1} are
bound with other nonlinear partial differential equation
$$
\frac{\partial^2\varphi}{\partial u^1\,\partial u^2}=e^{\varphi}-
e^{-2\varphi}.\hskip -2em
\tag2.8
$$
\adjustfootnotemark{-2}
Similar to \thetag{2.7}, this equation is integrable by inverse
scattering method. Tzitzeika equation \thetag{2.8} was first
discovered in \cite{14}. Later on it was rediscovered in \cite{20}
and \cite{21}. The analog of geometrical B\"acklund transformation
for Tzitzeika surfaces was found in original paper \cite {14}. 
In form of differential relationships similar to \thetag{2.6}
B\"acklund transformation for the equation \thetag{2.8} was 
rediscovered in \cite{22}\footnote{Other form of differential B\"acklund
transformation for the  equation \thetag{2.8}, different from that
of \cite{14} and \cite{22}, can be found in \cite{23}.}.
It was studied in details in paper \cite{24}. The relation of
geometric construction of Tzitzeika and differential B\"acklund
transformation for the equation \thetag{2.8} is discussed in \cite{25}.
In papers \cite{26} and \cite{27} some special classes of solutions
for the equation \thetag{2.8} are constructed.\par
    In a series of papers \cite{28}, \cite{29}, \cite{30} 
multidimensional generalizations for Bianchi-Lie and B\"acklund
transformations in Euclidean and affine spaces. In Euclidean case
generalized B\"acklund transformation binds two $n$-dimensional submanifolds
$S$ and $\tilde S$ in the space of odd dimension $\Bbb R^{2n-1}$. Each
of these two submanifolds $S$ and $\tilde S$ are described by some
system of nonlinear differential equations which is integrable by
means of inverse scattering method and therefore can be treated as
a generalization of Sin-Gordon equation \thetag{2.7}.\par
     In paper \cite{31} Euclidean space $\Bbb R^{2n-1}$ was replaced
by Riemannian manifold $M$ of constant sectional curvature, where
$\dim M=2n-1$. With respect to the generalized version of B\"acklund
transformation from \cite{28} and \cite{29} paper \cite{31} plays
the same role as paper \cite{11} respective to classical constructions
of Bianchi, Lie, B\"acklund and Darboux. For us it's important to 
emphasize that in papers \cite{11} and \cite{31} {\bf rectilinear} 
segment of shift was replaced by a segment of {\bf geodesic line}.
In local coordinates geodesic line is described by ordinary
differential equations of second order
$$
\ddot x^k+\sum^m_{i=1}\sum^m_{j=1}\Gamma^k_{ij}(x^1,\ldots,x^m)\,
\dot x^i\,\dot x^j=0.\hskip -2em
\tag2.10
$$
So this was first event, when rectilinear segment of shift was
replaced by curvilinear one. We should note that this replacement
did not cause crucial complication in the constructions. The Sin-Gordon
equation \thetag{2.7} and its multidimensional generalizations from
\cite{27} remain unchanged.
\head
\S\,3. Bonnet transformation and geodesic normal shift.
\endhead
     As well as Bianchi-Lie and B\"acklund transformations Bonnet
transformation from definition~1.1 has its own multidimensional
generalization for the case of hypersurfaces in Euclidean space
$\Bbb R^n$ ($\dim S=n-1$). This multidimensional generalization
preserves two properties, which are important for us:
\roster
\rosteritemwd=0pt
\item"1)" multidimensional Bonnet transformation can be defined
          on arbitrary sufficiently small part of any hypersurface
          $S$ in $\Bbb R^n$ (here we meet no restrictions like
          constancy of curvature);
\item"2)" multidimensional Bonnet transformation is defined for
          any sufficiently small value of shift parameter $t=
          |A\tilde A|$, and for any such value of $t$ hypersurface
          $S_t$ is orthogonal to the segment $[A\tilde A]$ binding
          two related points $A\in S$ and $\tilde A\in S_t$.
\endroster
Due to the second property multidimensional Bonnet transformation
is {\bf a normal shift}. If we replace Euclidean space $\Bbb R^n$
by some $n$-dimensional Riemannian manifold and rectilinear segments
by segments of geodesic lines, we obtain the construction of
{\bf geodesic normal shift}.
\definition{Definition 3.1}Geodesic normal shift to the distance $t$
in Riemannian manifold is a map $f_t\!:S\to S_t$, that maps each
point $A$ of some hypersurface $S$ onto corresponding point $\tilde A
=f_t(A)$ of the other hypersurface $S_t$ such that the following
conditions are fulfilled:
\roster
\rosteritemwd=0pt
\item"1)" points $A$ and $\tilde A=f_t(A)$ are bound by a segment
          $[A\tilde A]$ of geodesic line, which is perpendicular
          to the tangent plane $T_A(S)$ to $S$ at the point $A$;
\item"2)" length of the geodesic segment $[A\tilde A]$ is constant,
          which is equal to $t$ (it doesn't depend on particular
          choice of point $A$ on hypersurface $S$).
\endroster
\enddefinition
Construction of geodesic normal shift in differential geometry is well
known. It is used to define {\it semigeodesic coordinates} on manifold
(see in \cite{16} or in \cite{32}). In general relativity the same
construction of geodesic normal shift is used to define {\it synchronous
reference system} (see \cite{33}, \S\,97).\par
     Geodesic normal shift in Riemannian manifold $M$ possess same
two properties as classical Bonnet transformation in $\Bbb R^3$ and
its generalizations in multidimensional flat spaces $\Bbb R^n$
($n=2$ and $n>3$):
\roster
\rosteritemwd=0pt
\item"1)" geodesic normal shift can be defined on arbitrary sufficiently
          small part of any hypersurface $S$ in $M$ (here we meet no
          restrictions like constancy of curvature);
\item"2)" geodesic normal shift is defined for any sufficiently small
          value of shift parameter $t=|A\tilde A|$, and for any such
          value of $t$ hypersurface $S_t$ is orthogonal to the segment
          $[A\tilde A]$ binding two related points $A\in S$ and $\tilde
          A\in S_t$.
\endroster
\head
\S\,4. Normal shift along trajectories of Newtonian dynamical
systems.
\endhead
\rightheadtext{\S\,4. Normal shift along trajectories \dots}
    From what was said in previous three sections \S\S\,1--3 we conclude
that development and generalizations of all constructions considered
there came through the same two steps: increase of the dimension and
transfer from flat Euclidean spaces to non-flat Riemannian manifolds.
In doing last step we replace rectilinear segments by the segments of
curved lines. However, we can replace {\it rectilinear segments} by
{\it curved segments} in flat space too. This idea has played the role
of impetus that gave rise to {\bf the theory of dynamical systems
admitting the normal shift}. In order realize this idea we was to
do the following steps:
\roster
\rosteritemwd=0pt
\item"1)" choose one of the four above constructions (Bianchi-Lie
          transformation, B\"ack\-lund transformation, Darboux transformation
          or Bonnet transformation) as a base for generalization;
\item"2)" choose class of curves that should be used in generalized
          construction;
\item"3)" choose a space, where the generalized construction should
          be built.
\endroster
     Our joint paper \cite{34} in cooperation with A.~Yu.~Boldin
(see also \cite{35}) was starting point of the theory of Newtonian
dynamical systems admitting the normal shift. In this paper we choose
Bonnet transformation as a basic object for generalization, since it
is the simplest one and since it can be applied to any hypersurface.
The choice of curves was prompted by the equations \thetag{2.10}.
This is the system of differential equations of the second order. We
replace them by the autonomous system of differential equations of
more general form:
$$
\align
\ddot x^1&=F^1(x^1,\ldots,x^n,\dot x^1,\ldots,\dot x^n),
\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
.\ .\ .\ \hskip -2em\tag4.1\\
\ddot x^n&=F^n(x^1,\ldots,x^n,\dot x^1,\ldots,\dot x^n).
\hskip -2em
\endalign
$$
Thus we escape the situation leading to the geodesic normal shift,
which was already known, and the same time did not change class of
curves drastically. Describing a curve by differential equations
\thetag{4.1} we not only define them as the set of point, but also
fix a parameter on them. As a parameter we take independent variable
$t$ that define derivatives $\dot x^1,\,\ldots,\,\dot x^n$ and second
derivatives $\ddot x^1,\,\ldots,\,\ddot x^n$ in \thetag{4.1}.\par
\parshape 21
0cm 12.8cm 0cm 12.8cm 0cm 12.8cm 0cm 12.8cm 0cm 12.8cm 0cm 12.8cm
0cm 12.8cm 0cm 12.8cm 0cm 12.8cm 6.9cm 5.9cm 6.9cm 5.9cm
6.9cm 5.9cm 6.9cm 5.9cm 6.9cm 5.9cm 6.9cm 5.9cm
6.9cm 5.9cm 6.9cm 5.9cm 6.9cm 5.9cm 6.9cm 5.9cm
6.9cm 5.9cm 0cm 12.8cm
    The choice of space $M=\Bbb R^2$ in \cite{35} was determined by
the reason of maximal simplicity. This was important in the initial
stage of constructing the theory. Hypersurfaces in $\Bbb R^2$ are
curves, therefore in $\Bbb R^2$ we have the displacement of curves
along other curves, the latter being trajectories of dynamical system
\thetag{4.1}. The equations of dynamical system \thetag{4.1} here can
be written in vectorial form:
$$
\xalignat 2
&\dot\bold r=\bold v,&&\dot\bold v=\bold F(\bold r,\bold v).\hskip -2em
\tag4.2
\endxalignat
$$
These are the equations of dynamics for mass point with unit mass $m=1$
according to Newton's second law. \vadjust{\vskip 0.2cm \special{em:graph
edis-01c.gif}\vskip -0.2cm}Vector $\bold r=\bold r(t)$ in \thetag{4.2} is
a radius-vector of mass point moving along the trajectory, $\bold v$ is a
vector of velocity tangent to the trajectory, and $\bold F(\bold r,\bold v)$
is a vector of force that defines force field of the system \thetag{4.2}.
\par
\parshape 8
6.9cm 5.9cm 6.9cm 5.9cm 6.9cm 5.9cm 6.9cm 5.9cm
6.9cm 5.9cm 6.9cm 5.9cm 6.9cm 5.9cm 
0cm 12.8cm
     In order to draw trajectories of Newtonian dynamical system
\thetag{4.2} coming out from the curve $S$ (see Fig.~4.1) we should
define initial velocity at each point of $S$, i\. e\. if curve $S$
is given in parametric form by vectorial function $\bold r=\bold r(u)$,
then we are to consider the following Cauchy problem for the system
of equations \thetag{4.2}:
$$
\xalignat 2
&\quad\bold r\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=\bold r(u),
&&\bold v\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=
\bold w(u).\hskip -2em
\tag4.3
\endxalignat
$$
Here $\bold w(u)$ is a vector-function that determines the initial
velocity for trajectories starting from the curve $S$. Cauchy problem
\thetag{4.3} for the equations \thetag{4.2} is solvable, for any
sufficiently small $t$ its solution defines a map $f_t\!:S\to S_t$
from initial curve $S$ to the curve $S_t$. This map is to be a
curvilinear generalization for classical Bonnet transformation, with
trajectories of dynamical system \thetag{4.2} being used instead of
rectilinear segments $[A\tilde A]$ in definition~3.1. Therefore these
trajectories should be perpendicular to $S$. In other words, this means
that vector $\bold w(u)$ in \thetag{4.3} should be directed along the
normal vector $\bold n(u)$ of initial curve $S$:
$$
\xalignat 2
&\quad\bold r\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=\bold r(u),
&&\bold v\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=
\nu(u)\cdot\bold n(u).\hskip -2em
\tag4.4
\endxalignat
$$
Scalar function $\nu(u)$ in \thetag{4.4} determines the modulus of
initial velocity for trajectories starting from $S$.\par
     As well as \thetag{4.3} the Cauchy problem \thetag{4.4} is solvable,
and it determines a map $f_t\!:S\to S_t$, which has more reasons to
be considered as a proper curvilinear generalization of classical Bonnet
transformation. It certainly possess first of two properties of Bonnet
transformation, which are listed in \S\,3, but shouldn't ever possess
the second one. This means that trajectories of the shift $f_t\!:S\to S_t$,
being orthogonal to $S$ due to \thetag{4.4}, can be {\bf not orthogonal}
to $S_t$ for $t\neq 0$. Though for some special choice of initial curve
$S$ and some special choice of function $\nu(u)$ in \thetag{4.4} they
could be orthogonal to $S_t$ for all $t$. In more details this point
is discussed in thesis \cite{36} by A.~Yu.~Boldin. There one can find
some examples that illustrate both cases: when trajectories of
displacement are orthogonal to all curves $S_t$, and when not as well.
Taking into account all what was said above we come to the concept of
{\it normal shift along trajectories of Newtonian dynamical system}.
This concept was introduced in paper \cite{35}.
\definition{Definition 4.1} Let $f_t\!:S\to S_t$ be a map defined by
the displacement of a planar curve $S\subset\Bbb R^2$ along trajectories
of Newtonian dynamical system \thetag{4.2} that start on $S$ according
to initial data \thetag{4.4}. This map is called {\it the normal shift},
if for all sufficiently small values of $t$ all trajectories of
displacement are perpendicular to all curves $S_t$ obtained by
displacement.
\enddefinition
    Suppose that we choose and fix an arbitrary curve $S$ on the plane.
Suppose that vector-function $\bold F(\bold r,\bold v)$ in \thetag{4.2}
is also fixed. Then the only arbitrariness that we have at our disposal
in defining displacement map $f_t\!:S\to S_t$ is the choice of function
$\nu(u)$ in \thetag{4.4}. Can we choose this function so that the map
$f_t\!:S\to S_t$ would be a normal shift in the sense of the above
definition\,? In general case the answer to this question is {\bf negative}.
Examples confirming this answer are given in the thesis \cite{36} by
A.~Yu.~Boldin. However, if the curve $S$ or the force field
$\bold F(\bold r,\bold v)$ is properly chosen, then further choice
of function $\nu(u)$ can be made so that the displacement map $f_t\!:S\to
S_t$ would be a normal shift. An example of proper choice of force
field is $\bold F(\bold r,\bold v)=0$. Trajectories of Newtonian dynamical
system with identically zero force field are straight lines. In this case
for any choice of curve $S$ we choose function $\nu(u)$ being identically
equal to unity: $\nu(u)=1$. Then all trajectories of shift from $S$ to
$S_t$ will be segments of straight lines with the same length $t$. This
means that due to our choice we construct a shift $f_t\!:S\to S_t$, which
coincides with classical Bonnet transformation. Therefore it satisfies all
conditions needed to be a normal shift.\par
    The above example demonstrates that there are some {\bf force fields}
(at least one) such that for any choice of curve $S$ they let define
transformation $f_t\!: S\to S_t$, being a normal shift, at the expense of
proper choice of the function $\nu(u)$ in \thetag{4.4}. Such force fields
or (more exactly) dynamical systems with such force fields in paper
\cite{35} were called the systems {\bf admitting the normal shift}.
\definition{Definition 4.2} Newtonian dynamical system \thetag{4.2}
on the plane with force field $\bold F(\bold r,\bold v)$ is called
the system {\bf admitting the normal shift of curves}, if for any
sufficiently small part $S$ of any curve on the plane one can find a
function $\nu$ on $S$ such that the map $f_t\!:S\to S_t$ defined by
the solution of Cauchy problem \thetag{4.4} is a normal shift in the
sense of definition~4.1. 
\enddefinition
    If force field of Newtonian dynamical system satisfies the condition
of definition~4.2, then we can use it to define the construction being
curvilinear generalization of Bonnet transformation in $\Bbb R^2$. Such
construction reproduces both properties of classical construction
stated in \S\,3, i\. e\. it is applicable to a sufficiently small part
of any curve in $\Bbb R^2$, and constructed transformation $f_t$ 
is a normal shift in the sense of definition 4.1.\par\newpage
\head
\S\,5. Normality equations.
\endhead
     By means of definition~4.2 we have introduced central object of 
theory: {\bf class of Newtonian dynamical systems admitting the
normal shift}. This class is non-empty, since it contains trivial
system with identically zero force field $\bold F(\bold r,\bold v)=0$.
But how broad is this class, does it contain less trivial systems\,?
In order to find answer to this question we should find another
description for dynamical systems of this class, more effective than
the definition~4.2. First step in this direction was made in paper
\cite{35} for the dimension $n=2$. There the system of two partial
differential equations for the vector-function $\bold F(\bold r,\bold
v)$ was derived. These equations form sufficient condition for Newtonian
dynamical system with force field to belong to the class of systems
admitting the normal shift. They were called {\bf the equations of
normality}. Description of how these equations were derived an more
detailed report on initial period of development of the theory of
dynamical systems admitting the normal shift can be found in thesis
\cite{36} by A\.~Yu\.~Boldin. Here we will only introduce some
definitions and notations and then will write the normality equations
from \cite{35} themselves.\par
     Planar Newtonian dynamical systems \thetag{4.2} describe the
motion of the points in $\Bbb R^2$. The space $M=\Bbb R^2$ is called
{\bf configuration space} of such systems, points of this space are
marked by radius-vector $\bold r$. Usually configuration space $M=
\Bbb R^2$ is completed by one more copy of this space $\Bbb R^2$:
$$
TM=\Bbb R^2\oplus\Bbb R^2.\hskip -2em
\tag5.1
$$
Resulting space $TM$ in \thetag{5.1} is called {\bf phase space}
of planar dynamical system \thetag{4.2}. Points of $TM$ are marked
by pairs of vectors $(\bold r,\bold v)$. In our case both copies
of the space $\Bbb R^2$ are assumed to be equipped with standard
Euclidean scalar product, which defines lengths of vectors and
angles between them. Velocity vector $\bold v$ is in the second
summand $\Bbb R^2$ in \thetag{5.1}. Vector of acceleration
$\bold a=\dot\bold v$ is assumed to be in the second summand too.
Therefore we can define the angle between vectors $\bold a$ and
$\bold v$ (though, if we identify two copies of $\Bbb R^2$ in
\thetag{5.1}, we can define the angle between vectors $\bold r$
and $\bold v$ too).\par
     Force field $\bold F(\bold r,\bold v)$ of dynamical system
\thetag{4.2} is function of point of phase space, its values are
vectors that are in the second copy of $\Bbb R^2$ in \thetag{5.1}.
Another example of such function is given by velocity vector: we
can treat $\bold v$ as a map that maps pair of vectors $(\bold r,
\bold v)$ onto the vector $\bold v$. Let's normalize its length to
the unity. As a result we obtain vector $\bold N$ of unit length:
$$
\bold N=\frac{\bold v}{|\bold v|}.\hskip -2em
\tag5.2
$$
In two dimensional space $\Bbb R^2$ we can rotate unit vector
\thetag{5.2} by the angle $90^\circ$ (clockwise or counter-clockwise).
Let's fix on of these two direction of rotation and denote by $\bold M$
unit vector obtained by such rotation. Like vector $\bold v$, both vectors
$\bold N$ and $\bold M$ can be interpreted as vector-functions on a phase
space $TM$ with values in second copy of $\Bbb R^2$ in \thetag{5.1}. 
The only peculiarity of these two functions $\bold N$ and $\bold M$
is that they aren't defined everywhere in $TM$, one should exclude
that points, where $\bold v=0$. Pair of vectors $\bold N$ and $\bold M$
forms an orthogonal and normalized frame in $\Bbb R^2$. Therefore one
can define force field $\bold F(\bold r,\bold v)$ by the expansion
$$
\bold F=A(\bold r,\bold v)\cdot\bold N+B(\bold r,\bold v)\cdot\bold M.
\hskip -2em
\tag5.3
$$
Here $A$ and $B$ are scalar functions on phase space $TM$. First
summand in the expansion \thetag{5.3} is a vector directed along
trajectory of dynamical system \thetag{4.2}, this is {\bf tangential}
component of force field. Second summand is directed perpendicular to
the trajectory toward the curvature center, this is {\bf centripetal}
component of force field $\bold F$.\par
     Functions $A$ and $B$ depend on four scalar arguments, two components
$r^1$ and $r^2$ of radius vector $\bold r$ and two components $v^1$ and
$v^2$ of the velocity vector $\bold v$. Partial derivatives 
$$
\frac{\partial A}{\partial r^1}\text{\ \ and \ }\frac{\partial A}
{\partial r^2}
$$
form the vector of {\bf spatial gradient} of the function $A$, we denote
it by $\nabla A$. Other two partial derivatives
$$
\frac{\partial A}{\partial v^1}\text{\ \ and \ }\frac{\partial A}
{\partial v^2}
$$
form the vector of {\bf velocity gradient} of the function $A$, which we
denote by $\tilde\nabla A$. Similarly we can define gradients $\nabla B$
and $\tilde\nabla B$. Let's consider the following expansions for the
vectors of gradients $\nabla A$, $\tilde\nabla A$, $\nabla B$, and
$\tilde\nabla B$:
$$
\xalignat 2
&\nabla A=\alpha_1\cdot\bold N+\alpha_2\cdot\bold M,
&&\tilde\nabla A=\alpha_3\cdot\bold N+\alpha_4\cdot\bold M,
\hskip -2em\\
\vspace{-7pt}
&&&\tag5.4\\
\vspace{-7pt}
&\nabla B=\beta_1\cdot\bold N+\beta_2\cdot\bold M,
&&\tilde\nabla B=\beta_3\cdot\bold N+\beta_4\cdot\bold M,
\hskip -2em
\endxalignat
$$
These expansions are analogous to the expansion \thetag{5.3} for
the vector of force $\bold F$. Coefficients of the expansions
\thetag{5.4} can be calculated in form of scalar products:
$$
\xalignat 2
&\alpha_1=(\nabla A\,|\,\bold N),&&\alpha_2=(\nabla A\,|\,\bold M),\\
&\alpha_3=(\tilde\nabla A\,|\,\bold N),&&\alpha_4=(\tilde\nabla A\,|\,
\bold M),\\
&\beta_1=(\nabla B\,|\,\bold N),&&\beta_2=(\nabla B\,|\,\bold M),\\
&\beta_3=(\tilde\nabla B\,|\,\bold N),&&\beta_4=(\tilde\nabla B\,|\,
\bold M).
\endxalignat
$$
Now we are able to write down the system of {\bf normality equations}
derived in \cite{35}: 
$$
\cases
B=-|\bold v|\,\alpha_4,\\
\vspace{3ex}
\dfrac{B\,A}{|\bold v|^2}-\beta_1-\beta_3\dfrac{A}{|\bold v|}-
\beta_4\,\dfrac{B}{|\bold v|}=\alpha_2-\alpha_3\,\dfrac{B}{|\bold v|}.
\endcases\hskip -2em
\tag5.5
$$
Parameters $\alpha_2$, $\alpha_3$, $\alpha_4$, $\beta_1$, $\beta_3$,
$\beta_4$ in \thetag{5.5} can be expressed through gradients of $A$
and $B$. Therefore the equations \thetag{5.5} form the system of
partial differential equations with respect to the coefficients $A$
and $B$ in the expansion \thetag{5.3} of the force field.
\proclaim{Theorem 5.1} Planar Newtonian dynamical system \thetag{4.2}
with force field \thetag{5.3} satisfying normality equations is a system
admitting normal in the sense of definition~4.2.
\endproclaim
     The first equation in \thetag{5.5} expresses $B$ through vector
of velocity gradient $\tilde\nabla A$ in explicit form: $B=-|\bold v|\,
\alpha_4=-|\bold v|\,(\tilde\nabla A\,|\,\bold M)$. By substituting
this expression into second equation \thetag{5.5} and taking into
account above formulas for $\alpha_2$, $\alpha_3$, $\alpha_4$, $\beta_1$,
$\beta_3$, $\beta_4$ we reduce the system of equations \thetag{5.5} to
one nonlinear partial differential equation for the function $A(\bold r,
\bold v)$. Some simples particular solutions of this equation and
corresponding dynamical systems were found in \cite{35}. Their existence
showed us that class of dynamical systems admitting the normal shift is
{\bf non-trivial}. And this class is worth for further study. List of
examples was substantially enlarged in \cite{37} on the base of systematic
analysis of the above normality equations. Here we will not consider
examples from \cite{35}  and \cite{37}, since they are analyzed in the
other thesis \cite{36}.
\head
\S\,6. Generalization for $n$-dimensional case.
\endhead
     Further progress in theory of  dynamical systems admitting the
normal shift was bound with the growth of dimension. In paper \cite{38}
two-dimensional space $\Bbb R^2$ was replaced by $\Bbb R^n$ (see also
\cite{34} section 5). The equations of Newtonian dynamical system
in $\Bbb R^n$ is written in a form 
$$
\xalignat 2
&\dot\bold r=\bold v,&&\dot\bold v=\bold F(\bold r,\bold v)\hskip -2em
\tag6.1
\endxalignat
$$
quite similar to \thetag{4.2}. But phase  space for dynamical system
\thetag{6.1} is a sum
$$
TM=\Bbb R^n\oplus\Bbb R^n.\hskip -2em
\tag6.2
$$
One can easily reformulate the definitions~4.1 and 4.2 for multidimensional
case: simply curves should be replaced by hypersurfaces. On sufficiently
small part of any hypersurface $S$ in $\Bbb R^n$ one can define normal
vector of unit length such that it is smooth function of the points of $S$.
This vector is orthogonal to $S$ in standard scalar product of Euclidean
space $\Bbb R^n$. We can set up the following Cauchy problem
for the equations \thetag{6.1}:
$$
\xalignat 2
&\quad\bold r\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=\bold r(p),
&&\bold v\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=
\nu(p)\cdot\bold n(p).\hskip -2em
\tag6.3
\endxalignat
$$
Here $p$ is a point on $S$, $\bold n(p)$ --- is a normal vector at $p$,
and $\bold r=\bold r(p)$ is a vectorial parametric equation of hypersurface
$S$. Solution of Cauchy problem \thetag{6.1} is given by vector-function
$\bold r=\bold r(t,p)$. If we map point $p$ with radius-vector $\bold
r(p)$ onto the point $\bold r(t,p)$, we get the displacement $f_t\!:S\to
S_t$ that maps hypersurface $S$ onto another hypersurface $S_t$.
\definition{Definition 6.1} Let $f_t\!:S\to S_t$ be a displacement of
hypersurface $S\subset\Bbb R^n$ along trajectories of Newtonian dynamical
system \thetag{6.1} defined by the solution of Cauchy problem \thetag{6.3}.
This displacement is called {\it the normal shift}, \pagebreak if for all
sufficiently small values of $t$ all trajectories of displacement are
perpendicular to all hypersurfaces $S_t$ obtained by this displacement.
\enddefinition
\definition{Definition 6.2} Newtonian dynamical system \thetag{6.1}
with force field $\bold F(\bold r,\bold v)$ is called the system
{\bf admitting the normal shift}, if for any sufficiently small part $S$
of any hypersurface in $\Bbb R^n$ one can find a function $\nu$ on $S$
such that the map $f_t\!:S\to S_t$ defined by the solution of Cauchy
problem \thetag{6.3} is a normal shift in the sense of definition~6.1. 
\enddefinition
     The derivation of normality equations in multidimensional case
has some peculiarities compared to that of two-dimensional case. In
order to describe these peculiarities we should consider some details
of such derivation. Let $\bold r(t,p)$ be the solution of Cauchy
problem \thetag{6.3} for the equations \thetag{6.1}. If we choose
local (curvilinear) coordinates $u^1,\,\ldots,\,u^{n-1}$ on $S$, then
$\bold r(t,p)$ is represented by vector-function
$$
\bold r=\bold r(t,u^1,\ldots,u^{n-1}).\hskip -2em
\tag6.4
$$
For fixed (and sufficiently small) $t$ function \thetag{6.4} can be
understood as a vectorial parametric equation of hypersurface $S_t$,
which is close to $S$. In this case displacement $f_t\!:S\to S_t$ is
local diffeomorphism that transfer local coordinates $u^1,\,\ldots,
\,u^{n-1}$ from $S$ to $S_t$. Partial derivatives
$$
\boldsymbol\tau_i=\frac{\partial\bold r}{\partial u^i},\quad
i=1,\,\ldots,\,n-1,\hskip -2em
\tag6.5
$$
form a base in tangent hyperplane to $S$ at the point $p$ with
coordinates $u^1,\,\ldots,\,u^{n-1}$.\par
     Now let's unfix parameter $t$ in \thetag{6.4}. Varying of $t$
corresponds to the motion of the point with radius vector \thetag{6.4}
along trajectory of dynamical system \thetag{6.1}. Varying parameters
$\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ means that we
transfer from one trajectory to another. Therefore in this context
vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ are called
the {\bf vectors of variation} of trajectory. Let's denote by $\varphi_1,
\,\ldots,\,\varphi_{n-1}$ the following scalar products:
$$
\varphi_i=(\boldsymbol\tau_i\,|\,\bold v),\quad
i=1,\,\ldots,\,n-1.\hskip -2em
\tag6.6
$$
In situation of  normal shift all functions $\varphi_i(t,u^1,\ldots,
u^{n-1})$ are identically zero, and conversely, if they are not zero,
this means that we deviate from the situation of normal shift.
Therefore functions $\varphi_i(t,u^1,\ldots,u^{n-1})$ will be called
{\bf functions of deviation}.\par
     Let's consider initial data \thetag{6.3}. According to \thetag{6.3}
vector of initial velocity is directed along the normal vector to $S$.
Therefore all functions of deviation on initial hypersurface $S$ are zero:
$$
\varphi_i\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=0,\quad i=1,\,\ldots,\,n-1.\hskip -2em
\tag6.7
$$
In situation of normal shift their derivatives should be zero as well:
$$
\dot\varphi_i\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=0,\quad i=1,\,\ldots,\,n-1.\hskip -2em
\tag6.8
$$
We can hope to satisfy conditions \thetag{6.8} by means of proper choice
of function $\nu(p)=\nu(u^1,\ldots,u^{n-1})$ in \thetag{6.3} (in
two-dimensional case this is really so). However, vanishing of second and
third derivatives and identical vanishing of all functions of deviation
is possible only due to some special properties of force field
$\bold F(\bold r,\bold v)$.
    Note that \thetag{6.7} and \thetag{6.8} can be treated as initial
data of some Cauchy problem for the ordinary differential equation of
the second order for the function $\varphi_i$ in $t$. If this equation
is linear and homogeneous
$$
\ddot\varphi_i-\Cal A\,\dot\varphi_i-\Cal B\,\varphi_i=0,\hskip -2em
\tag6.9
$$
then initial data \thetag{6.7} and \thetag{6.8} provide identical
vanishing of $\varphi_i$. These considerations form the motivation
for the following definition.
\definition{Definition 6.3} We say that newtonian dynamical system
\thetag{6.1} in $\Bbb R^n$ satisfies {\bf weak normality condition},
if for each trajectory of this system there is an ordinary differential
equation $\ddot\varphi=\Cal A(t)\,\dot\varphi+\Cal B(t)\,\varphi$
such that arbitrary function of deviation on this trajectory satisfies
this differential equation.
\enddefinition
    The term ``{arbitrary function of deviation}'' in definition~6.3
requires special comment. The matter is that Cauchy problem \thetag{6.3}
with some special function $\nu(p)$ on hypersurface $S$ is not the only
way of defining the family of trajectories for dynamical system
\thetag{6.1}. If instead of \thetag{6.4} we consider an arbitrary
parametric family of trajectories given by a vector-function
$\bold r(t,u)$, and if we calculate partial derivative $\boldsymbol
\tau=\partial\bold r/\partial u$, then the scalar product $\varphi=
(\boldsymbol\tau\,|\,\bold v)$ will be that ``{arbitrary function of
deviation}''.\par
    On a base of weak normality condition in \cite{38} {\bf weak normality
equations} were derived. They can be understood as multidimensional
generalizations for the equations \thetag{5.5}. We shall write down
these equations later, when required notations will be introduced.
Now let's come back to the condition \thetag{6.8}. Taking into account
\thetag{6.5}, \thetag{6.6} and the equations of dynamical system
\thetag{6.1} we can bring \thetag{6.8} to the form of Pfaff equations
for the function  $\nu(p)=\nu(u^1,\ldots,u^{n-1})$:
$$
\frac{\partial\nu}{\partial u^i}=\psi_i(\nu,u^1,\ldots,u^{n-1}),\quad
i=1,\,\ldots,\,n-1.
\hskip -3em
\tag6.10
$$
Functions $\psi_i(\nu,u^1,\ldots,u^{n-1})$  in the right hand side of
\thetag{6.10} are determined by force field $\bold F(\bold r,\bold v)$
and by hypersurface $S$, i\. e\. by vector-function $\bold r=\bold
r(u^1,\ldots,u^{n-1})$. We shall not give their explicit expressions
(see in \cite{38} or \cite{34}). Its the equations \thetag{6.10} that
substantially differ multidimensional case from two-dimensional
case. {\bf In two-dimensional case} we had only one equation \thetag{6.10},
which is ordinary differential equation of the first order respective
one independent variable $u=u^1$. Such equation is always solvable (at
least locally). {\bf In multidimensional case} the equations \thetag{6.10}
form complete system of Pfaff equations, solvability of this system depends
on its compatibility (see definition of compatibility below in Chapter
\uppercase\expandafter{\romannumeral 5}, \S\,8, definition~8.1). If we want
that the equations \thetag{6.10} would be compatible for any choice of
hypersurface $S$, then we come to the following definition.
\definition{Definition 6.4} We say that Newtonian dynamical system 
\thetag{6.1}
in $\Bbb R^n$ with $n\geqslant 3$ satisfies {\bf additional normality
condition}, if for any hypersurface $S$ system of Pfaff differential
equations \thetag{6.10} for the function $\nu(p)$ is compatible.
\enddefinition
    On a base of additional normality condition in \cite{38} (see also
\cite{34}, section 5) {\bf additional normality equations} were derived.
In order to write these equations, and to write weak normality equations
as well, we need to introduce special ``{fiber spherical coordinates}''
in phase space \thetag{6.2}. Cartesian coordinates $r^1,\,\ldots,\,r^n$
in first copy ob $\Bbb R^n$ in \thetag{6.2} remain unchanged. Cartesian
coordinates $v^1,\,\ldots,\,v^n$ in second copy of $\Bbb R^n$ will be
replaced by spherical coordinates $v,\,u^1,\,\ldots,\,u^{n-1}$ (this
means that we introduce spherical coordinates in {\bf velocity space}).
Transfer from spherical coordinates $v,\,u^1,\,\ldots,\,u^{n-1}$ back
to cartesian coordinates $v^1,\,\ldots,\,v^n$ is defined by
vector-function $\bold v=v\cdot\bold N(u^1,\,\ldots,\,u^{n-1})$, where
$\bold N$ is a unit vector directed along the vector of velocity, or,
in other words, $\bold N$ is a radius-vector of a point on unit sphere
in velocity space. Taking $\bold v=\bold N(u^1,\,\ldots,\,u^{n-1})$ for
the vectorial parametric equation of unit sphere in velocity space we
can define tangent vectors to this sphere:
$$
\bold M_i=\frac{\partial\bold N}{\partial u^i},\quad i=1,\,\ldots,\,n-1.
$$
Vectors $\bold N,\,\bold M_1,\,\ldots,\,\bold M_{n-1}$ form moving frame
of spherical coordinates. One can expand force vector by this frame:
$$
\bold F=A\cdot\bold N+\sum^{n-1}_{i=1} B^i\cdot\bold M_i.\hskip -3em
\tag6.11
$$
Coefficients $A,\,B^1,\,\ldots,\,B^{n-1}$ in the expansion \thetag{6.11}
depend on spatial cartesian coordinates $r^1,\,\ldots,\,r^n$ and on
spherical coordinates $v,\,u^1,\,\ldots,\,u^{n-1}$ in velocity space.
Their spatial gradients can be expanded by the frame of vectors
$\bold N,\,\bold M_1,\,\ldots,\,\bold M_{n-1}$:
$$
\aligned
&\nabla A=a\cdot\bold N+\sum^{n-1}_{k=1}\alpha^k\cdot\bold M_k,\\
\vspace{1ex}
&\nabla B^i=b^i\cdot\bold N+\sum^{n-1}_{k=1}\beta^{ik}\cdot\bold M_k.
\endaligned\hskip -3em
\tag6.12
$$
Denote by $G_{ij}$ components of metric tensor of standard Euclidean
metric in moving frame $\bold M_1,\,\ldots,\,\bold M_{n-1}$ of
spherical coordinates on unit sphere:
$$
G_{ij}=(\bold M_i\,|\,\bold M_j).
$$
Let $G^{ij}$ be components of dual metric tensor, and let
$\vartheta^k_{ij}$ be
component of metric connection for metric $G_{ij}$ in spherical
coordinates. Cartesian components of vectors $\bold N,\,\bold M_1,\,
\ldots,\,\bold M_{n-1}$, and component $G_{ij}$, $G^{ij}$,
$\vartheta^k_{ij}$ of metric tensors and metric connection are the
functions of the following variables: $u^1,\,\ldots,\,u^{n-1}$. Upon
choosing some particular spherical coordinates in velocity space
one can calculate them explicitly. Coefficients of expansions
\thetag{6.12} are calculated as follows:
$$
\xalignat 2
&a=(\nabla A\,|\,\bold N),
&&\alpha^k=\sum^{n-1}_{q=1}G^{kq}\,(\nabla A\,|\,\bold M_q),
\hskip -3em\\
\vspace{-7pt}
&&&\tag6.13\\
\vspace{-7pt}
&b^i=(\nabla B^i\,|\,\bold N),
&&\beta^{ik}=\sum^{n-1}_{q=1}G^{kq}\,(\nabla B^i\,|\,\bold M_q).
\hskip -3em
\endxalignat
$$
Now we can write weak normality equations, which were mentioned above:
$$
\cases
B^i=-\dsize\sum^{n-1}_{k=1}G^{ik}\,\dfrac{\partial A}{\partial u^k},\\
\vspace{2ex}
\dsize\alpha^i-\frac{B^i\,A}{v^2}+\sum^{n-1}_{k=1}\frac{\bar\nabla_kB^i
\,B^k}{v^2}+b^i+\frac{A}{v}\,\frac{\partial B^i}{\partial v}
-\frac{B^i}{v}\,\frac{\partial A}{\partial v} = 0,
\endcases\hskip -3em
\tag6.14
$$
And we can write additional normality equation too. Here are they:
$$
\cases
\dfrac{1}{v}\,\dfrac{\partial B^i}{\partial v}\,B^k-\beta^{ik}=
\dfrac{1}{v}\,\dfrac{\partial B^k}{\partial v}\,B^i-\beta^{ki},\\
\vspace{2ex}
\bar\nabla_kB^i=\dsize\sum^{n-1}_{q=1}\frac{\bar\nabla_qB^q}{n-1}
\,\delta^q_k.
\endcases\hskip -3em
\tag6.15
$$
Covariant derivatives $\bar\nabla_kB^i$ in \thetag{6.14} and \thetag{6.15}
are defined by relationships
$$
\bar\nabla_kB^i=\frac{\partial B^i}{\partial u^k}+
\sum^{n-1}_{q=1}\vartheta^i_{kq}\,B^q.
$$
These are covariant derivatives with respect to spherical coordinates
$u^1,\,\ldots,\,u^{n-1}$ calculated in metric connection $\vartheta$
in velocity space. Substituting \thetag{6.13} into \thetag{6.14} and
\thetag{6.15} we see that weak and additional normality equations are
partial differential equations for the components of force vector
$\bold F$ in expansion  \thetag{6.11}.
\proclaim{Theorem 6.1} If force field of Newtonian dynamical system
\thetag{6.1} in Euclidean space $\Bbb R^n$ satisfies both weak and
additional normality equations \thetag{6.14} and \thetag{6.15},
then this system admits normal shift in the sense of definition~6.2.
\endproclaim
\head
\S\,7. The condition of normalization. Strong and complete
normality.
\endhead
\rightheadtext{\S\,7. The condition of normalization.}
    Let's unite weak normality condition from definition~6.3 and
additional normality condition from definition~6.4 into one
condition. It is called {\bf complete normality condition}.
Theorem~6.1, that was proved in \cite{38}, asserts that complete
normality condition is {\bf enough} for the dynamical system to
admit the normal shift of hypersurfaces in $\Bbb R^n$, i\.~e\.
for to satisfy {\bf normality condition} from definition~6.2.
Let's study how far is it from being {\bf necessary} condition
too. In order to do this we consider two examples.\par
    Simplest example of dynamical system in $\Bbb R^n$ that satisfies
normality condition from definition 6.2 is a system with identically
zero force field $\bold F(\bold r,\bold v)=0$. Normal shift along
trajectories of such system is reduced to multidimensional generalization
of Bonnet construction, For to provide normality of displacement
$f_t\!:S\to S_t$ in this case one can always choose $\nu(p)=1$ in
\thetag{6.3}. Therefore $|\bold v|=1$ on all trajectories of displacement,
which are rectilinear segments in this case. This means that in process
of displacement $f_t\!:S\to S_t$ we use not all points of phase space
\thetag{6.2}. From this fact we derive the idea of another example:
$$
\bold F(\bold r,\bold v)=\cases 0 &\text{for \ }|\bold v|<2,\\
\bold f(\bold r,\bold v)&\text{for  \ }|\bold v|\geqslant 2.
\endcases\hskip -2em
\tag7.1
$$
Its easy to understand that for any choice of $\bold f(\bold r,\bold v)$
in \thetag{7.1} dynamical system with such force field admits the normal
shift in the sense of definition~6.2. So, normality condition from
definition~6.2 is not very rigid. It doesn't contradict to the existence
of gaps in phase space, where force field can be given by absolutely
arbitrary function. In such gaps normality equations \thetag{6.14} and
\thetag{6.15} can be broken.\par
    Thus we can draw conclusion: complete normality isn't necessary
for the normality condition from definition~6.2 to be satisfied. This
is not good. In order to improve this situation in \cite{39} we
introduced the normalization for the function $\nu(p)$:
$$
\nu(p_0)=\nu_0.\hskip -2em
\tag7.2
$$
Now definition~6.2 can be reformulated as follows.
\definition{Definition 7.1} Newtonian dynamical system \thetag{6.1}
with force field $\bold F(\bold r,\bold v)$ is called the system
admitting the normal shift {\bf in strong sense}, if for any
sufficiently small part $S$ of any hypersurface in $\Bbb R^n$, for
any point $p_0$ on $S$, and for arbitrary nonzero number $\nu_0$
one can find a function $\nu$ on $S$ normalized by the condition
\thetag{7.2} such that the map $f_t\!:S\to S_t$ defined by the solution
of Cauchy problem \thetag{6.3} is a normal shift in the sense of
definition~6.1.
\enddefinition
Definition~7.1 excludes the existence of lacunas like in second example
above. The matter is that any nonzero vector $\bold v\neq 0$ at any
point $p_0$ in $\Bbb R^n$ can be represented as $\bold v=\nu_0\cdot
\bold n(p_0)$, where $\bold n(p_0)$ is a normal vector of some
hypersurface passing through the point $p_0$. Hence each point of
phase space can be involved into the process of displacement for some
hypersurface.\par
     Condition stated in definition~7.1 is called {\bf strong normality
condition}. The diagram on Fig 7.1 (see next page) shows the mutual
relation for various normality conditions. Implication 1 in this diagram
is obvious. Implications 4, 5, 6, and 7 are proved during derivation of
normality equation \thetag{6.14} and \thetag{6.15}. \pagebreak
Implications 2 and 3 are expressed by the following theorem proved
in \cite{40}, Implication 3 in this theorem is obvious.
\proclaim{Theorem 7.1} Conditions of {\bf complete} and {\bf strong}
normality for Newtonian dynamical systems are equivalent.
\endproclaim
\vskip 0.5cm\special{em:graph edis-01d.gif}\vskip 7.5cm
    Condition of $\bold v\neq 0$ is very important. In definition~7.1
it is represented by $\nu_0\neq 0$. In implicit form it is present in
definition~6.4 too, Indeed, functions $\psi_i(\nu,u^1,\ldots,u^{n-1})$
in right hand side of Pfaff equations \thetag{6.10}, when written in
explicit form, are not defined for $\nu=0$. Point with $\bold v=0$
are singular point on the trajectory of dynamical system \thetag{6.1}.
Use of trajectories with singular points is unreasonable, since it
may cause some extra difficulties. 
\head
\S\,8. Problem of metrizability and the test for
non-triviality.
\endhead
\rightheadtext{\S\,8. Problem of metrizability \dots}
     Normal shift of hypersurfaces in $\Bbb R^n$ along trajectories
of special Newtonian dynamical systems generalizes classical
construction of Bonnet transformation. But is this generalization
non-trivial\,? Or, may be, it coincides with generalizations which
were already known\,? Particularly with geodesic normal shift for
some metrics in $\Bbb R^n$\,? This question was studied in papers
\cite{41} and \cite{42}. The answer appeared favorable for the theory
of dynamical systems admitting the normal shift. In general case
normal shift along trajectories of Newtonian dynamical system does not
reduce to geodesic normal shift. Moreover:
\roster
\rosteritemwd=0pt
\item"1)" complete description of those systems, for which normal shift
          along their trajectories reduces to geodesic normal shift, was
          obtained;
\item"2)" some explicit examples of dynamical systems, for which normal
          shift can't be reduced to geodesic normal shift, were given.
\endroster
First was done in \cite{41}, second in \cite{42} respectively.\par
    Suppose that $f_t\!:S\to S_t$ is a normal shift of hypersurface $S$
along trajectories of dynamical system \thetag{6.1}. For this shift to
coincide with geodesic normal shift of some Riemannian metric $\bold g$
in $\Bbb R^n$ the following conditions should be fulfilled:
\roster
\rosteritemwd=0pt
\item"1)" trajectories of shift should coincide (as sets of points)
          with geodesic lines for the metric $\bold g$ (though they
          can differ in parametrization);
\item"2)" right angles between trajectories of dynamical system and
          hypersurfaces $S_t$ measured in standard Euclidean metric in
          $\Bbb R^n$ should be right angles in Riemannian metric
          $\bold g$ too.
\endroster
If we want that coincidence of $f_t\!:S\to S_t$ and geodesic normal shift
for Riemannian metric $\bold g$ not to be a casual event, we should require
the conditions 1) and 2) to be fulfilled for any hypersurface $S$. Note that
right angle defined by any two vectors can be represented as an angle
between a tangent vector for some hypersurface $S$ and the unit normal
vector of this hypersurface. Condition 2) then means that each right angle
in Euclidean metric in $\Bbb R^n$ is a right angle in Riemannian metric
$\bold g$. Here we can apply the following obvious lemma.
\proclaim{Lemma 8.1} If each Euclidean right angle in $\Bbb R^n$ is a right
angle in Riemannian metric $\bold g$, then $\bold g$ is a conformally
Euclidean metric: $\bold g=e^{-2f}\,\boldsymbol\delta$.
\endproclaim
    By $f=f(\bold r)$ in Lemma 8.1 we denote some scalar function
in $\Bbb R^n$. In Cartesian coordinates for the components of metric
$\bold g$ we have
$$
g_{ij}=e^{-2f}\,\delta_{ij}\text{, \ where \ }\delta_{ij}=\cases
1&\text{for \ }i=j,\\ 0&\text{for \ }i\neq j.\endcases\hskip -2em
\tag8.1
$$
Metric \thetag{8.1} generates a geodesic flow, which can be represented
as Newtonian dynamical system \thetag{6.1} with force field 
$$
\bold F_0=-|\bold v|^2\cdot\nabla f+2\,(\nabla f\,|\,\bold v)
\cdot\bold v.\hskip -2em
\tag8.2
$$
Trajectories of dynamical system with force field \thetag{8.2} are
geodesic lines for the metric \thetag{8.1}.\par
     Let's consider pair of Newtonian dynamical systems with force
fields $\bold F(\bold r, \bold v)$ and $\bold F_0(\bold r, \bold v)$
in $n$-dimensional Euclidean space $\Bbb R^n$:
$$
\xalignat 2
&\dot\bold r=\bold v,&&\dot\bold v=\bold F(\bold r,\bold v),\hskip -2em
\tag8.3\\ \vspace{1ex}
&\dot\bold r=\bold v,&&\dot\bold v=\bold F_0(\bold r,\bold v).\hskip -2em
\tag8.4
\endxalignat
$$
Trajectories of these two dynamical systems are defined by initial data
that fix starting  point and initial velocity for them:
$$
\xalignat 2
&\quad\bold r\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=\bold r_0,
&&\bold v\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=\bold v_0,\hskip -2em
\tag8.5\\ \vspace{1ex}
&\quad\bold r\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=\bold r_0,
&&\bold v\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=\bold w_0.\hskip -2em
\tag8.6
\endxalignat
$$
Denote by $\bold r=\bold R_1(t,\bold r_0,\bold v_0)$ solution of
Cauchy  problem \thetag{8.5} for the equations \thetag{8.3}. Let
$\bold r=\bold R_2(t,\bold r_0,\bold w_0)$ be a solution of Cauchy
problem \thetag{8.6} for the equations \thetag{8.4}. If normal
shift along trajectories of dynamical system \thetag{8.3} coincides
with geodesic normal shift in metric \thetag{8.1}, then according
to above condition 1) each trajectory of such shift should coincide
with some trajectory of other dynamical system \thetag{8.4} up to
a change of parametrization. Since hypersurface $S$, point $p_0$ on
$S$, and numeric parameter $\nu_0\neq 0$ in normalization condition
\thetag{7.2}, we are in a situation described by the following
definition.
\definition{Definition 8.1} We say that dynamical system \thetag{8.3}
{\bf inherits} trajectories of the system \thetag{8.4}, if for any pair
of vectors $\bold r_0$ and $\bold v_0\neq 0$ one can find a vector 
$\bold w_0$ and twice differentiable function $T(\tau)$ such that
$T(0)=0$ and the following equality
$$
\bold R_2(T(\tau),\bold r_0,\bold w_0)=\bold R_1(\tau,\bold r_0,
\bold v_0),
$$
holds identically for $\tau$ in some neighborhood of zero. 
\enddefinition
\definition{definition 8.2} Two Newtonian dynamical systems in $\Bbb R^n$
are called {\bf trajectory equivalent}, if they inherit trajectories of
each other in the sense of definition~8.1.
\enddefinition
    Note that above definition of trajectory equivalence differs from the
definition of papers \cite{43--49}, which was used for topological
classification of integrable Hamiltonian systems. Our definition is more
specific, it is applicable only for Newtonian dynamical systems with
common phase space \thetag{6.2}. Similar definition was stated in paper
\cite{50} by Levi-Civita, and was used in \cite{51--54}. In geometry
the concept of geodesic equivalence for two affine connections appears
to be a specialization of definition~8.2 (see \cite{55--57}). 
\definition{Definition 8.3} Newtonian dynamical system in $\Bbb R^n$
is called {\bf metrizable}, if it inherits trajectories of geodesic
flow for some conformally Euclidean metric \thetag{8.1}.
\enddefinition
\adjustfootnotemark{-1}
    In paper \cite{41} was considered the class of Newtonian dynamical
systems, which are both {\bf metrizable} and {\bf admitting the normal
shift\footnotemark} simultaneously. The following result was obtained.
\footnotetext{Systems admitting the normal shift here and in what follows
are understood in strong sense, i\. e\. in the sense of definition~7.1.}
\proclaim{Theorem 8.2} Newtonian dynamical system \thetag{8.3} admitting
the normal shift in $\Bbb R^3$ is metrizable if its force field is given
by formula
$$
\bold F(\bold r,\bold v)=-|\bold v|^2\cdot\nabla f+2\,(\nabla f\,|\,
\bold v)\cdot\bold v+\frac{\bold v}{|\bold v|}\cdot
H(|\bold v|e^{-f})e^f,\hskip -1em
\tag8.7
$$
where $f=f(\bold r)=f(r^1,\dots,r^n)$ and $H=H(v)$ are arbitrary functions.
Being metrizable, such dynamical system performs geodesic normal shift for
metric \thetag{8.1}.
\endproclaim
     Theorem~8.2 solves the problem of metrizability for dynamical
systems admitting the normal shift and gives examples of force fields
$\bold F(\bold r,\bold v)\neq 0$, satisfying normality equations
\thetag{6.14} and \thetag{6.15} in multidimensional case. 
But force fields \thetag{8.7} are assumed to be trivial, since
their existence is predictable on a base of simple geometric
considerations. The normal shift defined by such systems is reduced
to the construction of geodesic normal shift, which was already
known in geometry. Therefore we have a problem to find some
other solutions of normality equations that correspond to {\bf
non-metrizable} dynamical systems. The importance of this
problem was pointed out by academician A.~T.~Fomenko during my
report in the seminar at Moscow State University in the beginning
of 1994. Fortunately solution of this problem appeared to be
relatively simple. It was solved in 1994 in paper \cite{42}.
Force field that was found in paper \cite{42} is the following:
$$
\bold F(\bold x,\bold v)=\frac{A(|\bold v|)}{|\bold v|^2}\,
\bigl(2\,(\bold m\,|\,\bold v)\cdot\bold v-\bold m\cdot|\bold v|^2\bigr).
\hskip -2em
\tag8.8
$$
Here $A(v)$ is some arbitrary function of one variable, and $\bold m$
is an arbitrary constant vector. Existence of the solution \thetag{8.8}
showed that, in spite of being overdetermined, system of equations
\thetag{6.14} and \thetag{6.15} has nontrivial solutions in
multidimensional case. So in multidimensional case we also have the
examples of dynamical systems that can perform normal shift of
hypersurfaces along their trajectories, being different from geodesic
normal shift.
\head
\S\,9. Generalization for  Riemannian geometry. Tensorial form
of equations.
\endhead
\rightheadtext{\S\,9. Generalization for  Riemannian geometry.}
     Next step in development of the theory of dynamical systems
admitting the normal shift was due to the transfer to Riemannian
geometry. This was done in \cite{58}, \cite{39}, and \cite{59}.
In these papers configuration space $M=\Bbb R^n$ of dynamical system
\thetag{6.1} was replaced by an arbitrary $n$-dimensional Riemannian
manifold $M$. Newton's second law, by its origin, is written for
separate mass point or for system of several mass interacting mass
point in $\Bbb R^3$. Configuration space for the system of $N$
mass points is $n$-dimensional space $\Bbb R^n$, where $n=3N$.
Continuous rigid body can be modeled by $N$ mass points bound
with each other by hard weightless rods. If we take into account
these bounds we see that they diminish the degree of freedom from
$n=3N$ to $n=6$. This means that in $n$-dimensional space $\Bbb R^n$
we mark six-dimensional manifold $M$ isomorphic to $\Bbb R^3\times
\SO(3,\Bbb R)$. This is a configuration space for moving rigid body.
Here we have $3$ degrees of freedom for translational movement and $3$
degrees of freedom for rotational movement. This situation may get more
complicated if some bounds are flexible, e\. g\. if some hard rods
are replaced by cardan joint, thumbscrew, tooth gearing etc. But even in
this case we have some manifold $M$ embedded into $\Bbb R^n$. Thus
we can draw the following conclusion: dynamical systems on manifolds are
more typical in classical mechanics than systems in Euclidean space
$\Bbb R^n$.\par
     Suppose that $M$ is some manifold, being configuration space
for some system of rigid bodies with flexible bounds. Local coordinates
$x^1,\,\ldots\,x^n$ from some map of such manifold in mechanics are
called {\it generalized coordinates}, their derivatives $\dot x^1,\,
\ldots,\,\dot x^n$ are called {\it generalized velocities} (see \cite{60}). 
{\bf Kinetic energy} is a very important characteristics of mechanical
system. Its dependence on generalized coordinates $x^1,\,\ldots,\,x^n$ may
be very complicated, but (for realistic mechanical systems) it is always a
quadratic function of generalized velocities. Let's denote kinetic energy
of dynamical system by $G$ and let's write it in the following form:
$$
G=\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1} g_{ij}\,\dot x^i\,\dot x^j.
\hskip -2em
\tag9.1
$$
Coefficients $g_{ij}=g_{ij}(x^1,\ldots,x^n)$ in quadratic form
\thetag{9.1} define a metric in $M$ arranging the structure of
Riemannian manifold.\par
     Dynamical systems that appears in classical mechanics are often
{\it conservative}. This means that their behavior is described by
two functions: kinetic energy \thetag{9.1} and {\it potential energy}
$\varPi=\varPi(x^1,\ldots,x^n)$. They are used to define {\it the
function of Lagrange} $L=\varPi-G$. Then differential equations of
dynamics are derived from variational principle and written as
Euler-Lagrange equations:
$$
\frac{d}{dt}\left(\frac{\partial L}{\partial\dot x^k}\right)-
\frac{\partial L}{\partial x^k}=0\text{, \ \ where\ \ \ }k=1,\,
\ldots,\,n.
$$
Afterwards these equations are usually examined for the existence
of stationary solutions, for some asymptotics near such stationary
solutions, for stability or, may be, for complete integrability.
Kinetic energy by itself (and associated Riemannian metric on $M$)
appears to be of less importance on this stage. In the theory
of dynamical system admitting the normal shift, to the contrary,
it plays central role, while conservativity or non-conservativity
of the system doesn't matter at all.\par
    One can consider Newtonian dynamical systems in non-Riemannian
manifolds as well. In local coordinates such systems are written in
form of systems of $2n$ ordinary differential equations
$$
\xalignat 2
&\quad\dot x^k=v^i,&&\dot v^k=\Phi^k(x^1,\ldots,x^n,v^1,\ldots,v^n),
\hskip -2em
\tag9.2
\endxalignat
$$
where $k=1,\,\ldots,\,n$. But in such unstructured manifolds one
cannot define the force vector. Functions $\Phi^1,\,\ldots,\,\Phi^1$
in \thetag{9.2} have no direct vectorial interpretation. They form
a subset of coordinates for some vector field on tangent bundle $TM$:
$$
\boldsymbol\Phi=v^1\,\frac{\partial}{\partial x^1}+\ldots+
v^n\,\frac{\partial}{\partial x^n}+\varPhi^1\,\frac{\partial}
{\partial v^1}+\ldots+\varPhi^n\,\frac{\partial}{\partial v^n}.
\hskip -3em
\tag9.3
$$
If we have Riemannian metric $\bold g$ in $M$, then geodesic flow of
this metric defines another vector field of the form \thetag{9.3} on
$TM$:
$$
\tilde{\boldsymbol\Phi}=v^1\,\frac{\partial}{\partial x^1}+\ldots+v^n\,
\frac{\partial}{\partial x^n}-\sum^n_{i=1}\sum^n_{j=1}\sum^n_{k=1}
\Gamma^k_{ij}\,v^i\,v^j\,\frac{\partial}{\partial v^k}.\hskip -3em
\tag9.4
$$
Difference of vectors \thetag{9.3} and \thetag{9.4} is a vector
tangent to the fiber in $TM$:
$$
\boldsymbol\Phi-\tilde{\boldsymbol\Phi}=\sum^n_{k=1}\left(\varPhi^k+
\shave{\sum^n_{i=1}\sum^n_{j=1}}\Gamma^k_{ij}\,v^i\,v^j\right)
\frac{\partial}{\partial v^k}.\hskip -3em
\tag9.5
$$
By means of canonical vertical lift (see below in \S\,3 of Chapter
\uppercase\expandafter {\romannumeral 3}) we can identify \thetag{9.5}
with tangent vector $\bold F$ on initial manifold $M$:
$$
\bold F=\sum^n_{k=1}\left(\varPhi^k+\shave{\sum^n_{i=1}\sum^n_{j=1}}
\Gamma^k_{ij}\,v^i\,v^j\right)\frac{\partial}{\partial x^k}.\hskip -3em
\tag9.6
$$
Vector $\bold F$ from \thetag{9.6} generalizes the concept of {\bf force
vector} of classical mechanics for the case of Newtonian dynamics on an
abstract Riemannian manifold. In terms of components of such vector the
equations \thetag{9.2} are written as 
$$
\xalignat 2
&\quad\dot x^k=v^k,&&\dot v^k+\sum^n_{i=1}\sum^n_{j=1}\Gamma^k_{ij}\,
v^i\,v^j=F^k,\hskip -2em
\tag9.7
\endxalignat
$$
where $k=1,\,\ldots,\,n$. Force vector is a tangent vector to $M$.
However, its components in \thetag{9.7} depend not only on local
coordinates $x^1,\,\ldots,\,x^n$ of the point on $M$, but they depend
also on the components of velocity vector $v^1,\,\ldots,\,v^n$, which
are local coordinates on the fiber of tangent bundle $TM$.
In trying to comprehend this situation in \cite{58} we came to the
concept of {\bf extended vector field}.
\definition{Definition 9.1} Vector valued function $\bold F$ on $TM$
that for each point $q=(p,\bold v)$ on $TM$ puts into correspondence
some vector of tangent space $T_p(M)$ at the point $p=\pi(q)$ on $M$
is called the {\bf extended vector field} on $M$.
\enddefinition
     Extended scalar fields, extended covector fields and other extended
tensor field are defined in a similar way, they form {extended algebra
of tensor fields} on the manifold $M$ (see below in Chapter
\uppercase\expandafter {\romannumeral 2}). The concept of extended
algebra of tensor fields is not new, such fields are intensively
used in Finslerian geometry (see \cite{61}). In the theory of
dynamical systems admitting the normal shift in Riemannian
manifolds they appear to be to the point too.\par
     In extended algebra of tensor field on Riemannian manifold
we have two operation of covariant differentiation: {\bf spatial
gradient} $\nabla$ and {\bf velocity gradient} $\tilde\nabla$
(see Chapter \uppercase\expandafter {\romannumeral 3} below).
Weak normality equations in Riemannian geometry are written as
the equations for the components of force field $\bold F$ in terms
of gradients (covariant differentiations $\nabla$ and $\tilde\nabla$):
$$
\cases
\dsize\sum^n_{i=1}\left(v^{-1}\,F_i+\shave{\sum^n_{j=1}}
\tilde\nabla_i\left(N^j\,F_j\right)\right)P^i_k=0,\\
\vspace{2.5ex}
\aligned
 &\sum^n_{i=1}\sum^n_{j=1}\left(\nabla_iF_j+\nabla_jF_i-2\,v^{-1}
 \,F_i\,F_j\right)N^j\,P^i_k\,+\\
 \vspace{0.5ex}
 &+\sum^n_{i=1}\sum^n_{j=1}\left(\frac{F^j\,\tilde\nabla_jF_i}{v}
 -\sum^n_{r=1}\frac{N^r\,N^j\,\tilde\nabla_jF_r}{v}\,F_i\right)P^i_k=0.
 \endaligned
\endcases\hskip -3em
\tag9.8
$$
Here $v=|\bold v|$ is the modulus of velocity vector, \pagebreak $\bold N$
is the unit vector directed along $\bold v$, and $\bold P$ is the operator
field composed of orthogonal projectors to the hyperplanes perpendicular
to the vector of velocity $\bold v$. Components of $\bold P$ are denoted
by $P^i_k$, components of vector $\bold N$ are denoted by $N^1,\,\ldots,
\,N^n$, and finally $N^1,\,\ldots,\,N^n$ are covariant components of
force field. They are obtained by lowering the upper index:
$$
F_i=\sum^n_{j=1}g_{ij}\,F^j.
$$\par
    Additional normality equations are also written in terms of
covariant differentiations $\nabla$ and $\tilde\nabla$ (spatial
and velocity gradients):
$$
\cases
\aligned
&\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_i\,\tilde\nabla_mF_j}{v}
-\nabla_iF_j\right)=\\
\vspace{0.5ex}
&\quad=\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_j\,\tilde\nabla_mF_i}{v}
-\nabla_jF_i\right),
\endaligned\\
\vspace{2.5ex}
\dsize\sum^n_{i=1}\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF^i\,
P^\varepsilon_i=\sum^n_{i=1}\sum^n_{j=1}\sum^n_{m=1}
\frac{P^j_m\,\tilde\nabla_jF^i\,P^m_i}{n-1}\,P^\varepsilon_\sigma.
\endcases\hskip -3em
\tag9.9
$$
The above form of normality equations \thetag{9.8} and \thetag{9.9}
implies that we choose local coordinates $x^1,\,\ldots,\,x^n$ in $M$
and corresponding local coordinates $v^1,\,\ldots,\,v^n$ in fibers of
tangent bundle $TM$. However, transfer from one system of local
coordinates to another doesn't change the form of these equations.
This is their difference from the equations \thetag{6.14} and
\thetag{6.15}, which are strictly bound to Cartesian coordinates in
$\Bbb R^n$. Direct recalculation of \thetag{9.8} and \thetag{9.9}
to the form \thetag{6.14} and \thetag{6.15} for the case $M=\Bbb R^n$
was done in \cite{62}.\par
    The equations \thetag{9.8} and \thetag{9.9} were derived in \cite{58}
and \cite{39}. Transfer of basic definitions of strong and complete
normality (definitions~6.3, 6.4, and 7.1 above) to the Riemannian geometry
did not require much efforts (details see in Chapter \uppercase\expandafter
{\romannumeral 5} below). Diagram of implications on Fig\. 7.1 remained
unchanged thereby. One should note that implication 2 in that diagram
was proved immediately for the case of Riemannian manifolds skipping
preliminary steps of $\Bbb R^2$ and $\Bbb R^n$.\par
    In paper \cite{59} the problem of metrizability by means of
conformally equivalent metric $\tilde\bold g=e^{-2f}\,\bold g$
for the dynamical systems on Riemannian manifold with metric $\bold g$
was considered. Within the class of dynamical systems admitting the
normal shift subclass of metrizable systems was distinguished. Force
fields for such dynamical systems are given by the following explicit
formula:
$$
\bold F=-|\bold v|^2\cdot\nabla f+2\,(\nabla f\,|\,
\bold v)\cdot\bold v+\frac{\bold v}{|\bold v|}\cdot
H(|\bold v|e^{-f})e^f.\hskip -1em
\tag9.10
$$
Here $H(v)$ is an arbitrary function of one variable, \pagebreak 
and $f=
f(x^1,\ldots,x^n)$ is a scalar field on $M$. By  $\nabla f$
in \thetag{9.10} we denote the vector field with components 
$$
\nabla^if=\sum^n_{j=1}g^{ij}\,\frac{\partial f}{\partial x^j}.
$$
Formula \thetag{9.10} appeared to coincide with formula
\thetag{8.7} that corresponds to special case $M=\Bbb R^n$.\par
    In \S\,4 of Chapter \uppercase\expandafter {\romannumeral 6}
of this thesis wider treatment of the problem of
metrizability is suggested. The condition of conformal equivalence
$\bold g$ and $\tilde\bold g$ in its statement is eliminated.
Moreover, there Newtonian dynamical systems that inherit trajectories
of geodesic flow of an arbitrary affine connection $\Gamma$ in $M$ are
considered. But even in such deliberate approach, if we require that
dynamical system should admit the normal shift in metric $\bold g$,
then $\Gamma$ appears to be a metric connection for some conformally
equivalent metric $\tilde\bold g=e^{-2f}\,\bold g$.
\head
\S\,10. Reduction of normality equations for the case
$n\geqslant 3$.
\endhead
    In the dimension $\dim M=n\geqslant 3$ complete system of normality
equations consists of weak normality equations \thetag{9.8} and
additional normality equations \thetag{9.9}. It is highly overdetermined,
becoming more and more overdetermined with the growth of dimension.
In the study of overdetermined systems of equations we encounter the
problem of their compatibility. The existence of the solution \thetag{9.10}
an the solution \thetag{8.8} in the flat space $M=\Bbb R^n$ make this
problem less urgent as it could be. But this doesn't not solve the problem.
The matter is that systems of differential equations of some definite
order one can derive additional equations being differential consequences
of initial ones. Sometimes the order of such differential consequences
is equal to, or even less than maximal order of initial equations in the
system. For instance, well known Korteweg-de Vries equation
$u_t=6\,u\,u_x-u_{xxx}$ is the differential consequence in the system of
two Lax equations
$$
\cases
-\psi_{xx}+u\,\psi=\lambda\,\psi,\\
\vspace{1ex}
\psi_t=-4\,\psi_{xxx}+6\,u\,\psi_x+3\,u_x\,\psi,
\endcases
$$
where $\lambda=\const$. Complete analysis of the system of differential
equations implies that one should find all differential consequences of
the order less or equal to the maximal order of the equations in the
system, and possibly simplify the system by means of such differential
consequences. Such analysis for the complete system of normality
equations \thetag{9.8} and \thetag{9.9} was done in \cite{63}.\par
    Writing normality equations \thetag{6.14} and \thetag{6.15} in
Euclidean space $M=\Bbb R^n$ we used the expansion \thetag{6.11} for
vector of force. In this expansion we have one component $A\cdot\bold N$
directed along the vector of velocity, other components in this expansion
$B^1\cdot\bold M_1,\,\ldots,\,B^{n-1}\cdot\bold M_{n-1}$ are perpendicular
to $\bold v$. Due to the first equation \thetag{6.14} coefficients $B^1,\,
\ldots,\,B^{n-1}$ of perpendicular components are expressed through the
coefficient $A$. \pagebreak Something like this we have in general case
for dynamical systems in Riemannian manifolds. Here for the force vector
we can write the expansion
$$
F^k=A\,N^k+\sum^n_{j=1}P^k_j\,F^j,\hskip -3em
\tag10.1
$$
similar to \thetag{6.11}. Substituting \thetag{10.1} into first
normality equation from \thetag{9.8} we derive the following
expressions for the components of force vector $\bold F$:
$$
F_k=A\,N_k-|\bold v|\,\sum^n_{i=1}P^i_k\,\tilde\nabla_i A.\hskip -3em
\tag10.2
$$
Formula \thetag{10.2} is called the {\bf scalar substitution}, since
it expresses force vector $\bold F$ through one scalar field $A$
from extended algebra of tensor field in $M$.\par
    Let's substitute \thetag{10.2} into \thetag{9.8}. As a result first
equation in \thetag{9.8} arrears to be identically fulfilled, while
the second equation takes the form
$$
\gathered
\sum^n_{s=1}\left(\nabla_sA+|\bold v|\sum^n_{q=1}\sum^n_{r=1}
P^{qr}\,\tilde\nabla_qA\,\tilde\nabla_r\tilde\nabla_sA\right.-\\
\vspace{1ex}
-\left.\shave{\sum^n_{r=1}}N^r\,A\,\tilde\nabla_r\tilde\nabla_sA
-|\bold v|\shave{\sum^n_{r=1}}N^r\,\nabla_r\tilde\nabla_sA\right)
P^s_k=0.
\endgathered\hskip -3em
\tag10.3
$$
Making scalar substitution into first equation in \thetag{9.9} we get
$$
\aligned
\sum^n_{s=1}&\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\left(
\nabla_r\tilde\nabla_sA+\shave{\sum^n_{q=1}}\tilde\nabla_rA\,N^q\,
\tilde\nabla_q\tilde\nabla_sA\right)=\\
\vspace{1ex}
&=\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\left(
\nabla_s\tilde\nabla_rA+\shave{\sum^n_{q=1}}\tilde\nabla_sA\,N^q\,
\tilde\nabla_q\tilde\nabla_rA\right).
\endaligned\hskip -3em
\tag10.4
$$
Reduced form of second equation in the system \thetag{9.9} is especially
remarkable:
$$
\sum^n_{r=1}\sum^n_{s=1}P^r_\sigma\,\tilde\nabla_r\tilde\nabla_sA\,
P^{s\varepsilon}=\lambda\,P^\varepsilon_\sigma.\hskip -3em
\tag10.5
$$
Here $\lambda$ is a scalar parameter that can be determined by formula
$$
\lambda=\sum^n_{r=1}\sum^n_{s=1}\frac{P^{rs}\,\tilde\nabla_r
\tilde\nabla_sA}{n-1}.\hskip -3em
\tag10.6
$$
This equation was analyzed in \cite{63}. There the following
theorem was proved.
\proclaim{Theorem 10.1} Extended scalar field $A$ on riemannian
manifold $M$ satisfies normality equations \thetag{10.5} if and
only if it is given by formula
$$
A=a+\sum^n_{i=1}b_i\,v^i,\hskip -3em
\tag10.7
$$
where $a$ and $\bold b$ are special scalar and covectorial fields of
extended algebra such that they depend only on the modulus of velocity
vector $v=|\bold v|$.
\endproclaim
Then the expression \thetag{10.7} for scalar field $A$ should be
substituted into normality equations \thetag{10.3} and \thetag{10.4}.
This yields
$$
\gather
\left(\frac{\partial}{\partial x^s}+b_s\,\frac{\partial}{\partial v}
\right)a=\left(a\,\frac{\partial}{\partial v}\right)b_s,\hskip -3em
\tag10.8\\
\vspace{1ex}
\left(\frac{\partial}{\partial x^s}+b_s\,\frac{\partial}{\partial v}
\right)b_r=\left(\frac{\partial}{\partial x^r}+b_r\,\frac{\partial}
{\partial v}\right)b_s.\hskip -3em
\tag10.9
\endgather
$$
Unfortunately by substituting \thetag{10.7} into \thetag{10.4} in
\cite{63} we made a mistake. As a result nonlinear terms in
equation \thetag{10.9} were lost, and we obtained erroneous equations.
Further study of erroneous equations has led us to the equation
$$
y''=H_y(y'+1)+H_x\text{, \ where \ }H=H(x,y).\hskip -3em
\tag10.10
$$
Now we know that the equation \thetag{10.10} has no relation to
the theory of dynamical systems admitting the normal shift. When
this wasn't yet known, in \cite{64-67} we undertook broad study
of the following class of ordinary differential equations
$$
y''=P(x,y)+3\,Q(x,y)\,y'+3\,R(x,y)\,{y'}^2+S(x,y)\,{y'}^3
$$
that includes the equations \thetag{10.10} as a subclass. Now, due to
recent news, the results of \cite{64-67} are excluded from this
thesis.\par
    Displeasing error found in \cite{63} now is corrected. Corrected
results are given below in Chapter \uppercase\expandafter {\romannumeral
7}. Correct system of equations \thetag{10.8} and \thetag{10.9} appears
to be explicitly solvable. General solution of these equations is
defined by two arbitrary functions. First is a function of $(n+1)$
variables $W=W(x^1,\ldots,x^n,v)$ with 
$$
W_v=\frac{\partial W}{\partial v}=\sum^n_{i=1}N^i\,\,
\tilde\nabla_i W\neq 0.\hskip -3em
\tag10.11
$$
Second is a function of one variable $h=h(v)$. Then $a$ and $b_k$ are
the following:
$$
\pagebreak
\aligned
&a=\frac{h(W(x^1,\ldots,x^n,v))}{\partial W(x^1,\ldots,x^n,v)
/\partial v},\\
\vspace{2ex}
&b_k=-\frac{\partial W(x^1,\ldots,x^n,v)/\partial x^k}
{\partial W(x^1,\ldots,x^n,v)/\partial v}.
\endaligned\hskip -3em
\tag10.12
$$
Function $W$ is interpreted as a scalar field of extended algebra
of tensor fields on $M$ depending only on the modulus of velocity
vector $v=|\bold v|$. Substituting \thetag{10.12} into \thetag{10.7},
and then substituting \thetag{10.7} into \thetag{10.2} we make
scalar substitution \thetag{10.2} more specific. As a result we get
the following theorem proved in Chapter \uppercase\expandafter
{\romannumeral 7}.
\proclaim{Theorem 10.2} \ Newtonian dynamical system on $n$-dimensional
Riemannian ma\-nifold $M$ with $n\geqslant 3$ admits the normal shift
of hypersurfaces in $M$ if and only if its force field $\bold F$ is given
by the formula
$$
F_k=\frac{h(W)\,N_k}{W_v}-|\bold v|\sum^n_{i=1}\frac{\nabla_iW}{W_v}
\,\bigl(2\,N^i\,N_k-\delta^i_k\bigr).\hskip -2em
\tag10.13
$$
Here $W$ is arbitrary extended scalar field depending only on
$v=|\bold v|$ and satisfying condition \thetag{10.11}; $h=h(v)$ is
arbitrary scalar function of one variable.
\endproclaim
    Explicit formula \thetag{10.13} for the force field $\bold F$
makes possible detailed analysis of mechanism of normal shift for
any hypersurface (see in \S\,7 of Chapter \uppercase\expandafter
{\romannumeral 7} below). The inequality $n\geqslant 3$ in theorem~10.2
is essential. In two-dimensional case situation is quite different
(see \cite{36}).
\head
\S\,11. Some generalizations: generalization for Finslerian
geometry and higher order dynamical systems.
\endhead
\rightheadtext{\S\,11. Some generalizations \dots}
    Formula \thetag{10.13} yields explicit and complete description of
force fields for all Newtonian dynamical systems admitting the normal
shift in Riemannian manifolds of the dimension greater than $2$.
However, one shouldn't consider it as a final point of the theory in
whole. It was in 1993, when in \cite{68} theory of dynamical systems
admitting the normal shift was generalized  for  the  systems  on 
Finslerian
manifolds. Paper \cite{68} was submitted to one of journals, was being
refereed  for  the  long  time,  but,  nevertheless,  it   wasn't 
published.
Results of this paper is given in Chapter \uppercase\expandafter
{\romannumeral 8} of this thesis. Here we give brief description
of these results.\par
    Finslerian metric $\bold g$ in the manifold $M$, in contrast to
the Riemannian metric, is extended tensor field on $M$. Its components
depend not only on the point $p\in M$, but on the components of velocity
vector $\bold v$ too. This doesn't influence the properties of
scalar product of two tangent vectors
$$
(\bold X\,|\,\bold Y)=\sum^n_{i=1}\sum^n_{j=1}g_{ij}(p,\bold v)\,
X^i\,Y^j,\hskip -3em
\tag11.1
$$
except for the case when one of these two vectors $\bold X$ or $\bold Y$
coincides with the vector of velocity $\bold v$. It such case scalar
product is nonlinear respective to one of its multiplicands. And this
is the very situation that arises when we define the normal shift in
Finslerian geometry. Despite to this difficulty in \cite{68} the proper
generalization of the theory was found. There all normality equations
were rederived and the diagram of implications from Fig\.~7.1 was
completely reproduced.\par
     Its remarkable that weak normality equations and additional
normality equations in Finslerian manifolds have exactly the same form
as the equations \thetag{9.8} and \thetag{9.9} in Riemannian geometry.
Scalar substitution \thetag{10.2} here is also applicable. However,
this is the end point of coincidences. The matter is that proof of
theorem~10.2 in \cite{63} is based on the fact that Riemannian metric
$\bold g$ in $M$ induces flat Euclidean metric in each fiber of tangent
bundle $TM$. For Finslerian metric this is not true. Therefore problem
of reducing normality equations in the dimension $n\geqslant 3$ for
Finslerian manifolds still remains open.\par
    Another possible way of generalization for the theory of dynamical
systems admitting the normal shift consist in considering higher order
equations. Even in flat space $\Bbb R^n$ we can replace Newtonian
dynamics \thetag{6.1} by the equations of $k$-th order:
$$
\frac{d^k\bold r}{dt^k}=\bold F(\bold r,\bold v_1,\ldots,\bold
v_{k-1})\text{, \ where \ }\bold v_i=\frac{d^i\bold r}{dt^i},\ \
i=1,\,\ldots,\,k-1.
$$
These are the equations of {\bf higher} (non-Newtonian) dynamical system.
By analogy with Newtonian systems vector-function $\bold F$ is called the
{\bf force field} of higher dynamical system. Similar higher order
dynamical systems can be defined on an arbitrary Riemannian manifolds.
This was done in \cite{40}. In this paper one version of the theory of
higher dynamical systems admitting the normal shift was suggested. 
All necessary definitions were stated and normality equations for the
force field $\bold F$ were derived. However, for $k>2$ complete system
of normality equations appeared to be sufficient condition for the
dynamical system to admit the normal shift. But it is far from being
necessary condition too. And we have no examples of force fields
satisfying all normality equations from \cite{40} simultaneously.
Therefore all results for $k>2$ are not included into this thesis.
\head
\S\,12. Unsolved problems, possible applications
and further prospects.
\endhead
\rightheadtext{\S\,12. Unsolved problems \dots}
\adjustfootnotemark{-1}
    Explicit formula \thetag{10.13} for the force field $\bold F$ closes
the problem of studying normality equations for $n\geqslant 3$ in
Riemannian geometry\footnote{In Finslerian geometry such problem still
remains open.}. But at the same time this formula substantially simplifies
the use of dynamical systems admitting the normal shift for further
generalization of classical Bianchi-Lie, B\"acklund, and Darboux
constructions and their multidimensional analogs from \cite{28--31}.
This is the nearest and most realistic prospect of application for our
theory.\par
    We have some prospects within the theory of dynamical systems
admitting the normal shift. First is the study of normality equations
in Finslerian geometry and the theory of higher order systems. Then
one can replace hypersurfaces by submanifolds of higher codimension.
How does it change the theory\,? This question is still open. Note
also the problem of inheriting. Suppose, that $M$ is Riemannian or
Finslerian manifold equipped with dynamical system admitting the normal
shift, and let $M'\subset M$ be a submanifold in $M$. Can we restrict
dynamical system from $M$ to $M'$\,? Will this restriction be admitting
the normal shift or not\,?\par
    As a further prospect we should mark the desire to join this theory
with the theory of relativity and with other geometric theories popular
in modern physics. Finslerian version of our theory developed in \cite{68}
may be very opportunely here (see for instance \cite{69--73}).
\head
\S\,13. Structure of thesis: distribution of topics
by chapters.
\endhead
    In this thesis we state the theory of dynamical systems admitting the
normal shift starting from that point when it was generalized
for the case of systems on arbitrary Riemannian manifolds. Chapter
\uppercase\expandafter {\romannumeral 2}, Chapter \uppercase\expandafter
{\romannumeral 3}, and \uppercase\expandafter {\romannumeral 4} contain
some preliminaries. {\bf In Chapter \uppercase\expandafter {\romannumeral
2}} we derive the equations of Newtonian dynamics on Riemannian manifolds
proceeding from mechanics of a system of mass points with holonomic
bounds. Here we explain the origin of Riemannian metric in configuration
space of realistic mechanical systems. Moreover in Chapter
\uppercase\expandafter {\romannumeral 2} we introduce the concept of
extended algebra of tensor fields which is a basic tool for all other
Chapters.\par
     {\bf In Chapter \uppercase\expandafter {\romannumeral 3}} we build
the calculus of differentiations in extended algebra of tensor fields.
Here the covariant differentiations $\nabla$ and $\tilde\nabla$ are
introduced and their properties are studied. Structural theorem for
arbitrary differentiation in extended algebra of tensor fields is
proved.\par
     {\bf In Chapter \uppercase\expandafter {\romannumeral 4}} we
consider the curves and one-parametric families of curves on Riemannian
manifolds. Vector of variation is defined and tensor fields on curves
are studied.\par 
     {\bf Chapter \uppercase\expandafter {\romannumeral 5}} contains
main results of thesis. Here  basic definitions of the theory of
dynamical systems admitting the normal shift are given and normality
equations are derived. And here we prove all implications shown on
Fig\.~7.1.\par
    {\bf In Chapter \uppercase\expandafter {\romannumeral 6}} problem
of metrizability is studied. Explicit formula for the force field of
metrizable dynamical systems admitting the normal shift is derived,
and the existence of non-metrizable systems is shown.
    {\bf In Chapter \uppercase\expandafter {\romannumeral 7}} we study
the complete system of normality equations on Riemannian manifolds,
and find explicit formula for their general solution in the dimension
$n\geqslant 3$.\par
    {\bf In Chapter \uppercase\expandafter {\romannumeral 8}} theory
of dynamical systems admitting the normal shift is generalized for
the case of Finslerian manifolds.
\head
\S\,14. Acknowledgments.
\endhead
\adjustfootnotemark{-1}
     First papers on the theory of dynamical systems admitting the
normal shift was written about 7 years ago. I am grateful to all
people who helped me over these years. I am grateful to academician
R\.~I\.~Nigmatullin for the opportunity to announce our first results
in DAN (Reports of Russian Academy of Sciences, see paper \cite{74}),
to professors I\.~Sh\.~Akhatov, L\.~A\.~Kalyakin, V\.~Yu\.~Novokshenov,
Yu\.~S\.~Shatalov, V\.~A\.~Baikov, and N\.~Kh\.~Ibragimov for the
opportunity to make reports in seminars at Bashkir State University,
Ufa State University of Aviation and Technology, Institute of
Mathematics of Ural branch of Russian Academy of Sciences and
Computer Science Department of Moscow State University. I am grateful
to academician V\.~S\.~Vladimirov for the opportunity to make report
in the seminar of his department at Steklov Mathematical Institute in
Moscow. I am grateful to academician A\.~T\.~Fomenko, to professors
A\.~V\.~Bolsinov and A\.~I\.~Shafarevich for the opportunity to make
report in a seminar of the Chair of Differential Geometry and Applications
at Moscow State University. I am grateful to professor A\.~B\.~Shabat
and academician A\.~T\.~Fomenko for the help in settling conflict in
publishing series of papers \cite{39}, \cite{58}, and \cite{59}
in TMF (Russian journal on Theoretical and Mathematical Physics).
I am grateful to professor A\.~P\.~Veselov and academician
O\.~A\.~Oleynik for the invitation to Petrovsky memorial conference
at Moscow State University in 1994 (see \cite{75}). I am grateful to
professors E\.~Bedford and S\.~I\.~Pinchuk for the invitation to
Indiana University, USA, and for the opportunity to make reports in
several seminars there. I am especially grateful to professor
S\.~I\.~Pinchuk for financial support of my travel to USA from his
ISF\footnote{International Science Foundation of George Soros.} grant,
and for kind hospitality in Bloomington during my visit. I am grateful
to professors K\.~Zumbrun, A\.~R\.~Its, and Pu-Tak-Khan for the
opportunity to make report in IUPUI (Indianapolis). I am grateful to my
friends Alexander and Lyudmila Vishnevsky for hospitality during my visits
to Moscow. I am grateful to the dean of Mathematical Department of Bashkir
State University professor Ya\.~T\.~Sultanaev for financial support from
his RFBR\footnote{Russian Fund for Basic Research.} grant. I am grateful
to my students A\.~Yu\.~Boldin, V\.~V\.~Dmitrieva, A\.~A\.~Bronnikov, and
to the stuff of the Chair of Algebra and Geometry of Bashkir State
University for patience and support when I was writing this thesis.\par
    I am grateful to  George  Soros  foundation  ``{Open  society 
Institute}'',
to European foundation INTAS, to Russian Foundation for Basic Research,
and to Academy of Sciences of the Republic Bashkortostan for financial
support.
\enddocument
\end

ediss-02.tex
\input amstex
\documentstyle{amsppt}
%
% Author:   Ruslan Abdulovich Sharipov
% Title:    Dynamical systems admitting the normal shift
% Comment:  Thesis for the degree of Doctor of Sciences in Russia
% Address:  Rabochaya street 5, Ufa, 450003, Russia
% E-mail:   R_Sharipov@ic.bashedu.ru
%           ruslan-sharipov@usa.net
% Web-page: http://www.geocities.com/CapeCanaveral/Lab/5341
%
\nopagenumbers
\catcode`@=11
\redefine\logo@{\relax}
\catcode`\@=\active
\def\startpage#1{\pageno=#1}
\pagewidth{12.8cm}
\pageheight{20cm}
\accentedsymbol\tx{\tilde x}
\def\msum{\operatornamewithlimits{\sum\!{\ssize\ldots}\!\sum}}
\Monograph
\topmatter
\title\chapter{2} Dynamical systems on Riemannian manifolds.
\endtitle
\endtopmatter
\loadbold
\startpage{38}
\document
\head
\S\,1. Newton's equations and the change of variables in them.
\endhead
\rightheadtext{\S\,1. Newton's equations \dots}
     Let's consider a system of $N$ rigid bodies in classical mechanics.
If their sizes are small in comparison with the distances between them,
then these bodies can be considered as mass-points, and their positions
can be given by their radius-vectors. In such model we have $N$
radius-vectors $\bold r_1,\,\ldots,\,\bold r_{\sssize N}$. They depend
on $t$, describing the motion of these mass-points in the space.
According to Newton's second law vector-functions $\bold r_1(t),\,\ldots,
\,\bold r_{\sssize N}(t)$ are determined as the solutions of the system
of $N$ vectorial ordinary differential equations of the second order:
$$
\align
m_1\cdot\ddot\bold r_1&=\bold F_1(\bold r_1,\,\ldots,\,\bold r_{\sssize N},
\dot\bold r_1,\,\ldots,\,\dot\bold r_{\sssize N},t),\hskip -2em\\
.\ .\ .\ .\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
.\ .\ .\ .\hskip -2em\tag1.1\\
m_{\sssize N}\cdot\ddot\bold r_{\sssize N}&=\bold F_{\sssize N}(\bold r_1,\,
\ldots,\,\bold r_{\sssize N},\dot\bold r_1,\,\ldots,\,\dot
\bold r_{\sssize N},t).\hskip -2em
\endalign
$$
Here $m_1,\,\ldots,\,m_{\sssize N}$ is masses of points and $\bold F_1,
\,\ldots, \,\bold F_{\sssize N}$ are forces acting on them. Let's choose
some system of cartesian coordinates (which are possibly scalene). Then
each of the vectors $\bold r_1,\,\ldots,\,\bold r_{\sssize N}$ can be
represented by its three coordinates, the whole system of vectors being
given by the set of numbers $x^1,\,\ldots,\,x^n$, where $n=3N$. Lets
divide force vectors $\bold F_1,\,\ldots,\,\bold F_{\sssize N}$ by the
masses of the points. Resulting set of acceleration-vectors we denote by
$\Phi^1,\,\ldots,\,\Phi^n$:
$$
\boldsymbol\Phi_1=\frac{\bold F_1}{m_1},\ \boldsymbol\Phi_2=
\frac{\bold F_2}{m_2},\ \ldots,\ \boldsymbol\Phi_{\sssize N}=
\frac{\bold F_{\sssize N}}{m_{\sssize N}}.
$$
Then we can rewrite the equations \thetag{1.1} as follows:
$$
\align
\ddot x^1&=\Phi^1(x^1,\ldots,x^n,\dot x^1,\ldots,\dot x^n,t),
\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
.\ .\ .\ \hskip -2em\tag1.2\\
\ddot x^n&=\Phi^n(x^1,\ldots,x^n,\dot x^1,\ldots,\dot x^n,t).
\hskip -2em
\endalign
$$
In this form of equations \thetag{1.1} we loose the individuality
of each mass-point, we need not know even their masses. Equations
\thetag{1.2} coincides with the equations describing the dynamics
of one hypothetical mass-point of unit mass $m=1$ in $n$-dimensional
space. Therefore we say that the equations \thetag{1.2} define a
{\bf Newtonian dynamical system} in $\Bbb R^n$. Space $\Bbb R^n$ in
this case is called {\bf configuration space} of dynamical system
\thetag{1.2}, while variables $x^1,\,\ldots,\,x^n$ are coordinates
in configuration space $\Bbb R^n$.\par
    Let's do linear change of variables in configuration space, i\. e.\
in instead of $x^1,\,\ldots,\,x^n$ we introduce new variables $\tx^1,\,
\ldots,\,\tx^n$ defined as follows:
$$
\tx^i=\sum^n_{j=1} T^i_j\,x^j\text{, \ \ where\ \ \ }i=1,\,\ldots,\,n.
\hskip -2em
\tag1.3
$$
Here $T^i_j$ are constants, they can be arranged into a non-degenerate
square matrix $T$. Non-degeneracy of $T$ means that change of variables
\thetag{1.3} is invertible, and 
$$
x^j=\sum^n_{q=1} S^j_q\,\tx^q\text{, \ \ where\ \ \ }j=1,\,\ldots,\,n.
\hskip -2em
\tag1.4
$$
By $S^j_q$ in \thetag{1.4} we denote components of inverse matrix
$S=T^{-1}$. Traditionally matrix $S$ is called the matrix of {\bf direct
transition} and $T$ is called the matrix of {\bf inverse transition}
(see \cite{12}, \cite{32}, \cite{76}, or \cite{77}).\par
    Relying on \thetag{1.3} and \thetag{1.4}, one can do the change of
variables in equations  \thetag{1.2}. Then we get the system of
equations of the same form as the initial system \thetag{1.2}:
$$
\align
\ddot{\tx}\vphantom{x}^1&=\tilde\Phi^1(\tx^1,\ldots,\tx^n,
\dot{\tx}\vphantom{x}^1,\ldots,\dot{\tx}\vphantom{x}^n,t),\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
.\ .\ .\ \hskip -2em\tag1.5\\
\ddot{\tx}\vphantom{x}^n&=\tilde\Phi^n(\tx^1,\ldots,\tx^n,
\dot{\tx}\vphantom{x}^1,\ldots,\dot{\tx}\vphantom{x}^n,
t).\hskip -2em
\endalign
$$
Right hand sides of the equations \thetag{1.2} and \thetag{1.5} are bound by 
the relationships
$$
\Phi^k=\sum^n_{m=1} S^k_m\,\tilde\Phi^m\hskip -2em
\tag1.6
$$
that correspond to transformation rule for the components of $n$-dimensional
vector by the change of base (see \cite{78}). Therefore vector with
components $\Phi^1,\,\ldots,\,\Phi^n$ is called the {\bf force vector} or
{\bf force field} of Newtonian dynamical system \thetag{1.2}.\par
    Kinetic energy is an important characteristics of the system of $N$
mass-points that is described by the equations \thetag{1.2}:
$$
G=\sum^N_{i=1}\frac{m_i\,|\dot\bold r_i|^2}{2}.\hskip -2em
\tag1.7
$$
This is scalar quantity defined by velocities of mass-points. When expressed
in coordinates $x^1,\,\ldots,\,x^n$ kinetic energy \thetag{1.7} can be
written as follows:
$$
G=\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1} g_{ij}\,\dot x^i\,\dot x^j.
\hskip -2em
\tag1.8
$$
Matrix $g$ in \thetag{1.8} is block-diagonal. It is composed of
of $N$ copies of Gram matrix of that base, which is used to
represent radius-vectors of mass-points. Change of variables
\thetag{1.3} breaks the block structure of the matrix $g$. So
we have
$$
G=\frac{1}{2}\sum^n_{p=1}\sum^n_{q=1} \tilde g_{pq}\,
\dot{\tx}\vphantom{x}^p\,\dot{\tx}\vphantom{x}^q,\hskip -2em
\tag1.9
$$
where $\tilde g$ is not block-diagonal. Matrix $\tilde g$ in
\thetag{1.9} is symmetric, its components are bound with the
components of matrix $g$ in \thetag{1.8} by the relationships
$$
g_{ij}=\sum^n_{p=1}\sum^n_{q=1} T^p_i\,T^q_j\,\tilde g_{pq}\hskip -3em
\tag1.10
$$
that correspond to transformation rule for the components of quadratic
form by the change of base (see \cite{78}). Quadratic form of
kinetic energy $G$ is positively definite, since masses of points in
\thetag{1.7} are positive constants.\par
    Now let's do nonlinear change of variables in the equations
\thetag{1.2}. This is the same as if we introduce curvilinear coordinates
in configuration space $\Bbb R^n$:
$$
\align
\tx^1&=\tx^1(x^1,\ldots,x^n),\hskip -3em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
\hskip -3em\tag1.11\\
\tx^n&=\tx^n(x^1,\ldots,x^n).\hskip -3em
\endalign
$$
Supposing the change of variables \thetag{1.11} to be invertible and
non-singular, we consider Jacoby matrix $T$ for this change of
variables:
$$
T^i_j=\frac{\partial\tx^i}{\partial x^j}.\hskip -3em
\tag1.12
$$
For non-singular change of variables matrix $T$ with components
\thetag{1.12} is non-degene\-rate. It is called {\bf inverse transition}
matrix. {\bf Direct transition} matrix $S=T^{-1}$ can be obtained as
Jacoby matrix for inverse change of variables:
$$
\align
x^1&=x^1(\tx^1,\ldots,\tx^n),\hskip -3em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
\hskip -3em\tag1.13\\
x^n&=x^n(\tx^1,\ldots,\tx^n).\hskip -3em
\endalign
$$
For the components of direct transition matrix $S=T^{-1}$ then we can
write
$$
S^i_j=\frac{\partial x^i}{\partial\tx^j}.\hskip -3em
\tag1.14
$$
Let's apply nonlinear change of variables \thetag{1.11} to the
equations \thetag{1.2}. Transformed equations in curvilinear
coordinates $\tx^1,\,\ldots,\,\tx^n$  have the same form as
\thetag{1.5}. However, \pagebreak relationships binding $\Phi^1,
\,\ldots,\,\Phi^n$ and $\tilde\Phi^1,\,\ldots,\,\tilde\Phi^n$ now
is different
from \thetag{1.6}. Instead of \thetag{1.6} here we have the following
relationships for $\Phi^1,\,\ldots,\,\Phi^n$:
$$
\Phi^k=\sum^n_{m=1} S^k_m\,\tilde\Phi^m-\sum^n_{m=1}\sum^n_{i=1}
\sum^n_{j=1} S^k_m\frac{\partial T^m_i}{\partial x^j}\,
\dot x^i\,\dot x^j.\hskip -3em
\tag1.15
$$
Let's compare the relationships \thetag{1.15} with the transformation
rules for the components of affine connection in differential geometry:
$$
\Gamma^k_{ij}=\sum^n_{m=1}\sum^n_{p=1}\sum^n_{q=1} S^k_m\,T^p_i
\,T^q_j\,\tilde\Gamma^m_{pq}+\sum^n_{m=1} S^k_m\,\frac{\partial
T^m_i}{\partial x^j}\hskip -3em
\tag1.16
$$
(see \cite{12}, \cite{32}, \cite{76}, or \cite{77}). We should take into
account that
$$
\xalignat 2
&\qquad\dot{\tx}\vphantom{x}^p=\sum^n_{i=1}T^p_i\,\dot x^i,
&&\dot x^i=\sum^n_{q=1}S^i_q\,\dot{\tx}\vphantom{x}^q.
\tag1.17
\endxalignat
$$
Let's multiply \thetag{1.16} y $\dot x^i\,\dot x^j$ and let's
sum up with respect to the indices $i$ and $j$. As a result from
\thetag{1.16} and \thetag{1.17} we derive
$$
\sum^n_{i=1}\sum^n_{j=1}\Gamma^k_{ij}\,\dot x^i\,\dot x^j=
\sum^n_{m=1}\sum^n_{p=1}\sum^n_{q=1}S^k_m\,\tilde\Gamma^m_{pq}\,
\dot{\tx}\vphantom{x}^p\,\dot{\tx}\vphantom{x}^q+\sum^n_{m=1}
\sum^n_{i=1}\sum^n_{j=1}S^k_m\frac{\partial T^m_i}{\partial
x^j}\,\dot x^i\,\dot x^j.
$$
Last terms in \thetag{1.15} and in the above relationships differ
only in sign. Hence, if we add these two relationships, last terms
in them will cancel each other:
$$
\Phi^k+\sum^n_{i=1}\sum^n_{j=1}\Gamma^k_{ij}\,\dot x^i\,\dot x^j=
\sum^n_{m=1}S^k_m\left(\tilde\Phi^m+\shave{\sum^n_{p=1}\sum^n_{q=1}}
\tilde\Gamma^m_{pq}\,\dot{\tx}\vphantom{x}^p\,\dot{\tx}\vphantom{x}^q
\right).
$$
Denote by $F^k$ left hand side of the above relationship:
$$
F^k=\Phi^k+\sum^n_{i=1}\sum^n_{j=1}\Gamma^k_{ij}\,\dot x^i\,\dot x^j.
\hskip -3em
\tag1.18
$$
The relationships similar to \thetag{1.18} can be written in curvilinear
coordinates too. Here we define the quantities $\tilde F^q$. Then
$$
F^k=\sum^n_{q=1}S^k_q\,\tilde F^q.\hskip -3em
\tag1.19
$$
The relationships \thetag{1.19} coincide with transformation rule for the
components of some vector $\bold F$ by the change of base.\par
    Quantities $F^1,\,\ldots,\,F^n$ introduced in \thetag{1.18} are used
to simplify the relationships \thetag{1.15}, which are now written as
\thetag{1.19}. But to define them we need an affine connection. Where
could we get it from\,? The answer to this question is found in quadratic
form of {\bf kinetic energy} \thetag{1.7}. It is positively defined,
therefore it defines Riemannian metric $\bold g$ in configuration space
$\Bbb R^n$ of dynamical system \thetag{1.2}. Riemannian metric is
always associated with metric connection of Levi-Civita. Its components
are the following:
$$
\Gamma^k_{ij}=\frac{1}{2}\sum^n_{s=1}g^{ks}\left(\frac{\partial g_{sj}}
{\partial x^i}+\frac{\partial g_{is}}{\partial x^j}-\frac{\partial g_{ij}}
{\partial x^s}\right)\hskip -3em
\tag1.20
$$
(see more details in \cite{12}, \cite{32}, \cite{76}, or \cite{77}). In
Cartesian coordinates components of matrix $g$ are constants. Therefore
components of metric connection \thetag{1.20} in Cartesian coordinates are
identically zero. Hence formula \thetag{1.18} reduces to $F^k=\Phi^k$.\par
    In curvilinear coordinates components of metric connection \thetag{1.20}
are nonzero, and formula \thetag{1.18} here is non-trivial. The equations
of dynamical system \thetag{1.2} in curvilinear coordinates are written
as follows:
$$
\align
\ddot x^1&+\sum^n_{i=1}\sum^n_{j=1}\Gamma^1_{ij}\,\dot x^i\,\dot x^j=
F^1,\hskip -3em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
\hskip -3em\tag1.21\\
\ddot x^n&+\sum^n_{i=1}\sum^n_{j=1}\Gamma^n_{ij}\,\dot x^i\,\dot x^j=
F^n.\hskip -3em
\endalign
$$
Quantities $F^1,\,\ldots,\,F^n$ in \thetag{1.21} are components of some
$n$-dimensional vector $\bold F$, which is called the {\bf force vector}.
time derivatives $\dot x^1,\,\ldots,\,\dot x^n$ in \thetag{1.21} are
also the components of some $n$-dimensional vector $\bold v$, which is
called the {\bf velocity vector}. For the force vector $\bold F$ in
\thetag{1.21} we have
$$
\bold F=\bold F(x^1,\ldots,x^n,\bold v,t).\hskip -3em
\tag1.22
$$
In other words, this means that force vector $\bold F$ can depend on the
point of configuration space (i\.~e\. on coordinates $x^1,\,\ldots,
\,x^n$), it can depend on velocity vector $\bold v$, and it can
contain explicit dependence on time variable $t$.
\head
\S\,2. Dynamical systems with constraints.
\endhead
    Writing the equations of the system of $N$ bodies in form of
\thetag{1.21}, we unify these equations and introduce the structure
of Riemannian manifold in configuration space of the system. But
geometry of this manifold is trivial, since Euclidean metric
\thetag{1.8} generates metric connection \thetag{1.20} of zero
curvature. Much less trivial situation arises in considering
mechanical system with constraints. These constraints can be hard
(like welding joint) or flexible (like cardan joint, thumbscrew, tooth
gearing, hinge and so on). System of $N$ mass-points has $3N$ degrees
of freedom. Constraints imposed on this system diminish the number
of degrees of freedom.
\definition{Definition 2.1} Constraint imposed on the system of $N$
mass-points is called {\bf holonomic} constraint, if it is described
by one or several equations of the form $f(\bold r_1,\ldots,\bold
r_{\sssize N})=0$, where $\bold r_1,\,\ldots,\,\bold r_{\sssize N}$
are radius-vectors of mass-points.
\enddefinition
    Let's consider the system of $N$ mass-points with holonomic
constraints. Denote by $\bold r$ the radius-vector of a point in
configuration space $\Bbb R^n$ of the system of free mass-points
($n=3N$). Its components are $x^1,\,\ldots,\,x^n$. According to
the definition~2.1 constraints are given by the equations
$$
\align
&f_1(x^1,\ldots,x^n)=0,\hskip -2em\\
&.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
\hskip -2em\tag2.1\\
&f_{\sssize K}(x^1,\ldots,x^n)=0.\hskip -2em
\endalign
$$
In general case the equations \thetag{2.1} define some submanifold
$M$ of the dimension $m=3N-K$ in $\Bbb R^n$. It is called the
{\bf configuration space} of the system with constraints.\par
    Let $y^1,\,\ldots,\,y^m$ be local coordinates in some map on
of the manifold $M$. Suppose that embedding of $M$ into $\Bbb R^n$
is described by vector-function $\bold r=\bold R(y^1,\ldots,y^m)$:
$$
\align
x^1&=R^1(y^1,\ldots,y^m),\hskip -2em\\
.\ .&\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
\hskip -2em\tag2.2\\
x^n&=R^n(y^1,\ldots,y^m),\hskip -2em
\endalign
$$
Function \thetag{2.2} are the coordinate form of the vector-function
$\bold r=\bold R(y^1,\ldots,y^m)$. If we substitute them into \thetag{2.1},
these equations would be be identically fulfilled.\par
    In order to calculate the $n$-dimensional vector of velocity for
the system in $\Bbb R^n$ let's differentiate the function
$\bold r=\bold R(y^1,\ldots,y^m)$ in $t$:
$$
\bold v=\frac{d\bold r}{dt}=\sum^m_{i=1}\frac{\partial\bold R}
{\partial y^i}\,\dot y^i=\sum^m_{i=1}\bold E_i\,\dot y^i.\hskip -2em
\tag2.3
$$
Partial derivatives 
$$
\bold E_i(y^1,\ldots,y^m)=\frac{\partial\bold R}{\partial y^i}
\hskip -2em
\tag2.4
$$
that arise in \thetag{2.3} form the base in tangent space to $M$
embedded into $\Bbb R^n$.\par
     Quadratic form of kinetic energy \thetag{1.7} defines scalar
product $(\bold X\,|\,\bold Y)=2\,G(\bold X,\bold Y)$ in $\Bbb R^n$
and it induces Riemannian metric in submanifold $M$. Components
of metric tensor $\bold g$ in local coordinates $y^1,\,\ldots,\,y^m$
on $M$ are calculated by formula
$$
g_{ij}=(\bold E_i\,|\,\bold E_j).\hskip -2em
\tag2.5
$$
Let's differentiate vector $\bold E_i$ from \thetag{2.4} with respect
to $y_j$ \pagebreak and expand the derivative into a sum of two components,
one of them being in tangent space to $M$, and second being perpendicular
to the manifold $M$:
$$
\frac{\partial\bold E_i}{\partial y^j}=
\sum^m_{k=1}\Gamma^k_{ij}\,\bold E_k+\bold N_{ij}.\hskip -2em
\tag2.6
$$
In \thetag{2.6} tangent component is expanded in a base of vectors
$\bold E_1,\,\ldots,\,\bold E_m$, while perpendicular component is
denoted by $\bold N_{ij}$. Formulas like \thetag{2.6} are known as
{\bf derivational formulas of Weingarten} (see \cite{32}, \cite{76},
or \cite{77}). Coefficients in right hand side of such formulas
coincide with components of  metric connection for induced Riemannian
metric \thetag{2.5} on $M$:
$$
\Gamma^k_{ij}=\frac{1}{2}\sum^m_{s=1}g^{ks}\left(\frac{\partial g_{sj}}
{\partial x^i}+\frac{\partial g_{is}}{\partial x^j}-\frac{\partial g_{ij}}
{\partial x^s}\right).\hskip -2em
\tag2.7
$$\par
    Now let's calculate time derivative of velocity vector $\bold v$,
taking its components $\dot x^1,\,\ldots,\,\dot x^n$ from \thetag{2.3}.
Then let's substitute $\ddot x^1,\,\ldots,\,\ddot x^n$ into the equations
\thetag{1.2}:
$$
\bold a=\frac{d\bold v}{dt}=\sum^m_{i=1}\bold E_i\,\ddot y^i+
\sum^m_{i=1}\sum^m_{j=1}\frac{\partial\bold E_i}{\partial y^j}
\,\dot y^i\,\dot y^j.\hskip -2em
\tag2.8
$$
Taking into account derivational formulas \thetag{2.6}, we can
bring \thetag{2.8} to the form
$$
\bold a=\sum^m_{k=1}\left(\ddot y^k+\shave{\sum^m_{i=1}\sum^m_{j=1}}
\Gamma^k_{ij}\,\dot y^i\,\dot y^j\right)\bold E_k+\sum^m_{i=1}
\sum^m_{j=1}\bold N_{ij}\,\dot y^i\,\dot y^j.\hskip -2em
\tag2.9
$$
Vector of $n$-dimensional acceleration $\bold a$ in \thetag{2.9} is
expanded into a sum of two components: one is tangent to the manifold
$M$, second is perpendicular to it. Perpendicular component of $\bold a$
given by the formula
$$
\bold a_{\sssize\perp}=\sum^m_{i=1}\sum^m_{j=1}\bold N_{ij}\,
\dot y^i\,\dot y^j\hskip -3em
\tag2.10
$$
is called {\bf centripetal acceleration}, and tangent component
$\bold a_{\sssize\parallel}$ is called {\bf tangent acceleration}.
Force vector $\boldsymbol\Phi$ with components $\Phi^1,\,\ldots,\,
\Phi^n$ from \thetag{1.2} can also be represented as a sum of
tangent and perpendicular components: $\boldsymbol\Phi=\boldsymbol
\Phi_{\sssize\parallel}+\boldsymbol\Phi_{\sssize\perp}$. Tangent
component in turn can be expanded in the base of vectors 
$\bold E_1,\,\ldots,\,\bold E_m$ in tangent space to the manifold
$M$:
$$
\boldsymbol\Phi_{\sssize\parallel}=\sum^m_{k=1} F^k\,\bold E_k.
\hskip -3em
\tag2.11
$$
Taking into account \thetag{2.9}, \thetag{2.10}, and \thetag{2.11},
we can rewrite the equations of dynamics \thetag{1.2} in form of two
systems of equations. First arises from $\bold a_{\sssize
\parallel}=\boldsymbol\Phi_{\sssize\parallel}$:
$$
\qquad\ddot y^k+\sum^m_{i=1}\sum^m_{j=1}\Gamma^k_{ij}\,\dot y^i\,
\dot y^j=F^k\text{, \ \ where\ \ \ }k=1,\,\ldots,\,m.
\hskip -3em
\tag2.12
$$
This is the system of $m$ scalar differential equations of the second
order with respect to local coordinates $y^1,\,\ldots,\,y^m$. Second
system is written as an equality 
$$
\sum^m_{i=1}\sum^m_{j=1}\bold N_{ij}\,\dot y^i\,\dot y^j=
\boldsymbol\Phi_{\sssize\perp}\hskip -3em
\tag2.13
$$
of two vectors, which are perpendicular to $M$. Therefore it is
equivalent to $n-m$ scalar equations.\par
    The equations \thetag{2.12} play the main role in description of
dynamics for mechanical systems with constraints. As for the equations
\thetag{2.13}, they are often omitted at all. The matter is that in right
hand sides of initial Newton's equations \thetag{1.1}, apart from forces
defined by the nature of interacting mass-points, we have {\bf constraint
forces}. In most these are maintaining forces that provide the equations
\thetag{2.1} to be fulfilled. For instance, if we have two bodies
bound by a hard rod, then in dynamics there appears the force that
prevents tension and contraction of this rod. Let's denote by $\bold Q$ the
vector of constraint forces, and let's expand it into a sum of two
components: $\bold Q=\bold Q_{\sssize\parallel}+\bold Q_{\sssize\perp}$.
Then the equations \thetag{2.13} are written as follows:
$$
-\bold Q_{\sssize\perp}=\boldsymbol\Phi_{\sssize\perp}-\sum^m_{i=1}
\sum^m_{j=1}\bold N_{ij}\,\dot y^i\,\dot y^j.\hskip -3em
\tag2.14
$$
Force $Q_{\sssize\perp}$ is purely maintaining component of constraint force
$\bold Q$. And \thetag{2.14} is the very equation that defines the value of
this component. Vector 
$$
\bold N=-\sum^m_{i=1}\sum^m_{j=1}\bold N_{ij}\,\dot y^i\,\dot y^j,
$$
differs from \thetag{2.10} only by sign, it is called {\bf centrifugal
force}. One shouldn't be surprised that in our notations forces and
accelerations are measured by the same units. This is because we reduced
initial Newton's equations \thetag{1.1} to the form \thetag{1.2} that
corresponds to $n$-dimensional dynamics of a point with mass $m=1$.\par
     Force $\bold Q_{\sssize\parallel}$ is usually a dissipative
component of constraint force $\bold Q$. It is due to the friction in
joints (in cardan joints, thumbscrews, tooth gearings, hinges and so on).
It is defined by the load of joints (i\.~e\. by the force$Q_{\sssize
\perp}$) and by relative velocities of various parts of joints. Therefore
we can write 
$$
\bold Q_{\sssize\parallel}=\bold K(y^1,\ldots,y^n,\dot y^1,\ldots,\dot y^n,
\bold Q_{\sssize\perp}).\hskip -3em
\tag2.15
$$
Explicit expression for the function $\bold K$ in \thetag{2.15} depends
on particular structure of joints and materials they are made of.\par
    So, in order to describe mechanical system with constraints we should
find vector $\bold Q_{\sssize\perp}$ by means of the equation \thetag{2.14},
then we should substitute it into \thetag{2.15} and calculate total
vector of constraint force $\bold Q=\bold Q_{\sssize\parallel}+
\bold Q_{\sssize\perp}$. It appears to be depending on local coordinates
$y^1,\,\ldots,\,y^m$ and their time derivatives $\dot y^1,\,\ldots,\,
\dot y^m$.\par
     Let's expand vector $\bold Q_{\sssize\parallel}$ in the base of
vectors $\bold E_1,\,\ldots,\,\bold E_m$ in tangent space to the manifold
$M$ embedded into the space $\Bbb R^n$:
$$
\bold Q_{\sssize\parallel}=\sum^m_{k=1} Q^k\,\bold E_k.
\hskip -3em
\tag2.16
$$
Taking into account constraint force in the form of \thetag{2.16},
we can write the equations of dynamics \thetag{2.12} as 
$$
\quad\ddot y^k+\sum^m_{i=1}\sum^m_{j=1}\Gamma^k_{ij}\,\dot y^i\,
\dot y^j=F^k+Q^k\text{, \ \ where\ \ \ }k=1,\,\ldots,\,m.
\tag2.17
$$
Here $F^k$ are the components of force vector other than constraint
force. Force components $F^k$ and $Q^k$ in right hand sides of the
equations \thetag{2.17} differ only by their physical origin. Adding
$Q^k$ to $F^k$, we do not change the structure of differential equations
in whole, since
$$
\aligned
F^k&=F^k(y^1,\ldots,y^n,\dot y^1,\ldots,\dot y^n,t),\\
\vspace{1ex}
Q^k&=Q^k(y^1,\ldots,y^n,\dot y^1,\ldots,\dot y^n).
\endaligned\hskip -3em
\tag2.18
$$
Due to \thetag{2.18} we can redesignate $Q^k+F^k$ by $F^k$, and then
we come back to the equations \thetag{2.12}. However, the study of
constraint forces, as described above, is important from conceptual point
of view. It explains how the system of $n$ equations \thetag{1.2}
is reduced to the system of $m$ equations \thetag{2.12}.
\head
\S\,3. Newtonian dynamical systems in Riemannian manifolds.
\endhead
    As we see in \S\,1 and \S\,2, each mechanical system with $n$ degrees
of freedom is associated with some $n$-dimensional Riemannian manifold
$M$, which is called {\bf configuration space} of the system. This takes
place for the system of free mass-points and for the system with holonomic
constraints as well. Let $x^1,\,\ldots,\,x^n$ be local coordinates in
some map on the manifold $M$. Kinetic energy of mechanical system is defined
by metric tensor $\bold g$ according to the formula
$$
G=\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1} g_{ij}\,\dot x^i\,\dot x^j,
\hskip -2em
\tag3.1
$$
its dynamics is defined by Newton's equations written as 
$$
\qquad\ddot x^k+\sum^n_{i=1}\sum^n_{j=1}\Gamma^k_{ij}\,\dot x^i\,
\dot x^j=F^k\text{, \ \ where\ \ \ }k=1,\,\ldots,\,n.
\hskip -2em
\tag3.2
$$
Here $\Gamma^k_{ij}$ are the components of metric connection, which
can be calculated by standard formula \thetag{1.20}.\par
     Each solution of the equations \thetag{3.2} is a set of $n$
functions $x^1(t),\,\ldots,\,x^n(t)$ that can be interpreted as
parametric equations o some curve in $M$. This curve is called
the {\bf trajectory}. Let's recalculate the equations trajectory
to local coordinates in some other map on $M$ by means of change of
variables \thetag{1.11}. Resulting set of functions $\tx^1(t),\,\ldots,
\,\tx^n(t)$ forms the solution of the equations analogous to
\thetag{3.2}:
$$
\qquad\ddot{\tx}\vphantom{x}^k+\sum^n_{i=1}\sum^n_{j=1}
\tilde\Gamma^k_{ij}\,\dot{\tx}\vphantom{x}^i\,\dot{\tx}\vphantom{x}^j
=\tilde F^k\text{, \ \ where\ \ \ }k=1,\,\ldots,\,n.\hskip -2em
\tag3.3
$$
\definition{Definition 3.1} Systems of equations \thetag{3.2} and
\thetag{3.3} related to the pair of overlapping maps $(U,x^1,\ldots,
x^n)$ and $(\tilde U,\tx^1,\ldots,\tx^n)$ on $M$ are called
{\bf concordant systems}, if in the domain of intersection of maps
right hand sides of these equations are bound by the relationships
\thetag{1.19}.
\enddefinition
\definition{Definition 3.2} Newtonian dynamical system on Riemannian
manifold $M$ is a set of concordant systems of differential equations
of the form \thetag{3.2} associated to each map on this manifold.
\enddefinition
    Simplest example of Newtonian dynamical system is a {\bf geodesic
flow}, when right hand sides of the equations \thetag{3.2} for all maps
are identically zero. Trajectories of geodesic flow are geodesic lines
on $M$.\par
     Let's study the time dependence of kinetic energy \thetag{3.1}
on the trajectory of some dynamical system  \thetag{3.2}:
$$
G(t_2)-G(t_1)=\int\limits^{\ t_2}_{t_1}\frac{dG}{dt}\,dt.\hskip -2em
\tag3.4
$$
Time derivative under the sign of integral in \thetag{3.4} can be
evaluated by direct calculation. Taking into account the symmetry
of components of metric $\bold g$, we get:
$$
\gather
\frac{dG}{dt}=\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1}\left(g_{ij}\,\ddot x^i
\,\dot x^j+g_{ij}\,\dot x^i\,\ddot x^j+\shave{\sum^n_{k=1}}\frac{\partial
g_{ij}}{\partial x^k}\,\dot x^i\,\dot x^j\,\dot x^k\right)=\\
\vspace{1ex}
=\sum^n_{i=1}\sum^n_{j=1}g_{ij}\,F^i\,\dot x^j+\frac{1}{2}\sum^n_{i=1}
\sum^n_{j=1}\sum^n_{k=1}\nabla_kg_{ij}\,\dot x^i\,\dot x^j\,\dot x^k.
\endgather
$$
Second term in the obtained expression vanishes due to vanishing of
covariant derivative $\nabla_kg_{ij}$. The latter fact is the consequence
of concordance of metric and connection (see \cite{12}, \cite{32},
\cite{76}, or \cite{77}). Therefore we have
$$
\frac{dG}{dt}=\sum^n_{i=1}\sum^n_{j=1}g_{ij}\,F^i\,\dot x^j.\hskip -2em
\tag3.5
$$
Right hand side of \thetag{3.5} is a scalar product of two tangent
vectors on $M$: force vector $\bold F$ and velocity vector $\bold v$.
\pagebreak In inner geometry of the manifold $M$ they are represented
by two first order linear differential operators:
$$
\xalignat 2
&\quad\bold F=\sum^n_{i=1}F^i\,\frac{\partial}{\partial x^i},
&&\bold v=\sum^n_{i=1}\dot x^i\,\frac{\partial}{\partial x^i}.
\hskip -1em
\tag3.6
\endxalignat
$$
If $M$ is embedded into some Euclidean space (as it usually happens
for mechanical systems with constraints), then vectors \thetag{3.6}
in exterior geometry are represented by \thetag{2.11} and \thetag{2.3}.
Regardless to the form of representation the vectors $\bold F$ and
$\bold v$ the relationship \thetag{3.4} can be written as 
$$
G(t_2)-G(t_1)=\int\limits^{\ t_2}_{t_1}(\bold F\,|\,\bold v)\,dt.
\hskip -2em
\tag3.7
$$
Integral in right hand side of \thetag{3.7} is called the {work}
of the force $\bold F$ on the trajectory of dynamical system.\par
    Special subclass of dynamical systems \thetag{3.2} is formed by
system for which the value  of integral in \thetag{3.7} depends only
on initial an final points of trajectory. Force $\bold F$ for such
systems is determined by gradient of some scalar function $\varPi$
on $M$:
$$
F^i=-\sum^n_{k=1}g^{ik}\,\nabla_k\varPi=-\sum^n_{k=1}g^{ik}\,
\frac{\partial\varPi}{\partial x^k}.\hskip -2em
\tag3.8
$$
Function $\varPi=\varPi(x^1,\ldots,x^n)$ in \thetag{3.8} is called
{\bf potential energy} of the system, and sum $E=G+\varPi$ is called
{\bf total energy} of the system. From \thetag{3.7} and \thetag{3.8}
we can easily derive {\bf conservation law} for total energy of
the system. Dynamical systems with potential energy obeying the 
conservation law for their total energy are called {\bf conservative}
systems.\par
    For conservative mechanical systems \thetag{3.2} one can define the
Lagrange function $L=G-\varPi$. The equations \thetag{3.2} of dynamical
system in this case are written in form of Euler-Lagrange equations:
$$
\frac{d}{dt}\left(\frac{\partial L}{\partial\dot x^k}\right)-
\frac{\partial L}{\partial x^k}=0\text{, \ \ where\ \ \ }k=1,\,\ldots,
\,n.\hskip -2em
\tag3.9
$$
We shall not come deep into the Lagrangian mechanics any more
(see books \cite{60} or \cite{79}). Instead of this we shall formulate
the main conclusion from all what was said in \S\S\,1--3: {\it Newtonian
dynamical systems on Riemannian manifolds appears in a quite natural
way in the study of real mechanical systems}. {\it Corresponding
manifolds for such systems are obtained as their configuration spaces,
while metric are defined by kinetic energy of these systems}.
\head
\S\,4. Tangent bundle and the force field of Newtonian
dynamical system.
\endhead
\rightheadtext{\S\,4. Tangent bundle and the force field \dots}
    Having solved the problem of the origin of Newtonian dynamical
systems on Riemannian manifolds, let's proceed with the description
of mathematical structures related to such systems. Suppose that
some particular Newtonian dynamical system \thetag{3.2} is fixed.
Denote by $v^1,\,\ldots,\,v^n$ components of velocity vector $\bold
v$ in \thetag{3.6}. Then by means of $v^1,\,\ldots,\,v^n$ we can
rewrite the equations \thetag{3.2} as a system of $2n$ differential
equations of the first order:
$$
\xalignat 2
&\qquad\dot x^k=v^k,
&&\dot v^k+\sum^n_{i=1}\sum^n_{j=1}\Gamma^k_{ij}\,v^i\,v^j=F^k.
\hskip -1em
\tag4.1
\endxalignat
$$
Here $k$ runs from $1$ to $n$. The equations \thetag{4.1} correspond
to the dynamical system on tangent bundle $TM$ of the manifold $M$.
Let's recall that {\bf tangent bundle $TM$} is a set of pairs
$(p,\bold v)$, where $p$ is a point of $M$, and $\bold v$ is a vector
from tangent space $V=T_p(M)$ of manifold $M$ at the point $p$. Denote
by $\pi: TM\to M$ canonical projection that maps the pair $(p,\bold v)$
onto the point $p$. Structure of manifold in tangent bundle $TM$ is
defined by maps
$$
(\pi^{-1}(U),x^1,\ldots,x^n,v^1,\ldots,v^n),\hskip -2em
\tag4.2
$$
where $(U,x^1,\ldots,x^n)$ is a local map on base manifold $M$.
In other words, $x^1,\,\ldots,\,x^n$ are local coordinates of the
point $p$ from pair $(p,\bold v)$ in the map $(U,x^1,\ldots,x^n)$,
and $v^1,\,\ldots,\,v^n$ are components of the vector $\bold v$
in the expansion
$$
\bold v=\sum^n_{i=1}v^i\,\frac{\partial}{\partial x^i}.\hskip -2em
\tag4.3
$$
Suppose that two maps $(U,x^1,\ldots,x^n)$ and $(\tilde U,\tx^1,\ldots,
\tx^n)$ are overlapping, i\.~e\. $U\cap\tilde U\neq\varnothing$, and
suppose that transition functions are given:
$$
\xalignat 2
\quad\tx^1&=\tx^1(x^1,\ldots,x^n),&x^1&=x^1(\tx^1,\ldots,\tx^n),
\hskip -3em\\
\quad .\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ & .\ .\ &.\ .\
.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ \hskip -3em\tag4.4\\
\quad\tx^n&=\tx^n(x^1,\ldots,x^n),&x^n&=x^n(\tx^1,\ldots,\tx^n).
\hskip -3em
\endxalignat
$$
Then corresponding maps $\pi^{-1}(U)$ and $\pi^{-1}(\tilde U)$
of the form \thetag{4.2} on $TM$ are also overlapping. Transition
functions for such maps are given by formulas
$$
\xalignat 2
&\quad\left\{\aligned &\tx^k=\tx^k(x^1,\ldots,x^n),\\
\vspace{1ex}&\tilde v^k=\sum^n_{i=1}T^k_i\,v^i,\endaligned\right.
&&\left\{\aligned &x^k=x^k(\tx^1,\ldots,\tx^n),\\
\vspace{1ex}&v^k=\sum^n_{i=1}S^k_i\,\tilde v^i,\endaligned\right.
\hskip -3em
\tag4.5
\endxalignat
$$
where $k$ runs from $1$ to $n$. Transition matrices $S$ and $T=S^{-1}$
in \thetag{4.5} are composed by partial derivatives of transition
functions from \thetag{4.4}:
$$
\xalignat 2
&S^k_i=\frac{\partial x^k}{\partial\tx^i},
&&T^k_i=\frac{\partial\tx^k}{\partial x^i}.
\hskip -3em
\tag4.6
\endxalignat
$$
Tangent bundle $TM$ corresponds to physical concept of {\bf phase space}. 
Its dimension is twice as great than the dimension of {\bf configuration
space}.\par
    Now let's consider again the equations \thetag{4.1}. According to
the results of \S\S\,1--3 the quantities $F^1,\,\ldots,\,F^n$ in
\thetag{4.1} are the components of force vector $\bold F$ tangent to
$M$ at the point $p$ with local coordinates $x^1,\ldots,x^n$. This
vector depend on point $p$, on velocity vector $\bold v$, and it can
depend on $t$ explicitly:
$$
\bold F=\bold F(p,\bold v,t).\hskip -2em
\tag4.7
$$
\definition{Definition 4.1} Newtonian dynamical system \thetag{4.1}
is called the {\bf stationary system}, if its force field doesn't
depend on time variable explicitly.
\enddefinition
    Further we shall consider only stationary Newtonian dynamical
systems. Here instead of the formula \thetag{4.7} we have
$$
\bold F=\bold F(p,\bold v).\hskip -2em
\tag4.8
$$
Vectors depending on the points of manifold usually correspond to the
concept of vector fields. In order to be more precise let's remember
standard definition of vector field on the manifold.
\definition{Definition 4.2} Vector field $\bold F$ on the manifold
$M$ is a vector-valued function that to each point $p$ of $M$ puts
into correspondence some vector from tangent space $T_p(M)$ at this
point $p$.
\enddefinition
     Now its clear that definition~4.2 describes only some special
functions \thetag{4.8} when force doesn't depend on the velocity:
$\bold F=\bold F(p)$. In general case argument of the function
\thetag{4.8} is a point of tangent bundle $TM$. However these
functions cannot be treated as vector fields on $TM$ too. Their
values are tangent vectors to $M$, not to $TM$. Therefore we need
to formulate special definition to describe force fields of
Newtonian dynamical systems.
\definition{Definition 4.3} {\bf Extended} vector field $\bold F$
on the manifold $M$ is a vector-valued function that to each point
$q=(p,\bold v)$ of tangent bundle $TM$ puts into correspondence
some vector from tangent space $T_p(M)$ at the point $p$ of base
manifold $M$ being the image of $q$ by canonical projection:
$p=\pi(q)$.
\enddefinition
\parshape 12
6.0cm 6.8cm 6.0cm 6.8cm 6.0cm 6.8cm 6.0cm 6.8cm
6.0cm 6.8cm 6.0cm 6.8cm 6.0cm 6.8cm 6.0cm 6.8cm
6.0cm 6.8cm 6.0cm 6.8cm 6.0cm 6.8cm 0cm 12.8cm
\noindent\quad\! Schematically the concept of extended vector field is shown
on Fig\.~4.1. \vadjust{\vskip -0.4cm\special{em:graph edis-02a.gif}
\vskip 0.4cm}Let $q=(p,\bold v)$ be the point of tangent bundle $TM$.
By definition it is the pair including the point $p=\pi(q)$ and the
vector $\bold v$ tangent to $M$ at the point $p$. Function that maps
$q=(p,\bold v)$ into the vector $\bold v$ is an example of extended
vector field on $M$. It is called the {\bf ve\-lo\-ci\-ty field} on $M$.
Velocity field is defined on any manifold regardless to the existence
of Riemannian metric \pagebreak or Newtonian dynamical system on it.
As for the Riemannian manifolds, here stationary Newtonian dynamical
systems \thetag{4.1} are in one-to-one correspondence with extended
vector fields, i\.~e\. in order to define such dynamical system we
should only define its force field $\bold F$.
\head
\S\,5. Extended algebra of tensor fields.
\endhead
     Definition~4.3 extends the concept of vector field. Similarly we
can extend the concept of scalar field, of covectorial field, and of
arbitrary tensor field as well.
\definition{Definition 5.1} {\bf Extended} scalar field $\varphi$ on
the manifold $M$ is a scalar function on its tangent bundle $TM$.
\enddefinition
\definition{Definition 5.2} {\bf Extended} covectorial field $\bold F$
on the manifold $M$ is a covector-valued function that to each point
$q=(p,\bold v)$ of tangent bundle $TM$ puts into correspondence
some vector from cotangent space $T^*_p(M)$ at the point $p$ of base
manifold $M$ being the image of $q$ by canonical projection: $p=\pi(q)$.
\enddefinition
    In order to define extended tensor field first we consider the
following tensor products of $r$ copies of tangent space and $s$ copies
of cotangent space:
$$
T^r_s(p,M)=\overbrace{T_p(M)\otimes\ldots\otimes T_p(M)}^{\text{$r$
times}}\otimes\underbrace{T^*_p(M)\otimes\ldots\otimes T^*_p(M)}_{\text{$s$
times}}
$$
Tensor product $T^r_s(p,M)$ is known as a space of $(r,s)$-tensors at the
point $p$ of the manifold $M$. Pair of integer numbers $(r,s)$ determines
the type of tensors. Elements of the space $T^r_s(p,M)$ are called
{\it $r$-times contravariant and $s$-times covariant tensors}, or tensors
of the type $(r,s)$, or for brevity $(r,s)$-tensors.
\definition{Definition 5.3} {\bf Extended} tensor field $\bold X$
of the type $(r,s)$ on the manifold $M$ is a tensor-valued function
that to each point $q=(p,\bold v)$ of tangent bundle $TM$ puts into
correspondence some tensor from tensor space $T^r_s(p,M)$ at the point
$p$ on $M$.
\enddefinition
   Tensor field of the type $(1,0)$ is a vector field, tensor field of
the type $(0,1)$ is a covector field, and tensor field of the type $(0,0)$
is a scalar field. This is the consequence of the following relationships:
$$
\xalignat 3
&T^1_0(p,M)=T_p(M),&&T^0_1(p,M)=T^*_p(M),&&T^0_0(p,M)=\Bbb R.
\endxalignat
$$
Ordinary tensor fields form a subset in the set of extended tensor fields.
Most of properties of ordinary tensor fields are available for extended
fields too. First of all we consider algebraic properties. In the set of
tensors we have summation, multiplication by real numbers, and the
operation of tensor product:
\roster
\item \quad $T^r_s(p,M)+T^r_s(p,M)\longrightarrow T^r_s(p,M)$;
\item \quad $\Bbb R\cdot T^r_s(p,M)\longrightarrow T^r_s(p,M)$;
\item \quad $T^r_s(p,M)\otimes T^a_c(p,M)\longrightarrow T^{r+a}_{s
+c}(p,M)$.
\endroster
Due to the equality $T^0_0(p,M)=\Bbb R$ second operation is a special case
of the third one. Now let's consider the sum
$$
\bold T(p,M)=\bigoplus^\infty_{r=0}\bigoplus^\infty_{s=0}T^r_s(p,M).
\hskip -2em
\tag5.1
$$
Operations \therosteritem{1}--\therosteritem{3} equip the set of tensors
\thetag{5.1} with the structure of algebra over the field of real numbers.
For extended tensor fields these operations are implemented pointwise in
each point $p=\pi(q)$. Suppose that $T^r_s(M)$ is a set of extended tensor
fields of the type $(r,s)$. It is equipped with the structure of module
over the ring of extended scalar fields, while the following sum
$$
\bold T(M)=\bigoplus^\infty_{r=0}\bigoplus^\infty_{s=0}T^r_s(M)
\hskip -2em
\tag5.2
$$
is a graded algebra over that ring. Algebra \thetag{5.2} in what follows
will be called the {\bf extended algebra of tensor fields} on the manifold
$M$.\par
     Extension of the concept of tensor fields in the sense of
definition~5.3 isn't of common use in geometry. Probably it was first
used in Finslerian geometry (see \cite{61}), and here it is
presently in use (see \cite{69--73}). Concept of extended tensor field
can have various generalizations, one of them is considered in
\cite{40}. The concept of {\bf jets} is close relative to the concept
extended tensor fields, it is much more general and more popular now.
\head
\S\,6. Extended tensor fields in local coordinates.
\endhead
    Let's choose some map $(U,x^1,\ldots,x^n)$ on the manifold $M$.
In definition~5.3, apart from $M$, we have another manifold, it's
tangent bundle $TM$. Therefore to describe extended tensor field
$\bold X\in T^r_s(M)$ we might have to choose tho maps: one for to
map point $q\in TM$, and other for to express components of tensor
$\bold X(q)\in T^r_s(p,M)$. But in our terminology extended tensor
field $\bold X$ is treated as a field on $M$. Therefore we will
choose  only  one  map   $(U,x^1,\ldots,x^n)$  on  $M$  and  will 
construct
associated map on tangent bundle $TM$ according to \thetag{4.2}.
Local coordinates $x^1,\,\ldots,\,x^n,v^1,\,\ldots,\,v^n$ of such
map will be called {\bf canonically associated coordinates} for the
coordinates $x^1,\,,\ldots,\,x^n$ on $M$. Base in tangent space 
$T_p(M)$ to $M$ consists of vectors
$$
\frac{\partial}{\partial x^1},\ \ldots,\ \frac{\partial}{\partial
x^n},\hskip -2em
\tag6.1
$$
while base in dual space $T^*_p(M)$ consists of covectors
$$
dx^1,\ \ldots,\ dx^n.\hskip -2em
\tag6.2
$$
Two bases \thetag{6.1} and \thetag{6.2} are dual to each other.
Base in tensor space $T^r_s(p,M)$ is enumerated by $r+s$ indices:
$$
\bold E^{j_1\ldots\,j_s}_{i_1\ldots\,i_r}=\frac{\partial}
{\partial x^{i_1}}\otimes\ldots\otimes\frac{\partial}
{\partial x^{i_r}}\otimes dx^{j_1}\otimes\ldots\otimes dx^{j_s},
\hskip -3em
\tag6.3
$$
each index running from $1$ to $n$. For the tensor field $\bold X$
from the definition~5.3 we have the following expansion:
$$
\bold X=\sum^n_{i_1=1}\!...\!\sum^n_{i_r=1}\sum^n_{j_1=1}\!...\!
\sum^n_{j_r=1}X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}\,
\bold E^{j_1\ldots\,j_s}_{i_1\ldots\,i_r}.\hskip -3em
\tag6.4
$$
Here $X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}$ are the components of the
field $\bold X$ in the base \thetag{6.3}. They are numeric functions
of canonically associated coordinates $x^1,\,\ldots,\,x^n,v^1,\,\ldots,
\,v^n$. These functions are called {\bf components} of extended tensor
field $\bold X$ in the map $(U,x^1,\ldots,x^n)$.\par
    Suppose that two maps  $(U,x^1,\ldots,x^n)$ and $(\tilde U,\tx^1,
\ldots,\tx^n)$ on $M$ are overlapping. Then in the domain $U\cap\tilde U$
components of tensor field $\bold X$ related to these two maps are
bound with each other by standard relationships
$$
X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=\msum\Sb h_1,\,\ldots,\,h_r\\
k_1,\,\ldots,\,k_s\endSb S^{i_1}_{h_1}\ldots\,S^{i_r}_{h_r}T^{k_1}_{j_1}
\ldots\,T^{k_s}_{j_s}\,\tilde X^{h_1\ldots\,h_r}_{k_1\ldots\,k_s},
\hskip -3em
\tag6.5
$$
which are known as transformation rule for the components of tensor
field under the change or local coordinates. Components of transition
matrices $S$ and $T=S^{-1}$ in \thetag{6.5} are defined by \thetag{4.6}.
Operating with components of tensor fields in the base \thetag{6.3} and
using canonically associated coordinates for to map their arguments,
we could reach our goals without mention of tangent bundle $TM$ at
all.
\head
\S\,7. Contraction of extended tensor fields.
\endhead
     Cotangent space $T^*_p(M)$ consists of linear functionals on the
tangent space $T_p(M)$. Choosing vector $\bold X$ from $T_p(M)$ and
covector $\bold h$ from $T^*_p(M)$, we can apply linear functional
$\bold h$ to the vector $\bold X$, and we can write this as follows:
$$
\bold h(\bold X)=\langle\bold h\,|\,\bold X\rangle=C(\bold X,\bold h).
\hskip -2em
\tag7.1
$$
According to \thetag{7.1} we can write $\bold h(\bold X)$ in form of
scalar product $\langle\bold h\,|\,\bold X\rangle$, this style of
writing arose in quantum mechanics (see \cite{80} or \cite{81}).
In contrast to \thetag{2.5} here we use angular brackets. Functional
form $C(\bold X,\bold h)$ of writing $\bold h(\bold X)$ in \thetag{7.1}
defines the function $C$ bilinear in both its arguments, one of which
is a vector while the other is a covector. It can be extended up to the
linear map
$$
C\!:\,T_p(M)\otimes T^*_p(M)\longrightarrow\Bbb R.
\hskip -2em
\tag7.2
$$
The map \thetag{7.2} is known as the {\bf contraction} of tensors
of the type $(1,1)$. In tensor product of several copies of $T_p(M)$
and $T^*_p(M)$
$$
\align
&T_p(M)\otimes\ldots\, T_p(M)\ldots\otimes T_p(M)\otimes T^*_p(M)
\otimes\ldots\, T^*_p(M)\ldots\otimes T^*_p(M)\\ \vspace{-1.5ex}
&\hbox{\vrule height 0pt depth 0pt width 2.4cm \vrule height 6pt
depth 0pt width 0.5pt \vrule height 0.5pt depth 0pt width 5.4cm\vrule
height 6pt depth 0pt width 0.5pt}
\endalign
$$
one can define contraction of any copy of $T_p(M)$ with any
copy of $T^*_p(M)$. \pagebreak This is done by the map $C\!:\,
T^{r+1}_{s+1}(p,M)\longrightarrow T^r_s(p,M)$ that in local
coordinates is given by 
$$
C(\bold X)^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=\sum^n_{\rho=1}
X^{i_1\ldots\,i_{m-1}\,\rho\,i_m\ldots\,i_r}_{j_1\ldots
\,j_{k-1}\,\rho\,j_k\ldots\,j_s}.\hskip -2em
\tag7.3
$$
This map is known as {\bf contraction} of tensor $\bold X$ by
its $m$-th upper and $k$-th lower index. By contracting extended
tensor field this operation is executed pointwise in each point
$p=\pi(q)$ of the manifold $M$. Therefore
$$
C\left(\,\shave{\bigotimes^{r+1}_{i=1}\bold X_i\otimes
\bigotimes^{s+1}_{j=1}\bold h_j}\!\right)=\langle\bold h_k\,|\,
\bold X_m\rangle\,\cdot\,\left(\,\shave{\bigotimes^{r+1}_{i\neq m}
\bold X_i\otimes\bigotimes^{s+1}_{j\neq k}\bold h_j}\!\right).
\hskip -3em
\tag7.4
$$
Here $\bold X_1,\,\ldots,\,\bold X_{r+1}$ are some arbitrary vector
fields, and $\bold h_1,\,\ldots,\,\bold h_{s+1}$ are arbitrary
covectorial fields. Formula \thetag{7.4} yields one more way of
determining the contraction map \thetag{7.3}.
\enddocument
\end
ediss-03.tex
\input amstex
\documentstyle{amsppt}
%
% Author:   Ruslan Abdulovich Sharipov
% Title:    Dynamical systems admitting the normal shift
% Comment:  Thesis for the degree of Doctor of Sciences in Russia
% Address:  Rabochaya street 5, Ufa, 450003, Russia
% E-mail:   R_Sharipov@ic.bashedu.ru
%           ruslan-sharipov@usa.net
% Web-page: http://www.geocities.com/CapeCanaveral/Lab/5341
%
\nopagenumbers
\catcode`@=11
\redefine\logo@{\relax}
\catcode`\@=\active
\def\startpage#1{\pageno=#1}
\def\id{\operatorname{id}}
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\pagewidth{12.8cm}
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\accentedsymbol\tx{\tilde x}
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\Monograph
\topmatter
\title\chapter{3}
Differentiation in extended algebra\\ of tensor fields.
\endtitle
\endtopmatter
\loadbold
\startpage{55}
\document
\head
\S\,1. Differentiation of extended tensor fields.
\endhead
     We shall construct the theory of differentiation for extended
tensor fields following the scheme of \cite{12}, where it was done
for ordinary (non-extended) tensor fields. The operation of
differentiation for tensor fields assumes sufficient smoothness
of objects to be differentiated. Suppose that $M$ is smooth real
manifold.
\definition{Definition 1.1} Extended scalar field $\varphi$ is
{\bf smooth}, if it is represented by smooth function $\varphi(x^1,
\ldots,x^n,v^1,\ldots,v^n)$ in any local map $U$ of $M$.
\enddefinition
\noindent Remember that $x^1,\,\ldots,\,x^n,v^1,\,\ldots,\,v^n$ are
local coordinates in tangent bundle $TM$ canonically associated
with local coordinates $x^1,\,\ldots,\,x^n$ in the map $U$ (see
formula \thetag{4.2} in \S\,4 of Chapter
\uppercase\expandafter{\romannumeral 2}). Extended tensor field
$\bold X$ of the type $(r,s)$ in map $U$ is represented by its
components $X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}$ in the expansions
like \thetag{6.4} in Chapter
\uppercase\expandafter{\romannumeral 2}.
\definition{Definition 1.2} Extended tensor field $\bold X$ is
{\bf smooth}, if it is represented by smooth functions $X^{i_1\ldots\,
i_r}_{j_1\ldots\,j_s}(x^1,\ldots,x^n,v^1,\ldots,v^n)$ in any local map
$U$ on $M$.
\enddefinition
    Starting from here we shall understand $T^r_s(M)$ to be the space
of smooth extended tensor fields on $M$, and the sum of such spaces
will be called the {\bf extended algebra of tensor fields} (see formula
\thetag{5.2} in Chapter \uppercase\expandafter{\romannumeral 2}).
\definition{Definition 1.3} The map $D\!:\bold T(M)\to\bold T(M)$ is
called the {\bf differentiation} of extended algebra of tensor fields,
if the following conditions are fulfilled:
\roster
\rosteritemwd=10pt
\item concordance with grading: $D(T^r_s(M))\subset T^r_s(M)$;
\item $\Bbb R$-linearity: $D(\bold X+\bold Y)=D(\bold X)+D(\bold Y)$
      and $D(\lambda\bold X)=\lambda D(\bold X)$ for $\lambda\in\Bbb R$;
\item commutation with contractions: $D(C(\bold X))=
      C(D(\bold X))$;
\item Leibniz rule: $D(\bold X\otimes\bold Y)=D(\bold X)
      \otimes\bold Y+\bold X\otimes D(\bold Y)$.
\endroster
\enddefinition
    The subset $T^0_0(M)$ in extended algebra of tensor fields is
closed with respect to the operation of tensor product. It coincides
with the ring of smooth scalar functions on tangent bundle $TM$, we
denote this ring by $\goth F(TM)$. Here $\goth F$ is gothic version
of the letter $F$. Tensor product in $T^0_0(M)$ coincides with ordinary
multiplication of functions in the ring $\goth F(TM)$. Total set of
extended tensor fields $T(M)$ is an algebra over this ring.\par
    Let's consider the set of differentiations of extended algebra of
tensor fields $\bold T(M)$. One can easily check that
\roster
\item the sum of two differentiations is a differentiation of
      the algebra $\bold T(M)$;
\item product of any differentiation and any function from the ring
      $\goth F(TM)$ is a differentiation of the algebra $\bold T(M)$.
\endroster
Let's denote by $\goth D(M)$ the total set of differentiations of
extended algebra of tensor fields $\bold T(M)$. Here $\goth D$ is a
gothic version of the letter $D$. Above two properties mean that
$\goth D(M)$ is a module over the ring of functions $\goth F(TM)$.
    Composition of two differentiations is not a differentiation.
However, their commutator 
$$
[D_1,\,D_2]=D_1\compos D_2-D_2\compos D_1
$$
is a differentiation. Therefore we can say that $\goth D(M)$ is a Lie
algebra.\par
     Suppose that $D$ is some differentiation from Lie algebra $\goth
D(M)$. Denote by $\delta$ and $\xi$ restrictions of the map $D\!:\bold
T(M)\to\bold T(M)$ to the spaces $T^0_0(M)$ and $T^1_0(M)$:
$$
\xalignat 2
&\quad\delta\!:T^0_0(M)\to T^0_0(M), &&\xi\!:T^1_0(M)\to T^1_0(M).
\hskip -2em
\tag1.1
\endxalignat
$$
Due to the equality $T^0_0(M)=\goth F(TM)$ first map in \thetag{1.1}
is a differentiation of the ring of smooth functions on the tangent bundle
$TM$:
$$
\delta(\varphi\cdot\psi)=\delta(\varphi)\cdot\psi+\varphi\cdot\delta(\psi).
\hskip -2em
\tag1.2
$$
It is known (see \cite{12}, Chapter \uppercase\expandafter{\romannumeral
1}, \S\,1) that any differentiation of the ring $\goth F(TM)$ is
determined by some vector field $\bold Z$ in $TM$: $\delta(\varphi)=\bold
Z\varphi$. In local coordinates it is written in the following form:
$$
\delta(\varphi)=\bold Z\varphi=\sum^n_{i=1}Z^i\frac{\partial\varphi}
{\partial x^i}+\sum^n_{i=1}W^i\frac{\partial\varphi}{\partial v^i}.
\hskip -2em
\tag1.3
$$
From \thetag{1.3} we easily derive the following property of locality
for the map $\delta$.
\proclaim{Lemma 1.1} If the values of two functions $\varphi$ and $\psi$
coincide in some neighborhood of the point $q\in TM$, then the functions
$\delta(\varphi)$ and $\delta(\psi)$ are equal at that point $q$.
\endproclaim
In other words, all changes of value of $\varphi$ outside of some
arbitrarily small neighborhood of the point $q$ do not change the value
of $\delta(\varphi)$ at the point $q$. Similar property of locality takes
place for the initial map $D$ too, for which $\delta$ is a restriction
to the space $T^0_0(M)$.
\proclaim{Lemma 1.2} If two extended tensor fields $\bold X$ and $\bold Y$
coincide in some neighborhood of the point $q\in TM$, then the values of
extended tensor fields $D(\bold X)$ and $D(\bold Y)$ at the point $q$ are
equal.
\endproclaim
\demo{Proof} Denote by $\bold V$ the difference of fields $\bold X$
and $\bold Y$. Field $\bold V$ equals to zero identically in some
neighborhood $O$ of the point $q$. Let's choose some smooth scalar
function $\varphi$ on $TM$ such that it is equal to $1$ outside the
neighborhood $O$ and equal to $0$ at the point $q$. Then $\bold V=
\varphi\cdot\bold V=\varphi\otimes\bold V$. Therefore
$$
D(\bold V)(q)=D(\varphi)(q)\cdot\bold V(q)+\varphi(q)\cdot
D(\bold V)(q).
$$
We know that $\bold V(q)=0$ and $\varphi(q)=0$, hence $D(\bold V)(q)=0$.
This yields the required equality $D(\bold X)(q)=D(\bold Y)(q)$ at the
point $q$.\qed\enddemo
    Now let's consider the second map in \thetag{1.1}. It is
$\Bbb R$-linear map from $T^1_0(M)$ to $T^1_0(M)$ bounded with $\delta$
by the following relationship
$$
\xi(\varphi\cdot\bold X)=\delta(\varphi)\cdot\bold X+
\varphi\cdot\xi(\bold X).\hskip -2em
\tag1.4
$$
Denote by $\zeta$ the restriction of $D$ to the space $T^0_1(M)$.
Then $\zeta\!:T^0_1(M)\to T^0_1(M)$ is $\Bbb R$-linear map
satisfying the relationship $\zeta(\varphi\cdot\bold h)=\delta(\varphi)
\cdot\bold h+\varphi\cdot\zeta(\bold h)$, which is analogous to
\thetag{1.4}. Moreover, $\delta$, $\xi$, and $\zeta$ are bound
by the relationship
$$
\delta(\langle\bold h\,|\,\bold X\rangle)=\langle\zeta(\bold h)\,|\,
\bold X\rangle+\langle\bold h\,|\,\xi(\bold X)\rangle,
\hskip -2em
\tag1.5
$$
which is the consequence of commutation of $D$ with contractions
(see definition~1.3).\par
     Suppose that for some differentiation $D$ of extended algebra of
tensor fields $\bold T(M)$ we know only its restrictions $\delta$ and
$\xi$ to $T^0_0(M)$ and $T^1_0(M)$ respectively. By means of \thetag{1.5}
we can reconstruct its restriction $\zeta$ to the space $T^0_1(M)$.
Then we can reconstruct the whole map $D$ by setting
$$
\align
D(&\bold X_1\otimes\ldots\otimes\bold X_r\otimes\bold h_1\otimes\ldots
\otimes\bold h_s)=\\ \vspace{1ex}
&=\sum^r_{m=1}\bold X_1\otimes\ldots\,\xi(\bold X_m)\ldots\otimes
\bold X_r\otimes\bold h_1\otimes\ldots\otimes\bold h_s\,+\hskip -3em
\tag1.6\\
&+\sum^s_{m=1}\bold X_1\otimes\ldots\otimes\bold X_r\otimes\bold h_1
\otimes\ldots\,\zeta(\bold h_m)\ldots\otimes\bold h_s\hskip -3em
\endalign
$$
for its action upon the tensor fields from $T^r_s(M)$. This formula
\thetag{1.6} is easily derived from Leibniz rule by induction in
the numbers of multiplicands $r$ and $s$. Above arguments form the
proof of the following theorem.
\proclaim{Theorem 1.1} Each differentiation $D$ of extended algebra
of tensor fields $\bold T(M)$ is uniquely fixed by its restrictions 
to the spaces $T^0_0(M)$ and $T^1_0(M)$.
\endproclaim
     Theorem~1.1 means that if the restrictions of two differentiations
$D_1$ and $D_2$ to $T^0_0(M)$ and $T^1_0(M)$ coincide, then $D_1=D_2$.
This result can be strengthened as follows.
\proclaim{Theorem 1.2} Any two $\Bbb R$-linear maps $\delta$ and $\xi$
from \thetag{1.1} satisfying the relationships \thetag{1.2} and \thetag{1.4}
define some differentiation $D$ of extended algebra of tensor fields
$\bold T(M)$ on the manifold $M$.
\endproclaim
    Theorem~1.2 is proved by direct construction of the map $D$ relying
on the formulas \thetag{1.5} and \thetag{1.6}. Then one should only to
check up the properties \therosteritem{1}--\therosteritem{4} from the
definition~1.3. Theorem~1.1 and 1.2 can be united into one theorem.
\proclaim{Theorem 1.3} Defining the differentiation $D$ in extended
algebra of tensor fields $\bold T(M)$ is equivalent to defining the
vector field $\bold Z$ in $TM$ and some $\Bbb R$-linear map $\xi\!:T^1_0(M)
\to T^1_0(M)$ satisfying the relationship $\xi(\varphi\cdot\bold X)=
\bold Z\varphi\cdot\bold X+\varphi\cdot\xi(\bold X)$ for any extended
scalar field $\varphi\in T^0_0(M)$ and for any extended vector field
$\bold X\in T^1_0(M)$.
\endproclaim
\definition{Definition 1.4} We say that the differentiation $D$
of extended  algebra  of  tensor  fields  $\bold  T(M)$  is  {\bf 
degenerate},
if its restriction $\delta$ to $T^0_0(M)$ is identically zero map.
\enddefinition
    For degenerate differentiation $D$ corresponding vector field $\bold Z$
is identically zero, and the relationship \thetag{1.4} is written as
$$
\xi(\varphi\cdot\bold X)=\varphi\cdot\xi(\bold X).\hskip -2em
\tag1.7
$$
Relying on \thetag{1.7} we conclude that the map $\xi\!:T^1_0(M)\to
T^1_0(M)$ is the endomorphism of the module $T^1_0(M)$ over the ring
$\goth F(TM)$. It is known (see \cite{12}, Chapter
\uppercase\expandafter{\romannumeral 1}, \S\,2 and \S\,3) that any such
endomorphism is determined by some extended tensor field $\bold S$:
$$
\xi(\bold X)=C(\bold S\otimes\bold X).\hskip -2em
\tag1.8
$$
Here $\bold S\in T^1_1(M)$, and $C$ is the operation of contraction.
From \thetag{1.8} and \thetag{1.5} we derive how the degenerate
differentiation $D$ acts upon covectorial field $\bold h\in T^0_1(M)$:
$$
\zeta(\bold h)=C(\bold h\otimes\bold S).\hskip -2em
\tag1.9
$$
The relationships \thetag{1.8} and \thetag{1.9} together with $\delta=0$
and the formula \thetag{1.6} yield complete description of all degenerate
differentiations of extended algebra of tensor fields on the manifold $M$.
\head
\S\,2. Covariant differentiations.
\endhead
     The set of differentiations $\goth D(M)$ of the extended algebra
of tensor fields on the manifold $M$ is equipped the structure of module
over the ring $\goth F(TM)$ of smooth functions on tangent bundle $TM$.
The set of extended vector fields $T^1_0(M)$ is equipped with the same
structure of module over the ring $\goth F(TM)$. This coincidence motivates
the following definition.
\definition{Definition 2.1} {\bf Covariant differentiation} $\nabla$ in the
algebra of extended vector fields $\bold T(M)$ is the homomorphism of $\goth
F(TM)$-modules $\nabla\!:T^1_0(M)\to\goth D(M)$. Image of vector field
$\bold Y$ under such homomorphism denoted by $\nabla_{\bold Y}$ is called
{\bf covariant differentiation along the vector field $Y$}.
\enddefinition
    The set of differentiations $\goth D(M)$ has natural grading that is
defined by the grading of extended algebra of tensor fields $\bold T(M)$:
$$
\goth D(M)\varsubsetneq\bigoplus^\infty_{r=0}\bigoplus^\infty_{s=0}
\goth D^r_s(M).\hskip -2em
\tag2.1
$$
Therefore each covariant differentiation $\nabla$ generates the series
of homomorphisms of $\goth F(TM)$-modules numerated by grading numbers
$r$ and $s$:
$$
\nabla\!:T^1_0(M)\to\goth D^r_s(M).\hskip -2em
\tag2.2
$$
According to the theorem~1.1 only two homomorphisms in the series 
\thetag{2.2} play key role in defining covariant differentiation
$\nabla\!:T^1_0(M)\to\goth D(M)$. These are
$$
\align
&\nabla\!:T^1_0(M)\to\goth D^0_0(M),\hskip -2em
\tag2.3\\
\vspace{1ex}
&\nabla\!:T^1_0(M)\to\goth D^1_0(M).\hskip -2em
\tag2.4
\endalign
$$
Let's consider the homomorphism \thetag{2.3}. It maps extended vector
field $\bold Y$ into the differentiation of the ring $\goth F(TM)$,
which, as we mentioned in \S\,1, is defined by some vector field
$\bold Z$ on the tangent bundle $TM$. Therefore we can treat the map
\thetag{2.3} as a homomorphism $f$ from $\goth F(TM)$-module of extended
vector fields on $M$ into $\goth F(TM)$-module of ordinary vector fields
on $TM$. Suppose that $q=(p,\bold v)$ is a point of tangent bundle $TM$,
and $p=\pi(q)$ is its projection to $M$. Extended vector field $Y$ maps
$q$ to the vector $\bold Y=\bold Y(q)$ tangent to $M$ at the point $p$.
Homomorphism $f$ maps $\bold Y$ to the vector $\bold Z=f(\bold Y)$
tangent to $TM$ at the point $q$. Vector function $\bold Z=f(\bold Y)$
is linear in its argument.
\definition{Definition 2.2} Suppose that at each point $q$ of tangent
bundle $TM$ we have $\Bbb R$-linear map $f\!: T_p(M)\to T_q(TM)$ from
tangent space $T_p(M)$ at the point $p=\pi(q)$ to the tangent space $
T_q(TM)$ at the point $q$. Such construction is called the {\bf lift}
of vectors from $M$ to $TM$.
\enddefinition
\definition{Definition 2.3} Lift of vectors $f$ from definition~2.2
is called {\bf smooth lift}, if it maps each smooth extended vector
field on $M$ to the smooth vector field on $TM$.
\enddefinition
    Let's compare the definition~2.2 with previous analysis of
homomorphism \thetag{2.3}. By this comparison we draw {\bf very
important conclusion} that defining homomorphism of $\goth F(TM)$-modules
$\nabla\!:T^1_0(M)\to\goth D^0_0(M)$ is equivalent to defining some
smooth lift of vectors from $M$ to tangent bundle $TM$.\par
    Now consider canonical projection $\pi\!:TM\to M$. Differential of
this map $\pi_*$ acts in the direction opposite to any lift of vectors:
$\pi_*\!:T_q(TM)\to T_p(M)$. Therefore composition $\pi_*\compos f$ is a
map from $T_p(M)$ to $T_p(M)$. Such composition defines operator field
(field of type $(1,1)$) in extended algebra of tensor fields on $M$.
\definition{Definition 2.4} Lift of vectors $f$ from $M$ to $TM$
is called {\bf vertical lift}, if composition $\pi_*\compos f$ is
identically zero: $\pi_*\compos f=0$.
\enddefinition
\definition{Definition 2.5} Lift of vectors $f$ from $M$ to $TM$
is called {\bf horizontal lift}, if composition $\pi_*\compos f$
is the field of identical operators on $M$, i\.~e\. $\pi_*\compos
f=\id$.
\enddefinition
     Tangent bundle $TM$ is naturally subdivided into the fibers over
the points of base manifold $M$. Vectors tangent to the fiber form
$n$-dimensional subspace $V_q(TM)$ in $2n$-dimensional space $T_q(TM)$.
It is called {\bf vertical} subspace in $T_q(TM)$. Vertical lift $f$
defines the map from $T_p(M)$ to vertical space $V_q(TM)$.\par
\proclaim{Lemma 2.1} Difference of two horizontal lifts is a
vertical lift.
\endproclaim
    If we take two horizontal lifts $f_1$ and $f_2$, then $\pi_*\compos(f_1
-f_2)=\pi_*\compos f_1-\pi_*\compos f_2=\id-\id=0$. This means that the
difference $f_1-f_2$ is vertical lift according to the definition~2.4.
\definition{Definition 2.6} Covariant differentiation $\nabla$ in extended
algebra of tensor fields on $M$ is called {\bf horizontal differentiation}
(or {\bf vertical differentiation}), if corresponding lift of vectors is
{\bf horizontal} (or {\bf vertical}). 
\enddefinition
\proclaim{Lemma 2.2} Difference of two horizontal covariant differentiations
is vertical differentiation in extended algebra of tensor fields on $M$.
\endproclaim
    Lemma~2.2 is an obvious consequence of lemma~2.1.
\head
\S\,3. Canonical vertical lift of vectors and the velocity gradient.
\endhead
\rightheadtext{\S\,3. Canonical vertical lift of vectors and \dots}
    On any manifold $M$ one can define infinitely many vertical lifts,
but there is only one which is canonical. We define it as described
below. Let $\bold Y$ be some fixed vector at the point $p$ in $M$.
Suppose that $q=(p,\bold v)$ is a point of tangent bundle $TM$ from
the fiber over the point $p$. Let's choose numeric parameter $t$
and define parametric curve $q=q(t)$ in $TM$ as follows:
$$
q(t)=(p,\bold v+t\cdot\bold Y).\hskip -2em
\tag3.1
$$
Curve \thetag{3.1} lies within the fiber over the point $p$ and passes
through the point $q=(p,\bold v)$, when $t=0$. Denote by $\bold Z$
tangent vector to the curve \thetag{3.1} at the point $q$, and let's take
it for the lift of vector $\bold Y$:
$$
\bold Z=w(\bold Y).\hskip -2em
\tag3.2
$$
Vector $\bold Z$ can be treated as differential operator acting at the point
$q$ upon the functions from the ring $\goth F(TM)$. Then
$$
\bold Z\varphi=\frac{d\varphi(p,\bold v+t\cdot\bold Y)}{dt}\,\hbox{\vrule
height14pt depth8pt width 0.5pt}_{\,t=0}.\hskip -2em
\tag3.3
$$
\definition{Definition 3.1} Vertical lift $w$ from \thetag{3.2} defined
by formula \thetag{3.3} is called {\bf canonical vertical lift} of vectors
from $M$ to $TM$.
\enddefinition
    One can easily write canonical vertical lift of vectors in local
coordinates. Suppose that vector $\bold Y\in T_p(M)$ is given by
formula 
$$
\bold Y=Y^1\,\frac{\partial}{\partial x^1}+\ldots+
Y^n\,\frac{\partial}{\partial x^n},\hskip -2em
\tag3.4
$$
in local coordinates $x^1,\,\ldots,\,x^n$. Then its lift
is defined by formula
$$
\bold Z=w(\bold Y)\in T_q(TM)=
Y^1\,\frac{\partial}{\partial v^1}+\ldots+
Y^n\,\frac{\partial}{\partial v^n},\hskip -2em
\tag3.5
$$
where $x^1,\,\ldots,\,x^n,v^1,\,\ldots,\,v^n$ are coordinates in
$TM$ canonically associated to local coordinates $x^1,\,\ldots,
\,x^n$. Now one can see that canonical vertical lift $w$, which
takes vector \thetag{3.4} to the vector \thetag{3.5}, is
one-to-one map from $T_p(M)$ to the vertical subspace $V_q(TM)$
in $T_q(TM)$.\par
     Canonical vertical lift $w\!:T_p(M)\to T_q(TM)$ corresponds to
some homomorphism of $\goth F(TM)$-modules of the form \thetag{2.3}.
Let's denote it by $\tilde\nabla$. This homomorphism can be extended
up to a covariant differentiation of the whole extended algebra of
tensor fields $\bold T(M)$. In order to do it we should define
homomorphism $\tilde\nabla\!:T^1_0(M)\to\goth D^1_0(M)$ of the form
\thetag{2.4}. Suppose that $\bold X$ and $\bold Y$ are two vector
fields from $T^1_0(M)$. We should put them into correspondence third
vector field $\hat\bold X=\tilde\nabla_{\bold Y}\bold X$. Let's choose
a point $p\in M$ and a point $q=(p,\bold v)\in TM$, consider the
curve \thetag{3.1} passing through the point $q$, and then define 
vector $\hat\bold X$ as a differential operator in the ring of smooth
functions on $M$ by means of the following formula:
$$
\hat\bold X\varphi=\frac{d\,(\bold X(q(t))\varphi)}{dt}\,
\hbox{\vrule height14pt depth8pt width 0.5pt}_{\,t=0}.\hskip -2em
\tag3.6
$$
Here $\varphi\in\goth F(M)$ is a smooth function on $M$, and $\bold
X(q(t))\varphi$ is its derivative along the vector $\bold X(q(t)
\in T_p(M)$. If we know components of vector fields $\bold X$ and
$\bold Y$ in some map $(U,x^1,\ldots,x^n)$ on $M$, then we can calculate
components of vector field $\hat\bold X=\tilde\nabla_{\bold Y}\bold
X$ in \thetag{3.6} by means of formula
$$
\hat X^k=\sum^n_{m=1}Y^m\,\frac{\partial X^k}{\partial v^m},
\hskip -2em
\tag3.7
$$
where $x^1,\,\ldots,\,x^n,v^1,\,\ldots,\,v^n$ are local coordinates
on $TM$ canonically associated to $x^1,\,\ldots,\,x^n$ on $M$.
Formula \thetag{3.7} can be extended for the case of arbitrary
tensor field $\bold X\in T^r_s(M)$ with components $X^{i_1\ldots\,i_r}_{j_1
\ldots\,j_s}$ in local map $(U,x^1,\ldots,x^n)$. For the components
of the field $\hat\bold X=\tilde\nabla_{\bold Y}\bold X$ in that map
we have
$$
\hat X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=\sum^n_{m=1}Y^m\,
\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial v^m}.
\hskip -2em
\tag3.8
$$
Covariant differentiation $\tilde\nabla$ given by the relationships
\thetag{3.3} and \thetag{3.6}, and expressed by the formula \thetag{3.8}
in local coordinates is called {\bf canonical vertical covariant
differentiation} in extended algebra of tensor fields on the manifold
$M$. It is also called {\bf velocity covariant differentiation} or
{\bf velocity gradient}. The latter is motivated by formulas \thetag{3.7}
and \thetag{3.8}, which contain partial derivatives in components of
velocity vector only.
\head
\S\,4. Horizontal lifts of vectors and dynamic connections.
\endhead
     Canonical vertical lift of vectors considered in section \S\,3
above is related to canonical covariant differentiation $\tilde\nabla$.
In this section we consider covariant differentiations related to
horizontal lifts of vectors (see definition~2.5). Suppose that $\nabla$
is one of such differentiations, and let $f$ be corresponding lift of
vectors from $M$ to $TM$. Horizontality of $f$ means that image of
linear map $f\!:T_p(M)\to T_q(TM)$ is some $n$-dimensional space
$H_q(TM)$ in $2n$-dimensional tangent space $T_q(TM)$. It is called
{\bf horizontal subspace}. Due to the equality $\pi_*\compos f=\id$
following maps
$$
\pagebreak
\xalignat 2
&\quad f\!:T_p(M)\to H_q(TM),&&\pi_*\!:H_q(TM)\to T_p(M)\hskip -2em
\tag4.1
\endxalignat
$$
are inverse to each other. Sum of vertical and horizontal subspaces is
a direct sum:
$$
H_q(TM)\oplus V_q(TM)=T_q(TM).\hskip -2em
\tag4.2
$$
\proclaim{Theorem 4.1} Defining horizontal lift of vectors from $M$ to
$TM$ is equivalent to defining direct complement $H_q(TM)$ for vertical
subspace $V_q(TM)$ at each point $q$ of tangent space $TM$.
\endproclaim
\demo{Proof} If horizontal lift $f$ is given, then horizontal subspace
$H_q(TM)$ is defined as an image of the map $f\!:T_p(M)\to T_q(TM)$.
Then \thetag{4.2} is a consequence of the equalities $\pi_*\compos f=\id$
and $V_q(TM)=\Ker\pi_*$.\par
    And conversely, if at each point $q\in TM$ we have subspace $H_q(TM)$
satisfying the equality \thetag{4.2}, then we can restrict canonical
projection $\pi_*\!:T_q(TM)\to T_p(M)$ to $H_q(TM)$. Due to the equality
\thetag{4.2} from $\Ker\pi_*=V_q(TM)$ we derive that restriction
$\pi_*\!:H_q(TM)\to T_p(M)$ is one-to-one map. By inverting this map we
can define the lift of vectors $\pi_*^{-1}\!:T_p(M)\to H_q(TM)\subset
T_q(TM)$. One can easily check that this lift is horizontal in the sense
of definition~2.5.\qed\enddemo
     Let's consider some horizontal lift $f$ in local coordinates.
Let's choose some local map $(U,x^1,\ldots,x^n)$ on the base manifold
$M$ and take coordinate vector fields:
$$
\bold E_1=\frac{\partial}{\partial x^1},\ \ldots,\ \bold E_n=
\frac{\partial}{\partial x^n}.\hskip -3em
\tag4.3
$$
Applying lift $f$ to base vectors \thetag{4.3}, we get the following
expression:
$$
f(\bold E_i)=\frac{\partial}{\partial x^i}-\sum^n_{k=1}\Gamma^k_i\,
\frac{\partial}{\partial v^k}.\hskip -2em
\tag4.4
$$
Such structure of $f(\bold E_k)$ is the consequence of $\pi_*\compos
f=\id$ and obvious equalities
$$
\xalignat 2
&\quad\pi_*\left(\frac{\partial}{\partial x^i}\right)=\bold E_i,
&&\pi_*\left(\frac{\partial}{\partial v^k}\right)=0.\hskip -2em
\tag4.5
\endxalignat
$$
Parameters $\Gamma^k_i=\Gamma^k_i(x^1,\ldots,x^n,v^1,\ldots,v^n)$
in thetag{4.4} are called {\bf components of lift} $f$ in the map
$U$. Under the change of maps $(U,x^1,\ldots,x^n)$ for
$(\tilde U,\tx^1,\ldots,\tx^n)$ these components are transformed as
follows:
$$
\Gamma^k_i=\sum^n_{m=1}\sum^n_{a=1} S^k_m\,T^a_i\,\tilde\Gamma^m_a+
\sum^n_{j=1}\sum^n_{m=1}v^j\,S^k_m\,\frac{\partial T^m_i}{\partial
x^j}.\hskip -2em
\tag4.6
$$
In order to prove the relationships \thetag{4.6} for $\Gamma^k_i$
and $\tilde\Gamma^m_a$ let's remember that
$$
\bold E_i=\sum^n_{a=1}T^a_k\,\tilde\bold E_a
\text{\ \ leads to \ }f(\bold E_i)=\sum^n_{a=1}T^a_i\,
f(\tilde\bold E_a).\hskip -2em
\tag4.7
$$
Components of transition matrix $T=S^{-1}$ in \thetag{4.7} are
determined by formulas \thetag{4.6} in Chapter
\uppercase\expandafter{\romannumeral 2}. For $f(\tilde\bold E_a)$
in \thetag{4.7} we shall write formula similar to formula \thetag{4.4}:
$$
f(\tilde\bold E_a)=\frac{\partial}{\partial\tx^a}-\sum^n_{m=1}
\tilde\Gamma^m_a\,\frac{\partial}{\partial\tilde v^m}.\hskip -2em
\tag4.8
$$
In order to convert differential operators $\partial/\partial\tx^p$ and
$\partial/\partial\tilde v^m$ in \thetag{4.8} to the variables $x^1,\,
\ldots,\,x^n,v^1,\,\ldots,\,v^n$ we use the relationships \thetag{4.5}
from Chapter \uppercase\expandafter{\romannumeral 2}. Differentiating
these relationships, we obtain
$$
\quad\left\{\aligned
\frac{\partial}{\partial\tilde v^m}&=\sum^n_{p=1}S^p_m\,
\frac{\partial}{\partial v^p},\\
\frac{\partial}{\partial\tx^a}&=\sum^n_{k=1}S^k_a\,\frac{\partial}
{\partial x^k}+\sum^n_{k=1}\sum^n_{r=1}\tilde v^r\,\frac{\partial S^k_r}
{\partial\tx^a}\,\frac{\partial}{\partial v^k}.
\endaligned\right.\hskip -3em
\tag4.9
$$
Now, substituting \thetag{4.4} and \thetag{4.8} into the relationship
\thetag{4.7} and using \thetag{4.9}, after some calculations we get
\thetag{4.6}.\par
    Continuing the study of horizontal covariant differentiation
$\nabla$ in extended algebra of tensor fields $\bold T(M)$, consider
the homomorphism \thetag{2.4} for it. Let $q$ be some point of tangent
bundle $TM$, and let $p=\pi(q)$ be its projection to $M$. To each vector
$\bold Y$ at the point $p$ homomorphism \thetag{2.4} puts into
correspondence the differential operator acting upon extended tensor
fields at the point $q$. Let's take some local map $(U,x^1,\ldots,x^n)$
in the neighborhood of the point $p$ and, choosing one of base vector
fields from \thetag{4.3}, assume $\bold Y=\bold E_i$. Then we take
another vector field $\bold E_j$ from \thetag{4.3} and apply the operator
$\nabla_{\bold Y}$ to it. The resulting field $\nabla_{\bold E_{\ssize i}}
\bold E_j$ can be expanded in the base of fields \thetag{4.3}. We write
it as follows:
$$
\nabla_{\bold E_{\ssize i}}\bold E_j=\sum^n_{k=1}\Gamma^k_{ij}\,
\bold E_k.\hskip -3em
\tag4.10
$$
Vector field $\bold E_j$ is defined locally within the map $U$. But
due to the lemma~1.2 this is not an obstacle for applying an operator
$\nabla_{\bold E_{\ssize i}}$ to it.\par 
    Coefficients $\Gamma^k_{ij}=\Gamma^k_{ij}(x^1,\ldots,x^n,v^1,
\ldots,v^n)$ in expansion \thetag{4.10} are called {\bf components
of connection} or {\bf Christoffel symbols}. Under the change of local
coordinates in the domain of intersection of two local maps these
coefficients are transformed according to standard rule for connection
components
$$
\Gamma^k_{ij}=\sum^n_{m=1}\sum^n_{a=1}\sum^n_{c=1} S^k_m\,T^a_i
\,T^c_j\,\tilde\Gamma^m_{ac}+\sum^n_{m=1} S^k_m\,\frac{\partial
T^m_i}{\partial x^j}.\hskip -3em
\tag4.11
$$
Quantities $\Gamma^k_{ij}(x^1,\ldots,x^n,v^1,\ldots,v^n)$, which are
transformed according to \thetag{4.11}, define special geometric
structure on $M$. We call it {\bf extended} or {\bf dynamic} affine
connection. It differs from ordinary affine connection by twice as
more arguments in Christoffel symbols.
\definition{Definition 4.1} Let $M$ be some manifold. We say that
an extended affine connection $\Gamma$ on $M$ is defined, if for
each local map $(U,x^1,\ldots,x^2)$ on $M$ we have the set of smooth
functions
$$
\Gamma^k_{ij}=\Gamma^k_{ij}(x^1,\,\ldots,\,x^n,v^1,\,\ldots,\,v^n)
$$
which are transformed according to the rule \thetag{4.11} under the
change of local coordinates in the domain of intersection of any two
local maps.
\enddefinition
\proclaim{Theorem 4.2} On any smooth paracompact manifold $M$ at least
one extended affine connection can be defined.
\endproclaim
    Omitting details, we give only the scheme of proof for the theorem~3.2.
Proof is based on the use of locally finite open covering
$\{U_\alpha\}_{\alpha\in A}$ composed of local maps and subordinate
smooth partition of unity
$$
1=\sum_{\alpha\in A}\eta_\alpha
$$
(see details in \cite{12}, appendix 3). With each map $U_\alpha$ we
associate some arbitrary set of smooth functions $\Gamma^k_{ij}(\alpha)$
in corresponding local coordinates $x^1,\,\ldots,\,x^n,v^1,\,\ldots,
\,v^n$.\par
     Suppose that $p$ is some point on $M$. It has a neighborhood $V$
that intersects with finite number of maps $\{U_\alpha\}_{\alpha\in A}$.
Let's introduce some local coordinates $x^1,\,\ldots,\,x^n$ in $V$
and transform each set of functions $\Gamma^k_{ij}(\alpha)$ to these
local coordinates in $V$ according to the rule \thetag{4.11}. Then we
set
$$
\Gamma^k_{ij}=\sum_{\alpha\in A}\Gamma^k_{ij}(\alpha)\cdot\eta_\alpha.
\hskip -3em
\tag4.12
$$
Formula \thetag{4.12} determines the components of extended affine
connection that we need in local map $(V,x^1,\ldots,x^n)$. Such maps
cover the entire manifold $M$, therefore our connection is globally
defined. Now we are only to do some formalities, testing transformation
rules \thetag{4.11} for the connection components defined by \thetag{4.12}.
\definition{Definition 4.2} Horizontal covariant differentiation in
extended algebra of tensor fields is called {\bf space covariant
differentiation} or {\bf space gradient}, if the following condition
is fulfilled:
$$
\nabla_{\bold Y}\bold v=0.\hskip -3em
\tag4.13
$$
This means that covariant derivative $\nabla_{\bold y}$ applied to the
velocity field $\bold v$ is identically zero for any vector field
$\bold Y$ on $M$.
\enddefinition
     Let's express the condition \thetag{4.13} in local coordinates by
choosing some map $U$ on $M$. In order to do it we use the expansion
$$
\bold v=\sum^n_{j=1}v^j\,\bold E_j\hskip -3em
\tag4.14
$$
and we apply the relationship \thetag{4.10} by differentiating
\thetag{4.14}:
$$
\align
\nabla_{\bold Y}\bold v&=\sum^n_{j=1}(\nabla_{\bold Y}v^j)\,\bold E_j
+\sum^n_{j=1}v^j\,\nabla_{\bold Y}\bold E_j=\\
&=\sum^n_{j=1}f(\bold Y)v^j\,\bold E_j+\sum^n_{i=1}\sum^n_{j=1}
v^j\,Y^i\,\nabla_{\bold E_{\ssize i}}\bold E_j=\\
&=\sum^n_{j=1}f(\bold Y)v^j\,\bold E_j+\sum^n_{i=1}\sum^n_{j=1}\sum^n_{k=1}
v^j\,Y^i\,\Gamma^k_{ij}\,\bold E_k.
\endalign
$$
Here $Y^1,\,\ldots,\,Y^n$ are components of vector $\bold Y$. We
calculate the result of applying $f(\bold Y)$ to the function $v^j$
by means of \thetag{4.4}. Then
$$
\nabla_{\bold Y}\bold v=\sum^n_{i=1}\sum^n_{k=1}Y^i\,\Gamma^k_i\,
\bold E_k+\sum^n_{i=1}\sum^n_{j=1}\sum^n_{k=1}v^j\,Y^i\,\Gamma^k_{ij}\,
\bold E_k.\hskip -3em
\tag4.15
$$
Since components of the field $\bold Y$ in \thetag{4.15} are
absolutely arbitrary, the condition \thetag{4.13} from definition~4.12
reduces to the relationships 
$$
\Gamma^k_i=\sum^n_{j=1}v^j\,\Gamma^k_{ij}.\hskip -3em
\tag4.16
$$
The relationships \thetag{4.16} do not change their appearance under
the change of local coordinates by transferring from one map to another.
This can be seen by comparison the formulas \thetag{4.6} and
\thetag{4.11}. Thus, in case of space covariant differentiations the
structure of horizontal lift of vectors is completely defined by the
structure of extended affine connection. This result will be formulated
as a theorem.
\proclaim{Theorem 4.3} Defining space covariant differentiation
$\nabla$ in extended algebra of tensor fields on the manifold $M$ is
equivalent to defining some extended affine connection on $M$.
\endproclaim
     We shall terminate this section by writing an explicit formula
for space covariant derivative $\hat\bold X=\nabla_{\bold Y}\bold X$
for extended tensor field $\hat\bold X=\bold X\in T^r_s(M)$ in local
coordinates:
$$
\pagebreak
\aligned
\hat X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}&=\sum^n_{m=1}Y^m\,
\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial x^m}-\\
&-\sum^n_{m=1}\sum^n_{a=1}\sum^n_{b=1}Y^m\,v^a\,\Gamma^b_{ma}\,
\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial v^b}+\\
&+\sum^n_{m=1}\sum^r_{k=1}\sum^n_{a_k=1}Y^m\,\Gamma^{i_k}_{m\,a_k}
\,X^{i_1\ldots\,a_k\ldots\,i_r}_{j_1\ldots\,\ldots\,\ldots\,j_s}-\\
&-\sum^n_{m=1}\sum^s_{k=1}\sum^n_{b_k=1}Y^m\,\Gamma^{b_k}_{m\,j_k}
X^{i_1\ldots\,\ldots\,\ldots\,i_r}_{j_1\ldots\,b_k\ldots\,j_s}
\endaligned\hskip -3em
\tag4.17
$$
Formula \thetag{4.17} is analogous to \thetag{3.8}. It can be proved
by induction in the number of indices $r$ and $s$.
\head
\S\,5. Structural theorem for differentiations.
\endhead
    In sections 1--4 above we have defined and have completely described
three types of differentiations in extended algebra of tensor fields
$\bold T(M)$ for some manifold $M$. These are: {\bf degenerate
differentiation} $\bold S$, which is given by some tensor field $\bold S$
of type $(1,1)$, and {\bf two covariant differentiations} $\nabla_{\bold X}$
and $\tilde\nabla_{\bold Y}$. Here we shall prove that by means of
$\bold S$, $\nabla_{\bold X}$, and $\tilde\nabla_{\bold Y}$ we can
exhaust the whole variety of differentiations in extended algebra of tensor
fields $\bold T(M)$.
\proclaim{Theorem 5.1} Let $M$ be a real smooth manifold equipped with
extended affine connection $\Gamma$. Then each differentiation $D$ in
extended algebra of tensor fields $\bold T(M)$ is expanded into a sum
$$
D=\nabla_{\bold X}+\tilde\nabla_{\bold Y}+\bold S,\hskip -2em
\tag5.1
$$
where $\bold X$ and $\bold Y$ are some extended vector fields on $M$, and
$\bold S$ is a degenerate differentiation defined by some extended tensor
field $\bold S$ of type $(1,1)$.
\endproclaim
\demo{Proof} Take some arbitrary differentiation $D\in\goth D(M)$. Its
restriction to subspace $T^0_0(M)\subset\bold T(M)$, as we mentioned
in \S\,1, is defined by some vector field $\bold Z$ on tangent bundle
$TM$. Expanded affine connection $\Gamma$ defines some horizontal
lift of vectors $f$; its components in local coordinates are defined
by relationships \thetag{4.16}. Let's apply theorem~4.1 to the lift $f$.
This yields the expansion of vector field $\bold Z$ into a sum of two
components:
$$
\bold Z=\bold H+\bold V.\hskip -2em
\tag5.2
$$
Here $\bold H$ is horizontal vector field, and $\bold V$ is a vertical
vector field on $TM$. Vector field $\bold H$ from \thetag{5.2} is
projected into the expanded vector field $\bold X$ on $M$ 
$$
\xalignat 2
&\quad\bold X=\pi_*(\bold H)=\pi_*(\bold Z),
&&\bold H=f(\bold X).\hskip -2em
\tag5.3
\endxalignat
$$
Vertical component $\bold V$ in \thetag{5.3} is also related to
some extended vector field on $M$. In order to get it we should
remember the bijective linear map $w\!:T_p(M)\to V_q(TM)$. Then 
by inverting this map we get
$$
\xalignat 2
&\quad\bold Y=w^{-1}(\bold V),
&&\bold V=w(\bold Y),\hskip -2em
\tag5.4
\endxalignat
$$
where $\bold Y$ is an extended vector field on $M$. Taking into
account \thetag{5.3} and \thetag{5.4} we rewrite \thetag{5.2}
as follows: 
$$
\bold Z=f(\bold X)+w(\bold Y).\hskip -2em
\tag5.5
$$\par
     As a next step let's consider the differentiation $\hat D=
\nabla_{\bold X}+\tilde\nabla_{\bold Y}$. Its restriction to the
subspace $T^0_0(M)$ is defined by the vector field which coincides
with \thetag{5.5}. Therefore difference $D-\hat D$ is a degenerate
differentiation in the extended algebra of tensor fields. \pagebreak
According to the results of \S\,1 it is defined by some extended tensor
field $\bold S$ of the type $(1,1)$. Now from $D-\hat D=\bold S$ and
$\hat D=\nabla_{\bold X}+\tilde\nabla_{\bold Y}$ for the
differentiation $D$ we get required expansion \thetag{5.1}. Theorem
is proved.\qed\enddemo
\head
\S\,6. Commutational relationships and curvature tensors.
\endhead
     The set of differentiations $\goth D(M)$ of extended algebra of
tensor fields $\bold T(M)$ is an infinite-dimensional Lie algebra
over the field of real numbers $\Bbb R$. Theorem~5.1 can give
more detailed description of this Lie algebra. Let's begin with
degenerate differentiations. Let $\bold S_1$ and $\bold S_2$
be some two degenerate differentiations defined by operator fields
$\bold S_1$ and $\bold S_2$ from extended algebra. Then 
$$
[\bold S_1,\,\bold S_2]=C(\bold S_1\otimes\bold S_2-\bold S_2\otimes
\bold S_1).\hskip -2em
\tag6.1
$$
Formula \thetag{6.1} means that commutator of two degenerate
differentiations is degenerate differentiation defined by operator
field being the pointwise commutator for operator fields $\bold S_1$
and $\bold S_2$.\par
     Suppose that $M$ is equipped with extended affine connection
$\Gamma$. Let's commutate space covariant differentiation
$\nabla_{\bold X}$ and velocity covariant differentiation
$\tilde\nabla_{\bold Y}$  with degenerate differentiation $\bold S$:
$$
\xalignat 2
&\quad [\nabla_{\bold X},\,\bold S]=\nabla_{\bold X}\bold S,
&&[\tilde\nabla_{\bold Y},\,\bold S]=\tilde\nabla_{\bold Y}\bold S.
\hskip -2em
\tag6.2
\endxalignat
$$
Formulas \thetag{6.2} show that such commutators are degenerate
differentiations defined by operator fields $\nabla_{\bold X}\bold S$
and $\tilde\nabla_{\bold Y}\bold S$ respectively. As a consequence
of \thetag{6.1} and \thetag{6.2} we get that the set of degenerate
differentiation $\goth S(M)$ is an ideal in $\goth D(M)$.\par
    Commutator of two  velocity covariant differentiations
$\tilde\nabla_{\bold X}$ and $\tilde\nabla_{\bold Y}$ is given by
the following rather simple formula
$$
[\tilde\nabla_{\bold X},\,\tilde\nabla_{\bold Y}]=\tilde
\nabla_{\bold U}\text{, \ where \ }\bold U=\tilde\nabla_{\bold X}
\bold Y-\tilde\nabla_{\bold Y}\bold X.\hskip -2em
\tag6.3
$$
In calculating the commutator $[\nabla_{\bold X},\,\tilde \nabla_{\bold Y}]$
we find some new extended tensor field $\bold D$ of the type $(1,3)$. Here
we have formula
$$
[\nabla_{\bold X},\,\tilde
\nabla_{\bold Y}]=\tilde\nabla_{\bold U}-\nabla_{\bold W}+
\bold D(\bold X,\bold Y)\hskip -2em
\tag6.4
$$
with $\bold U=\nabla_{\bold X}\bold Y-\bold D(\bold X,\bold Y)\,
\bold v$ and $\bold W=\tilde\nabla_{\bold Y}\bold X$. Components
of the tensor field $\bold D$ are expressed through the components
of connection $\Gamma$ by formula
$$
D^k_{rij}=-\frac{\partial\Gamma^k_{ir}}{\partial v^j}.\hskip -2em
\tag6.5
$$
By $\bold S=\bold D(\bold X,\bold Y)$ in thetag \thetag{6.4}
we denote tensor field of type $(1,1)$ constructed by $\bold D$,
$\bold X$, and $\bold Y$ as a result of contraction:
$\bold S=C(\bold D\otimes\bold X\otimes\bold Y)$. Being more precise,
components of the field $\bold S$ are expressed through
\thetag{6.5} as follows:
$$
S^k_r=\sum^n_{i=1}\sum^n_{j=1}D^k_{rij}\,X^i\,Y^j.\hskip -2em
\tag6.6
$$
Vector field $\bold D(\bold X,\bold Y)\,\bold v$ is obtained by
contracting operator field $\bold S$ from \thetag{6.6} with vector
field of velocity $\bold v$.\par
    Tensor field $\bold D$ with components given by \thetag{6.6} is
called the {\bf dynamic curvature tensor}. In calculating commutator
$[\nabla_{\bold X},\,\nabla_{\bold Y}]$ we find another curvature
tensor $\bold R$: 
$$
[\nabla_{\bold X},\,\nabla_{\bold Y}]=\nabla_{\bold U}-
\tilde\nabla_{\bold W}+\bold R(\bold X,\bold Y),\hskip -2em
\tag6.7
$$
Here $\bold U=\nabla_{\bold X}\bold Y-\nabla_{\bold Y}\bold X-
\bold T(\bold X,\bold Y)$ and  $\bold W=-\bold R(\bold X,\bold Y)
\,\bold v$. Components of curvature tensor $\bold R$  is given by
formula 
$$
\aligned
R^k_{rij}&=\frac{\partial\Gamma^k_{jr}}{\partial x^i}-
\frac{\partial\Gamma^k_{ir}}{\partial x^j}+\sum^n_{m=1}
\Gamma^k_{im}\,\Gamma^m_{jr}-\sum^n_{m=1}\Gamma^k_{jm}\,
\Gamma^m_{ir}-\\
\vspace{1ex}
&-\sum^n_{m=1}\sum^n_{h=1}v^m\,\Gamma^h_{im}\,\frac{\partial
\Gamma^k_{jr}}{\partial v^h}+\sum^n_{m=1}\sum^n_{h=1}v^m\,
\Gamma^h_{jm}\,\frac{\partial\Gamma^k_{ir}}{\partial v^h}.
\endaligned\hskip -1em
\tag6.8
$$
Extended tensor field $\bold T$ of type $(1,2)$ is called
the {\bf torsion field}. Components of torsion field $\bold T$
are calculated by formula
$$
T^k_{ij}=\Gamma^k_{ij}-\Gamma^k_{ji}.\hskip -2em
\tag6.9
$$
They are skew symmetric in pair of lower indices. Components of
$\bold Z=\bold T(\bold X,\bold Y)$ are expressed through \thetag{6.9}
as follows:
$$
Z^k=\sum^n_{i=1}\sum^n_{j=1}T^k_{ij}\,X^i\,Y^j.\hskip -2em
\tag6.10
$$
Operator field $\bold S=\bold R(\bold X,\bold Y)$ in \thetag{6.7}
is obtained by contracting $\bold X$ and $\bold Y$ with
curvature tensor: $\bold S=C(\bold R\otimes\bold X\otimes\bold Y)$. Its
components are given by formula
$$
S^k_r=\sum^n_{i=1}\sum^n_{j=1}R^k_{rij}\,X^i\,Y^j\hskip -3em
\tag6.11
$$
similar to \thetag{6.6} and \thetag{6.10}. Vector field $\bold R(\bold X,
\bold Y)\,\bold v$ in turn is obtained by contracting \thetag{6.11} and
the vector field of velocity
$\bold v$.
\head
\S\,7. Coordinate form of covariant differentiations.
\endhead
    Keeping in mind theorem~5.1 further we shall use only two types
of covariant differentiations in extended algebra of tensor fields.
These are space covariant differentiation $\nabla$ and velocity
covariant differentiation $\tilde\nabla$. Let $\bold X$ be a tensor
field of the type $(r,s)$. Derivatives $\nabla_{\bold Y}\bold X$ and
$\tilde\nabla_{\bold Y}\bold X$ are tensor fields of the same type.
They are $\goth F$-linear functions of $\bold Y$ (see definition~2.1
above). Therefore they can be expressed as the results of contracting
$\bold Y$ with tensor fields of the type $(r,s+1)$:
$$
\xalignat 2
&\quad\nabla_{\bold Y}\bold X=C(\bold Y\otimes\nabla\bold X),
&&\tilde\nabla_{\bold Y}\bold X=C(\bold Y\otimes\tilde
\nabla\bold X).\hskip -2em
\tag7.1
\endxalignat
$$
Fields $\nabla\bold X$ and $\tilde\nabla\bold X$ of type $(r,s+1)$
in \thetag{7.1} are called {\bf covariant differentials} of the field
$\bold X$: field $\nabla\bold X$ is a space covariant differential,
and field $\tilde\nabla\bold X$ is a velocity covariant differential.
According to what was said just above any covariant differentiation
could be understood as a map from $T^r_s(M)$ to $T^r_{s+1}(M)$.
However, definition~2.1 appears to be more convenient.\par
    Let $X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}$ be the components
of the field $\bold X$. For to numerate components of $\nabla\bold
X$ and $\tilde\nabla\bold X$ we need an extra index, say, the index
$m$. Components of the fields $\nabla\bold X$ and $\tilde\nabla\bold X$
are denoted as follows:
$$
\xalignat 2
&\quad\nabla_mX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s},
&&\tilde\nabla_mX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}.
\hskip -2em
\tag7.2
\endxalignat
$$
In \thetag{7.2} signs $\nabla$ and $\tilde\nabla$ are used to place
an extra index $m$ near them. In systematic work with tensors this
trick is permanently used, and therefore signs $\nabla_m$ and $\tilde
\nabla_m$ are understood as maps, that take the array of components
of the field $\bold X$ to arrays of components of the fields 
$\nabla\bold X$ and $\tilde\nabla\bold X$:
$$
\xalignat 2
&\nabla_m\!:X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}\to\nabla_mX^{i_1
\ldots\,i_r}_{j_1\ldots\,j_s},
&&\tilde\nabla_m\!:X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}\to
\tilde\nabla_mX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}.
\endxalignat
$$
These maps are called covariant derivatives. From \thetag{4.17} we
extract the following rule for calculating covariant derivative
$\nabla_m$:
$$
\aligned
&\nabla_mX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=\frac{\partial
X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial x^m}
-\sum^n_{a=1}\sum^n_{b=1}v^a\,\Gamma^b_{ma}\,\frac{\partial
X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial v^b}\,+\\
&+\sum^r_{k=1}\sum^n_{a_k=1}\!\Gamma^{i_k}_{m\,a_k}\,X^{i_1\ldots\,
a_k\ldots\,i_r}_{j_1\ldots\,\ldots\,\ldots\,j_s}
-\sum^s_{k=1}\sum^n_{b_k=1}\!\Gamma^{b_k}_{m\,j_k}
X^{i_1\ldots\,\ldots\,\ldots\,i_r}_{j_1\ldots\,b_k\ldots\,j_s}.
\endaligned\hskip -2em
\tag7.3
$$
The rule for calculating $\tilde\nabla_m$ is extracted from
\thetag{3.8}. It is much more simple:
$$
\tilde\nabla_mX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=\frac{\partial
X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial v^m}.
\hskip -2em
\tag7.4
$$\par
    Due to \thetag{7.4} covariant derivative $\tilde\nabla_m$ is
associated with $\partial/\partial v^m$. It is called covariant
derivative with respect to $m$-th velocity variable. In similar
way, due to \thetag{7.3} covariant derivative $\nabla_m$ is
associated with $\partial/\partial x^m$. It is called covariant
derivative with respect to $m$-th space variable $x^m$.
     Covariant derivatives $\tilde\nabla_m$ and $\nabla_m$
obey Leibniz rule. It is expressed by formulas 
$$
\pagebreak 
\aligned
\nabla_m(A^{\cdots}_{\cdots}\cdot B^{\,\cdots}_{\cdots})=
\nabla_mA^{\cdots}_{\cdots}\cdot B^{\,\cdots}_{\cdots}
+A^{\cdots}_{\cdots}\cdot\nabla_mB^{\,\cdots}_{\cdots},\\
\vspace{1ex}
\tilde\nabla_m(A^{\cdots}_{\cdots}\cdot B^{\,\cdots}_{\cdots})=
\tilde\nabla_mA^{\cdots}_{\cdots}\cdot B^{\,\cdots}_{\cdots}
+A^{\cdots}_{\cdots}\cdot\tilde\nabla_mB^{\,\cdots}_{\cdots}.
\endaligned\hskip -2em
\tag7.5
$$
Moreover, they commutate with the operation of contraction:
$$
\xalignat 2
&\qquad\sum^n_{k=1}\nabla_mA^{..k\ldots}_{\ldots k..}=\nabla_m\left(\,
\shave{\sum^n_{k=1}}A^{..k\ldots}_{\ldots k..}\right),
&&\sum^n_{k=1}\tilde\nabla_mA^{..k\ldots}_{\ldots k..}=\tilde\nabla_m
\left(\,\shave{\sum^n_{k=1}}A^{..k\ldots}_{\ldots k..}\right).
\hskip -2em
\tag7.6
\endxalignat
$$
From \thetag{6.3} we extract commutator of covariant derivatives
$\tilde\nabla_i$ and $\tilde\nabla_j$:
$$
[\tilde\nabla_i,\,\tilde\nabla_j]=0.\hskip -2em
\tag7.7
$$
The commutational relationship \thetag{6.4}, when applied to scalar and
vectorial fields $\varphi$ and $\bold X$, is written as follows:
$$
\aligned
&[\nabla_i,\,\tilde\nabla_j]\varphi=-\sum^n_{m=1}\sum^n_{k=1}
v^m\,D^k_{mij}\,\tilde\nabla_k\varphi,\\
&[\nabla_i,\,\tilde\nabla_j]X^r=-\sum^n_{m=1}\sum^n_{k=1}
v^m\,D^k_{mij}\,\tilde\nabla_kX^r+\sum^n_{m=1}D^r_{mij}\,X^m.
\endaligned\hskip -2em
\tag7.8
$$
For the case of covectorial field $\bold h$ from \thetag{7.5},
\thetag{7.6}, and  \thetag{7.8} we derive
$$
[\nabla_i,\,\tilde\nabla_j]h_r=-\sum^n_{m=1}
\sum^n_{k=1}v^m\,D^k_{mij}\,
\tilde\nabla_kh_r-\sum^n_{m=1}D^m_{rij}\,h_m.
\tag7.9
$$
The relationships \thetag{7.8} and \thetag{7.9} can be expanded for
the case of arbitrary tensor field of type $(r,s)$, if we remember
that $[\nabla_i,\,\tilde\nabla_j]$ acts as first order derivative.
\par
     From \thetag{6.7} we derive commutational relationships for
two spatial covariant derivatives. When applied a scalar field
$\varphi$ they look like
$$
[\nabla_i,\,\nabla_j]\varphi=-\sum^n_{m=1}\sum^n_{k=1}v^m\,R^k_{mij}
\,\tilde\nabla_k\varphi-\sum^n_{k=1}T^k_{ij}\,\nabla_k\varphi.
\hskip -3em
\tag7.10
$$
For the case of vectorial field $\bold X$ and covectorial field
$\bold h$ we have 
$$
\align
&\aligned
[\nabla_i,\,\nabla_j]X^r=-\sum^n_{m=1}&\sum^n_{k=1}v^m\,R^k_{mij}\,
\tilde\nabla_kX^r-\\
&-\sum^n_{k=1}T^k_{ij}\,\nabla_kX^r+\sum^n_{m=1}R^r_{mij}\,X^m,
\endaligned\hskip -3em
\tag7.11\\
&\aligned
[\nabla_i,\,\nabla_j]h_r=-\sum^n_{m=1}&\sum^n_{k=1}v^m\,R^k_{mij}\,
\tilde\nabla_kh_r-\\
&-\sum^n_{k=1}T^k_{ij}\,\nabla_kh_r-\sum^n_{m=1}R^m_{rij}\,h_m.
\endaligned\hskip -3em
\tag7.12
\endalign
$$
Further the relationships \thetag{7.10}, \thetag{7.11}, and \thetag{7.12}
can be expanded for the case of arbitrary tensor field of type $(r,s)$.
But we shall not write this formula.
\enddocument
\end

ediss-04.tex
\input amstex
\documentstyle{amsppt}
%
% Author:   Ruslan Abdulovich Sharipov
% Title:    Dynamical systems admitting the normal shift
% Comment:  Thesis for the degree of Doctor of Sciences in Russia
% Address:  Rabochaya street 5, Ufa, 450003, Russia
% E-mail:   R_Sharipov@ic.bashedu.ru
%           ruslan-sharipov@usa.net
% Web-page: http://www.geocities.com/CapeCanaveral/Lab/5341
%
\nopagenumbers
\catcode`@=11
\redefine\logo@{\relax}
\catcode`\@=\active
\def\startpage#1{\pageno=#1}
\pagewidth{12.8cm}
\pageheight{20cm}
\accentedsymbol\tx{\tilde x}
\def\msum{\operatornamewithlimits{\sum\!{\ssize\ldots}\!\sum}}
\Monograph
\topmatter
\title\chapter{4}
Tensor fields on curves and their differentiation.
\endtitle
\endtopmatter
\loadbold
\startpage{71}
\document
%\def\msum{\operatornamewithlimits{\sum\!{\ssize\ldots}\!\sum}}
\head
\S\,1. Vectorial and tensorial field on curves.
\endhead
     Let $M$ be real smooth manifold. By assigning to each point
$p\in M$ some tensor at this point we obtain a tensor field,
i\.~e\. tensor field is a tensor-valued function on $M$. By extending
the domain of such function up to a tangent bundle we come to the
concept of extended tensor field (see definition~5.3 in Chapter
\uppercase\expandafter{\romannumeral 2}). However, sometimes we need
to consider tensor-valued functions with very narrow domains. In the
theory of dynamical systems admitting the normal shift we consider
tensor fields on curves in $M$. Curves here arise in a natural way
as trajectories of Newtonian dynamical systems.\par
     Suppose that $p(t)$ is smooth parametric curves on the manifold
$M$. This is smooth map from some open interval $(a,b)$ on real
axis $\Bbb R$ to the manifold $M$. In geometry, considering
parametric curves, one often reparameterizes them, changing initial
parameter $t$ for some new parameter $t'=f(t)$, where $f$ is some
smooth monotonic function. Here we shall not yet consider the
reparametrization of curves, since in theory of Newtonian dynamical
systems we have canonical parameter on all trajectories. This is
time variable $t$.\par
     Let $p(t)$ be smooth parametric curve on the manifold $M$. Upon
choosing some local map $(U,x^1,\ldots,x^n)$ in $M$ it is represented
by numeric functions 
$$
x^1(t),\ \ldots,\ x^n(t).\hskip -2em
\tag1.1
$$
Let's differentiate these functions with respect to $t$. Their
time derivatives
$$
\dot x^1(t),\ \ldots,\ \dot x^n(t).\hskip -2em
\tag1.2
$$
are the components of tangent vector to this parametric curve at the point
$p(t)$. We denote it by $\bold v(t)=\dot p(t)$. Vector $\bold v(t)$ is
called {\bf velocity} vector of the point moving along the curve.\par
     In geometry the points, where $\bold v(t)=0$, are called {\bf singular}
point of the curve. At these points smoothness of the curve $p(t)$ could be
broken. In describing the motion of the point along the curve this is not
crucial, if the functions \thetag{1.1} are smooth. For instance, if we
consider a circle rolling along the straight line on the plane, each
point of the circle is moving along a cycloid, which has singular points.
\par
     Velocity vector $\bold v(t)$ is a simplest example of vector field
on the curve. It is defined only at the points of curve\footnote{It may
be defined on $TM$, when treated as an extended vector field, but it is not
a vector field on the manifold $M$ itself in the usual sense.}. Let $p(t)$
be a parametric curve on the manifold $M$.
\definition{Definition 1.1} If each value of parameter $t$ is associated
with some vector $\bold X(t)$ from the tangent space $T_p(M)$ to $M$ at
the point $p=p(t)$, then we say that the {\it vector field $\bold X$ on the
curve} $p(t)$ is defined.
\enddefinition
\definition{Definition 1.2} If each value of parameter $t$ is associated
with some tensor $\bold X(t)$ from tensor space $T^r_s(p,M)$ at the point
$p=p(t)$, then we say that the {\it tensor field $\bold X$ on the curve}
$p(t)$ is defined.
\enddefinition
    Note that if curve $p(t)$ has self intersection, i\.~e\. if $p(t_1)=
p(t_2)$, then the values of field $\bold X(t_1)$ and $\bold X(t_2)$
shouldn't coincide. This means that curve here is treated not only
as geometric set of points, but as an object with fixed parametrization,
where parameter is a time variable $t$.\par
    Vectorial and tensorial fields on curves can arise in many ways.
Let's consider one of them. Suppose that tensor field $\bold X$ on the
manifold $M$ is given, i\.~e\. at each point $p\in M$ we have a tensor
$\bold X(p)$. Then for the points of curve we take $\bold X(t)=\bold
X(p(t))$. This yields a tensor field on the curve $p(t)$. Such field
is called the {\bf restriction} of tensor field $\bold X$ from $M$ to
the curve.
\definition{Definition 1.3} Tensor field $\bold X(t)$ of type $(r,s)$
on the curve $p(t)\in M$ is called {\bf smooth field}, if for any
local map $U$ on $M$ its components $X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}
(t)$ in this map are smooth functions of parameter $t$.
\enddefinition
    Its evident, that by restricting smooth tensor field $\bold X$ from
$M$ to the curve we get smooth tensor field on this curve. This fact is
easily proved by considering the components of $\bold X$ in local maps on
$M$.
\head
\S\,2. Lift of curves to tangent bundle.
\endhead
    Suppose that $\bold X(t)$ is some smooth vector field on the curve
$p(t)\in M$. Pair $q(t)=(p(t),\bold X(t))$ is a point of tangent bundle
depending on $t$. Such pairs define smooth parametric curve $q(t)$ in
tangent bundle $TM$. This curve is called the {\bf lift} of curve $p(t)$
from $M$ to $TM$. Canonical projection $\pi\!:TM\to M$ maps it back to the
initial curve $p(t)$. Curve $p(t)$ has infinitely many lifts to tangent
bundle, but one of them is especially important.
\definition{Definition 2.1} Lift $q(t)=(p(t),\bold v(t))$ of curve
$p(t)$ defined by vector field of velocity $\bold v(t)=\dot p(t)$ is called
{\bf natural lift} of this curve from $M$ to $TM$.
\enddefinition
     Consider some tensor field $\bold Y$ from extended algebra $\bold T(M)$.
It can't be restricted to the curve $p(t)\in M$ immediately, since tensor
$\bold Y$ at the point $p$ for such field is defined not by $p$, but by
the point $q$ in the fiber $\pi^{-1}(p)$ of tangent bundle over the point
$p$ (see definition~5.3 in Chapter \uppercase\expandafter{\romannumeral 2}).
\pagebreak
However, if we have the lift of curve $q(t)=(p(t),\bold X(t))$, then we can
restrict $\bold Y$ to such lift:
$$
\bold Y(t)=\bold Y(q(t))=\bold Y(p(t),\bold X(t)).\hskip -2em
\tag2.1
$$
Tensor field $\bold Y(t)$ on the curve $p(t)$ defined by \thetag{2.1}
is called the {\bf restriction} of extended tensor field $\bold Y$ to
the curve $p(t)$ {\bf due to its lift $q(t)=(p(t),\bold X(t))$}.
\definition{Definition 2.2} If in restriction \thetag{2.1} we use
natural lift of curve $p(t)$ given by vector field of velocity
$\bold v(t)$, then such restriction is called {\bf natural restriction}
of extended tensor field $\bold Y$ to the curve $p(t)$ on $M$.
\enddefinition
    Velocity vector $\bold v$ can be treated as extended vector field
on $M$. If we take $\bold Y=\bold v$, then natural restriction of
$\bold Y$ to the curve $p(t)$ on $M$ coincides with vector field
of velocity $\bold v(t)=\dot p(t)$.
\head
\S\,3. Differentiation of tensor fields\\with respect to the parameter
along the curves.
\endhead
\rightheadtext{\S\,3. Differentiation of tensor fields \dots}
     Let $p(t)$ be smooth parametric curve on real smooth manifold $M$.
Denote by $T^r_s(t)$ the set of smooth tensor fields  of type $(r,s)$ on
this curve and take 
$$
\bold T(t)=\bigoplus^\infty_{r=0}\bigoplus^\infty_{s=0}T^r_s(t).
\hskip -2em
\tag3.1
$$
The set $T^0_0(t)$ is a ring of smooth functions with numeric argument $t$.
Denote it by $\goth F(t)$. Each set $T^r_s(t)$ possess the structure of
module over the ring $\goth F(t)$. Their direct sum $\bold T(t)$ in
\thetag{3.1} is an algebra over this ring. It is called the {\bf algebra
of tensor fields} on the curve $p(t)$.
\definition{Definition 3.1} The map $D\!:\bold T(t)\to\bold T(t)$ is called
the {\bf differentiation} in algebra of tensor fields on the
curve $p(t)$, if the following conditions are fulfilled:
\roster
\rosteritemwd=10pt
\item concordance with grading: $D(T^r_s(t))\subset T^r_s(t)$;
\item $\Bbb R$-linearity: $D(\bold X+\bold Y)=D(\bold X)+D(\bold Y)$
      and $D(\lambda\bold X)=\lambda D(\bold X)$ for $\lambda\in\Bbb R$;
\item commutation with contractions: $D(C(\bold X))=C(D(\bold X))$;
\item Leibniz rule: $D(\bold X\otimes\bold Y)=D(\bold X)
      \otimes\bold Y+\bold X\otimes D(\bold Y)$.
\endroster
\enddefinition
    Theory of differentiations in algebra $\bold T(t)$ is analogous to
the theory of differentiations for extended tensor fields, though it is
substantially more simple. Suppose that $D$ is a differentiation in
algebra $\bold T(t)$. Denote by $\delta$ and $\xi$ the restrictions
of $D$ to $T^0_0(t)$ and $T^1_0(t)$ respectively:
$$
\xalignat 2
&\quad\delta\!:T^0_0(t)\to T^0_0(t), &&\xi\!:T^1_0(t)\to T^1_0(t).
\hskip -2em
\tag3.2
\endxalignat
$$
The map $\delta$ is a differentiation in the ring $\goth F(t)$:
$$
\pagebreak
\delta(\varphi\cdot\psi)=\delta(\varphi)\cdot\psi+\varphi\cdot
\delta(\psi).\hskip -2em
\tag3.3
$$
Another map $\xi$ is bound with $\delta$ by the following relationship:
$$
\xi(\varphi\cdot\bold X)=\delta(\varphi)\cdot\bold X+
\varphi\cdot\xi(\bold X).\hskip -2em
\tag3.4
$$
any differentiation \thetag{3.3} in the ring of smooth functions
$\goth F(t)$ is given by formula 
$$
\delta=f(t)\,\frac{d}{dt},\hskip -2em
\tag3.5
$$
\i\.~e\., in main, it reduces to the operator of differentiation
with respect to time variable $t$. For non-restricted operator $D$
in whole we have the following propositions similar to lemma~1.2
and theorem~1.3 in Chapter \uppercase\expandafter{\romannumeral 3}.
\proclaim{Lemma 3.1} If two tensor fields $\bold X(t)$ and $\bold Y(t)$
on the curve $p(t)$ coincide with each other in some neighborhood of
some point $t=t_0$, then tensor fields $D(\bold X)$ and $D(\bold Y)$
have equal values for $t=t_0$.
\endproclaim
\proclaim{Theorem 3.1} Defining the differentiation $D$ in the algebra
of tensor fields $\bold T(t)$ on the curve is equivalent to defining
the function $f(t)$ in \thetag{3.5} and the $\Bbb R$-map $\xi\!:T^1_0(t)
\to T^1_0(t)$ satisfying the relationships \thetag{3.4}. 
\endproclaim
    If we take $f(t)=0$ in \thetag{3.5}, then we get the definition
degenerate differentiation.
\definition{Definition 3.2} Differentiation $D$ in the algebra of tensor
fields $\bold T(t)$ is called {\bf degenerate differentiation}, if its
restriction $\delta$ to subspace $T^0_0(t)$ is identically zero.
\enddefinition
    Any degenerate differentiation $D$ in $\bold T(t)$ is defined by some
tensor field $\bold S(t)$ of the type $(1,1)$. This can be done my means
of formulas similar to formulas \thetag{1.8} and \thetag{1.9} in Chapter
\uppercase\expandafter{\romannumeral 3}.
\definition{Definition 3.3} Differentiation $D$ in the algebra of tensor
fields $\bold T(t)$ is called {\bf covariant differentiation} with respect
to parameter $t$ along the curve $p(t)\in M$, if function $f(t)$ in
\thetag{3.5} for it is identically equal to unity: $f(t)=1$.
\enddefinition
    Covariant differentiation with respect to parameter $t$ along the
curve is denoted by $\nabla_t$. Condition $f(t)=1$ determines the
restriction of $\nabla_t$ to $T^0_0(t)$. In order to describe restriction
of $\nabla_t$ to $T^1_0(t)$ let's choose some local map $(U,x^1,\ldots,
x^n)$ and consider coordinate vector fields $\bold E_1,\,\ldots,\,\bold
E_n$ in this map (see \thetag{4.3} in Chapter
\uppercase\expandafter{\romannumeral 3}). Upon restricting them to the
curve we get the fields $\bold E_1(t),\,\ldots,\,\bold E_n(t)$. Let's
apply $\nabla_t$ to $\bold E_j(t)$ and expand resulting field in the
base of vector fields $\bold E_1(t),\,\ldots,\,\bold E_n(t)$:
$$
\nabla_t\bold E_j=\sum^n_{k=1}\Gamma^k_j(t)\,\bold E_k.\hskip -2em
\tag3.6
$$
Coefficients $\Gamma^k_j(t)$ in \thetag{3.6} play the same role as the
components of affine connection in formula \thetag{4.10} from Chapter
\uppercase\expandafter{\romannumeral 3}). In transferring from one local
map to another they are transformed according to the following rule:
$$
\Gamma^k_{j}=\sum^n_{m=1}\sum^n_{a=1} S^k_m\,T^a_j\,\tilde\Gamma^m_{a}
+\sum^n_{m=1} S^k_m\,\frac{dT^m_j}{dt}.\hskip -3em
\tag3.7
$$
Components of transition matrices $S$ and $T$ here are function of $t$,
since their natural arguments $x^1\,,\ldots,\,x^n$ and $\tx^1\,,\ldots,
\,\tx^n$ depend on $t$ due to \thetag{1.1}. Therefore
$$
\frac{dT^m_j}{dt}=\sum^n_{i=1}\frac{\partial T^m_j}{\partial x^i}
\,\dot x^i=\sum^n_{i=1}\frac{\partial T^m_i}{\partial x^j}\,\dot x^i.
\hskip -2em
\tag3.8
$$
Time derivatives $\dot x^1,\,\ldots,\,\dot x^n$ in \thetag{3.8} are
the components of velocity vector $\bold v(t)$ on the curve. Substituting
\thetag{3.8} into \thetag{3.7}, now we obtain
$$
\Gamma^k_{j}=\sum^n_{m=1}\sum^n_{a=1} S^k_m\,T^a_j\,\tilde\Gamma^m_{a}
+\sum^n_{m=1}\sum^n_{i=1} v^i\,S^k_m\,\frac{\partial T^m_i}{\partial x^j}.
\hskip -2em
\tag3.9
$$
Let's compare \thetag{3.9} with formula \thetag{4.11} in Chapter
\uppercase\expandafter{\romannumeral 3}). As a result of this comparison
we find that quantities $\Gamma^k_j$ in \thetag{3.6} could be defined
by formula
$$
\Gamma^k_{j}=\sum^n_{i=1}v^i\,\Gamma^k_{ij}(x^1,\ldots,x^n,v^1,\ldots,
v^n),\hskip -3em
\tag3.10
$$
where $\Gamma^k_{ij}$ are components of some extended connection in the
manifold $M$, and their arguments  $x^1,\,\ldots,\,x^n,v^1,\,\ldots,\,v^n$
depend on $t$ due to the natural lift of curve $p(t)$ from $M$ to $TM$.
\proclaim{Theorem 3.2} Any extended affine connection $\Gamma$ on the
manifold $M$ defines some covariant differentiation $\nabla_t$ of tensor
fields with respect to parameter $t$ on curves.
\endproclaim
\demo{Proof} In the set of scalar fields we define differentiation
$\nabla_t$ to be coincident with  ordinary  differentiation  with 
respect
to parameter $t$:
$$
\nabla_t\varphi=\frac{d\varphi}{dt}.\hskip -3em
\tag3.11
$$
This is in accordance with definition~3.3. Then we calculate parameters
$\Gamma^k_i$ by formula \thetag{3.10}, and for vector field $\bold X$
with components $X^1,\,\ldots,\,X^n$ in the base of coordinate vectors
$\bold E_1(t),\,\ldots,\,\bold E_n(t)$ we set
$$
\nabla_t\bold X=\sum^n_{k=1}\left(\frac{dX^k}{dt}+\sum^n_{i=1}
\Gamma^k_i\,X^i\right)\bold E_k.\hskip -3em
\tag3.12
$$
Formulas \thetag{3.11} and \thetag{3.12} define two maps $\delta$
and $\xi$ of the form \thetag{3.2}. Due to the theorem~3.1 these
maps completely define the differentiation $\nabla_t$. As for the
relationships \thetag{3.3} and \thetag{3.4}, one can easily check
them to be fulfilled for the maps $\delta$ and $\xi$ defined by
\thetag{3.11} and \thetag{3.12}.\qed\enddemo
    Let $\bold X(t)$ be a tensor field with components $X^{i_1
\ldots\,i_r}_{j_1\ldots\,j_s}$ on  the curve $p(t)$. Let's apply the
differentiation $\nabla_t$, which was defined just above, to the field
$\bold X(t)$. And denote by $\nabla_tX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}$
components of resulting tensor field $\nabla_t\bold X(t)$. Relying on
\thetag{3.11} and \thetag{3.12} we can derive the following formula
for these components
$$
\align
\nabla_t&X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=
\frac{dX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{dt}+\\
\vspace{1ex}
&+\sum^r_{k=1}\sum^n_{m=1}\sum^n_{a_k=1}\dot x^m\,\Gamma^{i_k}_{m\,a_k}
\,X^{i_1\ldots\,a_k\ldots\,i_r}_{j_1\ldots\,\ldots\,\ldots\,j_s}-
\hskip -3em
\tag3.13\\
\vspace{1ex}
&-\sum^s_{k=1}\sum^n_{m=1}\sum^n_{b_k=1}\dot x^m\,\Gamma^{b_k}_{m\,j_k}
X^{i_1\ldots\,\ldots\,\ldots\,i_r}_{j_1\ldots\,b_k\ldots\,j_s}.
\hskip -3em
\endalign
$$
Formula \thetag{3.13} is an analog of formula \thetag{4.17} in Chapter
\uppercase\expandafter{\romannumeral 3}. The sign $\nabla_t$ further will
be used in two cases. For the tensor field $\bold X(t)$ by $\nabla_t\bold
X(t)$ we shall denote the result of differentiating it with respect to
parameter $t$ along the curve. In local coordinates by $\nabla_tX^{i_1
\ldots\,i_r}_{j_1\ldots\,j_s}$ we shall denote the array of components
for the field $\nabla_t\bold X(t)$. Therefore in second case $\nabla_t$
is understood as an operation transforming one array into another array.
For this operation we have the following relationships:
$$
\gather
\nabla_t(A^{\cdots}_{\cdots}\cdot B^{\,\cdots}_{\cdots})=
\nabla_tA^{\cdots}_{\cdots}\cdot B^{\,\cdots}_{\cdots}
+A^{\cdots}_{\cdots}\cdot\nabla_tB^{\,\cdots}_{\cdots},
\hskip -2em
\tag3.14\\
\vspace{1ex}
\sum^n_{k=1}\nabla_tA^{..k\ldots}_{\ldots k..}=\nabla_t\left(\,
\shave{\sum^n_{k=1}}A^{..k\ldots}_{\ldots k..}\right).
\hskip -2em
\tag3.15
\endgather
$$
These relationships \thetag{3.14} and \thetag{3.15} are the analogs of
the relationships \thetag{7.5} and \thetag{7.6} in Chapter
\uppercase\expandafter{\romannumeral 3}.\par
     As a final result in this section we formulate the structural theorem
for differentiations in the algebra of tensor fields $\bold T(t)$.
\proclaim{Theorem 3.3} Let $M$ be real smooth manifold equipped with
extended affine connection $\Gamma$, and let $p(t)$ be smooth parametric
curve on $M$. Then any differentiation in the algebra of tensor fields
on this curve can be expressed as a sum
$$
D=f(t)\cdot\nabla_t+\bold S,\hskip -2em
\tag3.16
$$
where $f(t)$ is some smooth function on the curve $p(t)$, and
$\bold S$ is some degenerate differentiation defined by tensor field
$\bold S(t)$ of the type $(1,1)$.
\endproclaim
\noindent Theorem~3.3 is an analog of theorem~5.1 in Chapter
\uppercase\expandafter{\romannumeral 3}, and the expansion
\thetag{3.16} is analogous to the expansion \thetag{5.1}
in Chapter \uppercase\expandafter{\romannumeral 3}.
\head
\S\,4. Vector of acceleration. Differentiation of extended tensor
fields along the curves.
\endhead
\rightheadtext{\S\,4. Vector of acceleration.}
    Suppose that $M$ is a real smooth manifold with extended affine
connection $\Gamma$. In this situation we have two covariant
differentiations $\nabla$ and $\tilde\nabla$ in extended algebra of
tensor fields $\bold T(M)$ and differentiation $\tilde\nabla$ with
respect to parameter $t$ on curves. Let $p(t)$ be smooth parametric
curve on $M$ with parameter $t$ understood as a time variable.
Then we can apply $\nabla_t$ to the vector field of velocity
$\bold v(t)=\dot p(t)$ on this curve. As a result we get vector field
$\bold a(t)=\nabla_t\bold v$ with the following components:
$$
a^k=\ddot x^k+\sum^n_{i=1}\sum^n_{j=1}\Gamma^k_{ij}\,\dot x^i\,\dot x^j
\text{, \ where \ }k=1,\,\ldots,\,n.\hskip -2em
\tag4.1
$$
Vector $\bold a(t)$ with components \thetag{4.1} is called the
{\bf acceleration vector} for the point moving on $M$ along the
curve $p(t)$. This terminology is in concordance with the results
of Chapter \uppercase\expandafter{\romannumeral 2}, where we considered
constrained mechanical systems.\par
    Now let $\bold X$ be some extended tensor field on the manifold $M$.
It has natural restriction $\bold X(t)$ to the curve $p(t)$ due to the
natural lift of this curve from $M$ to $TM$ (see definition~2.1 and 
definition~2.2 in \S\,2 above). Let's apply $\nabla_t$ to the field
$\bold X(t)$ and let's calculate components of resulting field
$\nabla_t\bold X(t)$ by means of formula \thetag{3.13}.
To find time derivative of $X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}$ in
formula \thetag{3.13} we should differentiate composite function of
the following form:
$$
X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=X^{i_1\ldots\,i_r}_{j_1\ldots
\,j_s}(x^1(t),\ldots,x^n(t),\dot x^1(t),\ldots,\dot x^n(t)).
$$
Upon differentiating this composite function let's substitute the
resulting expression into \thetag{3.13} and rearrange terms in this formula
$$
\align
\nabla_t&X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=
\sum^n_{m=1}\dot x^m\left(\frac{\partial X^{i_1\ldots\,i_r}_{j_1
\ldots\,j_s}}{\partial x^m}+\shave{\sum^r_{k=1}\sum^n_{a_k=1}}
\Gamma^{i_k}_{m\,a_k}\,X^{i_1\ldots\,a_k\ldots\,i_r}_{j_1\ldots\,
\ldots\,\ldots\,j_s}\,-\right.\\
\vspace{1ex}
&\left.-\sum^s_{k=1}\shave{\sum^n_{b_k=1}}\Gamma^{b_k}_{m\,j_k}
X^{i_1\ldots\,\ldots\,\ldots\,i_r}_{j_1\ldots\,b_k\ldots\,j_s}\right)+
\sum^n_{m=1}\ddot x^m\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots
\,j_s}}{\partial v^m}.
\endalign
$$
Now compare the expression in round brackets here with the formula
\thetag{7.3} from Chapter \uppercase\expandafter{\romannumeral 3}.
The difference is only in one term
$$
-\sum^n_{a=1}\sum^n_{b=1}\dot x^a\,\Gamma^b_{ma}\frac{\partial X^{i_1
\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial v^b}.
$$
Let's add and then subtract this term. After opening brackets and
redesignating some summation indices we get
$$
\align
\nabla_t&X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=
\sum^n_{m=1}\dot x^m\,\nabla_mX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}+\\
&+\sum^n_{m=1}\left(\ddot x^m
+\shave{\sum^n_{a=1}\sum^n_{b=1}}\Gamma^m_{ab}\,\dot x^a\,\dot x^b
\right)\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial v^m}.
\endalign
$$
Let's compare this expression with formula \thetag{4.1} \pagebreak for
the components of acceleration vector, and with formula \thetag{7.4}
in Chapter \uppercase\expandafter{\romannumeral 3}. This yields
$$
\nabla_tX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=\sum^n_{m=1}v^m\,\nabla_m
X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}+\sum^n_{m=1}a^m\,\tilde\nabla_mX^{i_1
\ldots\,i_r}_{j_1\ldots\,j_s}.\hskip -2em
\tag4.2
$$
Formula \thetag{4.2} is the rule for differentiating natural restriction
of extended tensor field $\bold X$ with respect to parameter $t$ along
the curve. In coordinate-free form this rule is written as follows:
$$
\nabla_t\bold X=C(\bold v(t)\otimes\nabla\bold X)+
C(\bold a(t)\otimes\tilde\nabla\bold X).\hskip -2em
\tag4.3
$$
If we take into account the relationships \thetag{7.1} from Chapter
\uppercase\expandafter{\romannumeral 3}, then formula \thetag{4.3}
can be brought to the following equality:
$$
\nabla_t\bold X=\nabla_{\bold v}\bold X+\tilde\nabla_{\bold a}\bold X.
\hskip -2em
\tag4.4
$$
The relationships \thetag{4.2}, \thetag{4.3}, and \thetag{4.4} can be
understood as particular subcase of the rule of differentiating composite
functions written in covariant form and applied to tensor fields.
\head
\S\,5. Deformation of curves on manifold. Vector of variation.
\endhead
\rightheadtext{\S\,5. Deformation of curves on manifold.}
    Let $p(t)$ be smooth parametric curve on real smooth manifold $M$.
In order to describe the deformation of this curve lets consider the
function of two numeric arguments $p(t,u)$ with values being point
of manifold $M$. Let's denote
$$
p(t)=p(t,u)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,u=0}.
\hskip -2em
\tag5.1
$$
If condition \thetag{5.1} is fulfilled, we say that $p(t,u)$ defines
the {\bf deformation} of curve $p(t)$ on the manifold $M$. Parameter
$t$ is main parameter. Further it will be understood as time variable.
Parameter $u$ is auxiliary  parameter. It is called the {\bf parameter
of deformation}.\par
    Let's choose some local map  $(U,x^1,\ldots,x^n)$ on the manifold
$M$. Deformations of curve $p(t)$ in local map $U$ is expressed by the
functions
$$
x^1(t,u),\ \ldots,\ x^n(t,u).\hskip -2em
\tag5.2
$$
We say that $p(t,u)$ is a smooth deformation of the curve $p(t)$ if
it is expressed by smooth functions \thetag{5.2} in any local map.\par
    In case of smooth deformation $p(t,u)$ we can differentiate functions
\thetag{5.2} with respect to the auxiliary  parameter $u$. Derivatives
$$
\tau^i(t)=\frac{\partial x^i(t,u)}{\partial u}\,\hbox{\vrule height 14pt
depth 8pt width 0.5pt}_{\,u=0}\text{, \ where \ }i=1,\,\ldots,\,n,
\hskip -2em
\tag5.3
$$
are the components of some vector $\boldsymbol\tau(t)$ on the curve
$p(t)$. This vector is called the {\bf vector of variation}. Its
components define first order terms in Taylor series expansions for
the functions \thetag{5.2} in auxiliary variable $u$ about the point
$u=0$:
$$
\align
&x^1(t,u)=x^1(t)+\tau^1(t)\cdot u+O(u^2),\hskip -2em\\
&\,.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
.\ .\ .\ \hskip -2em\tag5.4\\
&x^n(t,u)=x^n(t)+\tau^n(t)\cdot u+O(u^2).\hskip -2em
\endalign
$$
Usually in considering deformations of curves only starting terms
in the series expansions \thetag{5.4} appear to be of importance.
In this case all results are formulated in terms of variation
vector $\boldsymbol\tau(t)$.\par
    Now suppose that manifold $M$ is equipped with extended affine
connection $\Gamma$, and suppose that smooth deformation of curve
$p(t,u)$ on $M$ is given. This means that for each fixed value of
$u$ we have some smooth curve with parameter $t$. Let's calculate
velocity vector $\bold v(t,u)=\dot p(t,u)$ on each such curve and
define its natural lift $q(t,u)=(p(t,u),\bold v(t,u))$ to $TM$. This
construction is called {\bf natural lift for deformation of curves
$p(t,u)$}. Some asymmetry between main and auxiliary parameters $t$
and $u$ is prebuilt into this construction.\par
    Suppose that for each fixed $u$ on each curve $p(t,u)$ the
tensor field $\bold X(t,u)$ of type $(r,s)$ is defined. If $p(t)=
p(t,u)$ and $\bold X(t)=\bold X(t,0)$, then we say that the
{\bf deformation} of tensor field $\bold X(t)$ is given. This
deformation is called {\bf smooth}, if it is represented by
smooth tensor-valued function $\bold X(t)$ of two numeric
arguments.\par
   Let $\bold X(t,u)$ be smooth deformation of tensor field
corresponding to the smooth deformation of the curve $p(t,u)$.
Having fixed the parameter $u$, we can apply covariant
differentiation $\nabla_t$ defined in \S\,3 to $\bold X(t,u)$.
However, this is not enough. In what follows we sometimes have
to differentiate $\bold X(t,u)$ with respect to its second
parameter $u$ for fixed $t$. Therefore we define covariant
differentiation with respect to parameter of deformation. For
the deformation of scalar field $\varphi(t,u)$ we set
$$
\nabla_u\varphi=\frac{d\varphi}{du}.\hskip -2em
\tag5.5
$$
According to the results of \S\,3, in order to expand \thetag{5.5}
up to a covariant differentiation with respect to parameter $u$
we should associate with each local map on $M$ the set of
functions $\Gamma^k_j=\Gamma^k_j(t,u)$. Under the change of local maps
on $M$ these function should obey the transformation rules
$$
\Gamma^k_{j}=\sum^n_{m=1}\sum^n_{a=1} S^k_m\,T^a_j\,\tilde\Gamma^m_{a}
+\sum^n_{m=1} S^k_m\,\frac{dT^m_j}{du}\hskip -2em
\tag5.6
$$
similar to \thetag{3.7}. Let's define them by the following formula:
$$
\Gamma^k_{j}=\sum^n_{i=1}\tau^i\,\Gamma^k_{ij}(x^1,\ldots,x^n,v^1,\ldots,
v^n).\hskip -2em
\tag5.7
$$
Here $\tau^1,\,\ldots,\,\tau^n$ are the components \pagebreak of variation
vector $\boldsymbol\tau(t,u)$, and $v^1,\,\ldots,\,v^n$ are the components of
velocity vector $\bold v(t,u)$. It is not difficult to check up that
the relationships \thetag{5.6} for the quantities \thetag{5.7} are
fulfilled.   Note   that   the   quantities   \thetag{5.7}    are 
intentionally 
chosen in a slightly different way than the quantities \thetag{3.10}.
This is due to different roles of the variables $t$ and $u$.\par
    Differentiation $\nabla_u$ defined by extended affine connection
$\Gamma$ due to \thetag{5.5} and \thetag{5.7} is called {\bf covariant
differentiation with respect to the parameter of deformation}. For the
components of the tensor field $\nabla_u\bold X(t,u)$ we have formula
analogous to formula \thetag{3.13} for $\nabla_t\bold X(t,u)$:
$$
\align
\nabla_u&X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=
\frac{dX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{du}+\\
\vspace{1ex}
&+\sum^r_{k=1}\sum^n_{m=1}\sum^n_{a_k=1}\tau^m\,\Gamma^{i_k}_{m\,a_k}
\,X^{i_1\ldots\,a_k\ldots\,i_r}_{j_1\ldots\,\ldots\,\ldots\,j_s}-
\hskip -2em
\tag5.8\\
\vspace{1ex}
&-\sum^s_{k=1}\sum^n_{m=1}\sum^n_{b_k=1}\tau^m\,\Gamma^{b_k}_{m\,j_k}
X^{i_1\ldots\,\ldots\,\ldots\,i_r}_{j_1\ldots\,b_k\ldots\,j_s}.
\hskip -2em
\endalign
$$
For the fields corresponding to deformations of curves the derivative
in $u$ is calculated for the fixed value of $t$, and time derivative
in \thetag{3.13} is calculated for the fixed value of $u$. These are
partial derivatives. However, in \thetag{5.8} we used the symbol of
ordinary derivative in order to keep similarity of formulas
\thetag{3.13} and \thetag{5.8}.\par
    The map $\nabla_u\!:\bold T(t,u)\to \bold T(t,u)$ is the
differentiation of tensor fields on deformations of curves (in
the sense of definition analogous to the definition~3.1). Therefore
in local coordinates  we have the relationships 
$$
\gather
\nabla_u(A^{\cdots}_{\cdots}\cdot B^{\,\cdots}_{\cdots})=
\nabla_uA^{\cdots}_{\cdots}\cdot B^{\,\cdots}_{\cdots}
+A^{\cdots}_{\cdots}\cdot\nabla_uB^{\,\cdots}_{\cdots},
\hskip -2em
\tag5.9\\
\vspace{1ex}
\sum^n_{k=1}\nabla_uA^{..k\ldots}_{\ldots k..}=\nabla_u\left(\,
\shave{\sum^n_{k=1}}A^{..k\ldots}_{\ldots k..}\right)
\hskip -2em
\tag5.10
\endgather
$$
similar to formulas \thetag{3.14} and \thetag{3.15}. For the components
of vectors $\bold v(t,u)$ and $\boldsymbol\tau(t,u)$ we have the obvious
equality
$$
\frac{\partial\tau^k}{\partial t}=\frac{\partial^2x^k}{\partial t\,
\partial u}=\frac{\partial^2x^k}{\partial u\,\partial t}=
\frac{\partial v^k}{\partial u},\hskip -3em
\tag5.11
$$
which is the consequence of permutability of partial derivatives in
$u$ and $t$. Written in covariant derivatives $\nabla_u$ and and
$\nabla_t$,  the  equality  \thetag{5.11}  takes   less   trivial 
appearance:
$$
\nabla_t\tau^k-\nabla_uv^k=-\sum^n_{i=1}\sum^n_{j=1}
T^k_{ij}\,\tau^i\,v^j.\hskip -3em
\tag5.12
$$
And finally, in coordinate-free form \thetag{5.12} is written
as follows:
$$
\nabla_t\boldsymbol\tau-\nabla_u\bold v=-\bold T(\boldsymbol\tau,
\bold v).\hskip -3em
\tag5.13
$$
Tensor field $\bold T$ in \thetag{5.13} is a tensor field of
torsion, its components $T^k_{ij}$ are calculated by formula
\thetag{6.8} from Chapter \uppercase\expandafter{\romannumeral 3}.
\par
    Now  let's  find  formula   for   commutator   of   covariant 
derivatives 
$\nabla_u$ and $\nabla_t$. It is very important for further
development of the theory. For $[\nabla_u,\,\nabla_t]$ we get 
$$
[\nabla_u,\,\nabla_t]=\bold S,\hskip -3em
\tag5.14
$$
where $\bold S$ is degenerate differentiation defined by tensor field
$\bold S$ of type $(1,1)$:
$$
\bold S=\bold R(\boldsymbol\tau,\bold v)+\bold D(\boldsymbol\tau,
\bold a)-\bold D(\bold v,\nabla_t\boldsymbol\tau)-
\bold D(\bold v,\bold T(\boldsymbol\tau,\bold v)).\hskip -3em
\tag5.15
$$
Here $\bold T$ is a tensor field of torsion, while $\bold D$ and
$\bold R$ are tensor fields of curvature. Components of $\bold D$
and $\bold R$ are calculated by formulas \thetag{6.5} and \thetag{6.8}
from Chapter \uppercase\expandafter{\romannumeral 3}. Vector $\bold a=
\nabla_t\bold v$ in formula \thetag{5.15} is a vector of acceleration
with components calculated by formula \thetag{4.1}.\par
    Degeneracy of differentiation $\bold S$ in right hand side
of \thetag{5.14} means that commutator of derivatives $\nabla_u$
and $\nabla_t$ is zero when applied to a scalar field
$\varphi(t,u)$:
$$
[\nabla_u,\,\nabla_t]\varphi=0\hskip -3em
\tag5.16
$$
Let's apply commutator \thetag{5.14} to vector field $\bold X(t,u)$
and let's write the result in local coordinates. Taking into account
\thetag{5.15}, we get
$$
\aligned
[&\nabla_u,\,\nabla_t]X^k=-\sum^n_{r=1}\left(\shave{\sum^n_{i=1}
\sum^n_{j=1}\sum^n_{m=0}\sum^n_{h=0}}v^m\,T^h_{im}\,D^k_{rjh}\,
\tau^i\,v^j\right.+\\
\vspace{1ex}
&+\left.\shave{\sum^n_{i=1}\sum^n_{j=1}}\left(D^k_{rij}\,v^i\,
\nabla_t\tau^j-R^k_{rij}\,\tau^i\,v^j-D^k_{rij}\,\tau^i\,a^j\right)
\right)X^r.
\endaligned
\tag5.17
$$
Now let's apply commutator \thetag{5.14} to covector field $\bold h(t,u)$.
The result is written as 
$$
\aligned
[&\nabla_u,\,\nabla_t]h_k=\sum^n_{r=1}\left(\shave{\sum^n_{i=1}
\sum^n_{j=1}\sum^n_{m=0}\sum^n_{h=0}}v^m\,T^h_{im}\,D^r_{kjh}\,
\tau^i\,v^j\right.+\\
\vspace{1ex}
&+\left.\shave{\sum^n_{i=1}\sum^n_{j=1}}\left(D^r_{kij}\,v^i\,
\nabla_t\tau^j-R^r_{kij}\,\tau^i\,v^j-D^r_{kij}\,\tau^i\,a^j\right)
\right)h_r.
\endaligned
\tag5.18
$$
Formulas \thetag{5.16}, \thetag{5.17}, and \thetag{5.18} are analogs
of formulas \thetag{7.10}, \thetag{7.11}, and \thetag{7.12}
from Chapter \uppercase\expandafter{\romannumeral 3}.\par
    Let $\bold X$ be extended tensor field on the manifold $M$, and
let $p(t,u)$ be the deformation of curves on this manifolds. For each
fixed $u$ we have parametric curve $p(t,u)$ with parameter $t$. Natural
lift of such curve, as we mentioned above, define the deformation of
curves $q(u,t)$ on tangent bundle $TM$. This construction, which is called
{\bf natural lift} for $p(t,u)$, can be used for restricting extended
tensor fields to the deformation of curves on $M$. For the field
$\bold X(t,u)$ obtained as a result of such {\bf natural restriction}
we have the formula
$$
\nabla_u\bold X=C(\boldsymbol\tau\otimes\nabla\bold X)+
C(\nabla_t\boldsymbol\tau\otimes\tilde\nabla\bold X)+
C(\bold T(\boldsymbol\tau,\bold v)\otimes\tilde\nabla\bold X),
\hskip -2em
\tag5.19
$$
being analog of formula \thetag{4.3}. By denoting $\bold b=\nabla_t
\boldsymbol\tau+\bold T(\boldsymbol\tau,\bold v)$ we can write 
formula \thetag{5.19} in the form similar to \thetag{4.4}:
$$
\nabla_u\bold X=\nabla_{\boldsymbol\tau}\bold X+
\tilde\nabla_{\bold b}\bold X.\hskip -3em
\tag5.20
$$
Being written in local coordinates, formula \thetag{5.20} looks like
$$
\nabla_uX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=\sum^n_{m=1}\tau^m\,\nabla_m
X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}+\sum^n_{m=1}b^m\,\tilde\nabla_mX^{i_1
\ldots\,i_r}_{j_1\ldots\,j_s}.\hskip -3em
\tag5.21
$$
It's easier first to prove formula \thetag{5.21} by direct calculations
in local coordinates, then back step to \thetag{5.20} is obvious.
\enddocument
\end

ediss-05.tex
\input amstex
\documentstyle{amsppt}
%
% Author:   Ruslan Abdulovich Sharipov
% Title:    Dynamical systems admitting the normal shift
% Comment:  Thesis for the degree of Doctor of Sciences in Russia
% Address:  Rabochaya street 5, Ufa, 450003, Russia
% E-mail:   R_Sharipov@ic.bashedu.ru
%           ruslan-sharipov@usa.net
% Web-page: http://www.geocities.com/CapeCanaveral/Lab/5341
%
\nopagenumbers
\catcode`@=11
\redefine\logo@{\relax}
\catcode`\@=\active
\def\startpage#1{\pageno=#1}
\pagewidth{12.8cm}
\pageheight{20cm}
\def\tr{\operatorname{tr}}
\def\const{\operatorname{const}}
\def\compos{\,\raise 1pt\hbox{$\sssize\circ$} \,}
\accentedsymbol\DotTildePhi{\Dot{\Tilde\varphi}}
\accentedsymbol\tx{\tilde x}
\Monograph
\topmatter
\title\chapter{5}
Normal shift in Riemannian manifolds.
\endtitle
\endtopmatter
\loadbold
\startpage{83}
\document
\head
\S\,1. Geodesic normal shift.
\endhead
     In this Chapter we return to the case of Riemannian manifolds
and derive main results of thesis. They consist in developing the
theory of {\bf Newtonian dynamical systems admitting the normal
shift} on such manifolds. As a starting point for this theory we
choose well-known classical construction of {\bf geodesic normal
shift}.\par
     Let $M$ be Riemannian manifold with metric tensor $\bold g$.
Tensor field $\bold g$ defines scalar product of tangent vectors
on the manifold $M$. Let's denote it as
$$
\bold g(\bold X,\bold Y)=(\bold X\,|\,\bold Y).\hskip -2em
\tag1.1
$$
Here $\bold X$ and $\bold Y$ are two vectors at one point $p\in M$.
Metric $\bold g$ defines metric connection $\Gamma$ with identically
zero torsion field $\bold T=0$. Its components in local map
$(U,x^1,\ldots,x^n)$ are determined by components of metric tensor
$\bold g$ according to the following standard formula (see \cite{12},
\cite{32}, \cite{76}, or \cite{77}):
$$
\Gamma^k_{ij}=\frac{1}{2}\sum^n_{s=1}g^{ks}\left(\frac{\partial g_{sj}}
{\partial x^i}+\frac{\partial g_{is}}{\partial x^j}-\frac{\partial g_{ij}}
{\partial x^s}\right),\hskip -2em
\tag1.2
$$
Quantities $g_{ij}$ and $\Gamma^k_{ij}$ in \thetag{1.2} do not depend
on the components of velocity vector  $v^1,\,\ldots,\,v^n$, i\.~e\. we
have ordinary (not extended) tensor field $\bold g$ and ordinary (not
extended) connection $\Gamma$. Applying velocity gradient $\tilde\nabla$
to the field $\bold g$, we get zero, while space gradient $\nabla$
applied to $\bold g$ coincides with traditional covariant differential of
this field. Therefore we have the equalities 
$$
\xalignat 2
&\nabla_\bold g=0,&&\tilde\nabla\bold g=0.\hskip -2em
\tag1.3
\endxalignat
$$
First equality in \thetag{1.2} is standard (see \cite{12}, \cite{32},
\cite{76}, or \cite{77}). It expresses the {\bf concordance of} metric
and connection).\par
    {\bf Geodesic lines} play important role in geometry. In local
coordinates  the are defined by ordinary differential equations
$$
\qquad\ddot x^k+\sum^n_{i=1}\sum^n_{j=1}\Gamma^k_{ij}\,\dot x^i\,
\dot x^j=0\text{, \ \ where\ \ \ }k=1,\,\ldots,\,n,
\hskip -2em
\tag1.4
$$
provided special parametrization is used, where $t$ is a length
of the curve measured from some fixed reference point on it. Let's
write the equations \thetag{1.4} in terms of covariant differentiation
with respect to parameter $t$ along the curve (such covariant
differentiations were considered in Chapter
\uppercase\expandafter{\romannumeral 4}):
$$
\xalignat 2
&\dot x^k=v^k,&&\nabla_tv^k=0.\hskip -2em
\tag1.5
\endxalignat
$$
The equations \thetag{1.5} form the system of $2n$ ordinary
differential equations of the first order. In setting up Cauchy
problem for these equations initial values of local coordinates
$x^1,\,\ldots,\,x^n$ and components of velocity vector
$v^1,\,\ldots,\,v^n$ are given:
$$
\xalignat 2
&\quad x^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=x^k(0),
&&v^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=v^k(0).
\hskip -2em
\tag1.6
\endxalignat
$$
This means that we fix initial point $p=p(0)$ and initial vector
of velocity $\bold v(0)$ at the point $p(0)$. Note that equations
\thetag{1.4} are invariant under the following transformation of
independent variable: $t\to\const\cdot\,t$. Therefore without loss
of generality we can assume the initial vector $\bold v(0)$ in
\thetag{1.6} to be of unit length: $|\bold v(0)|=1$. Once fulfilled
for $t=0$ this condition remains fulfilled for nonzero $t$ as well:
$$
|\bold v(t)|=1.\hskip -2em
\tag1.7
$$
This is the very case when parameter $t$ is a length of curve
measured from some fixed reference point, it is called {\bf natural
parameter} on geodesic line.\par
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6.0cm 6.8cm 6.0cm 6.8cm 6.0cm 6.8cm 6.0cm 6.8cm 6.0cm 6.8cm
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     Let $S$ be some hypersurface in $M$. For the sake of simplicity
$S$ assumed to be connected and simply connected smooth submanifold
of codimension $1$ without boundary such that the closure of $S$
is compact. \vadjust{\vskip -1.5cm\special{em:graph edis-05a.gif}
\vskip 1.5cm}Moreover, we assume that $S$ is compactly imbedded into
some other open hypersurface, i\.~e\. $S\Subset S'$. This strong
requirement eliminates most difficulties that could arise on the
boundary $\partial S$. Under the above assumptions hypersurface $S$
is orientable, and one can choose smooth field of unitary normal
vectors on it, which has smooth continuation to any point on the
boundary $\partial S$. This field defines unitary normal vector
$\bold n(p)$ at each point $p\in S$. Let's choose components of
$\bold n(p)$ as the components of initial velocity in setting up
initial data for Cauchy problem \thetag{1.6}. Then the condition
$|\bold v(0)|=1$ is fulfilled due to $|\bold n(p)|=1$. Let's write
\thetag{1.6} as
$$
\pagebreak
\xalignat 2
&\quad x^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=x^k(p),
&&v^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=n^k(p).
\hskip -2em
\tag1.8
\endxalignat
$$
Solution of the equations \thetag{1.5} satisfying initial data \thetag{1.8}
defines the family of geodesic lines beginning at the points of hypersurface
$S$ and being perpendicular to $S$ at that points (See Fig\. 1.1). The value
of parameter $t$ on these lines coincides with their length measured from
starting points on $S$.\par
     Let $p_0$ be a pint on the hypersurface $S$, and let $U$ be some map
on $S$ covering this point. Suppose that $u^1,\,\ldots,\,u^{n-1}$ are local
coordinates of the point $p$ in the map $U$. Then geodesic lines obtained
as the solution of Cauchy problem \thetag{1.8} for the system of equations
\thetag{1.5} are represented by the set of $n$ functions
$$
\align
x^1&=x^1(u^1,\ldots,u^{n-1},t),\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .
\hskip -2em\tag1.9\\
x^n&=x^n(u^1,\ldots,u^{n-1},t),\hskip -2em
\endalign
$$
where $x^1,\,\ldots,\,x^n$ are local coordinates in $M$. According
to well-known ``{existence and uniqueness}'' theorem (see \cite{82} or
\cite{83}) the domain of functions \thetag{1.9} is such that they
define the map $U'\times I_\delta\to M$, where $U'$ is an open
subset of local map $U$ containing the point $p_0$, and $I_\delta=
(-\delta,+\delta)$ is open $\delta$-neighborhood of zero on real
axis $\Bbb R$. Due to the compactness of the closure of $S$, or, being
more exact, due to $S\Subset S'$ local maps $U'\times I_\delta\to M$
can be glued into one global map $S\times I_\varepsilon\to M$.\par
     Let's differentiate the functions \thetag{1.9} with respect to
parameters $u^1,\,\ldots,\,u^{n-1}$, and let's form the following
vectors by the obtained derivatives:
$$
\bold K_i=\sum^n_{j=1}\frac{\partial x^j}{\partial u^i}\,
\frac{\partial}{\partial x^j}\text{, \ where \ }i=1,\,\ldots,\,n-1.
\hskip -3em
\tag1.10
$$
Differentiating the functions \thetag{1.9} with respect to their last
argument $t$, we form one more vector $\bold N$. It is defined by
the formula similar to \thetag{1.10}:
$$
\bold N=\sum^n_{j=1}\frac{\partial x^j}{\partial t}\,
\frac{\partial}{\partial x^j}.\hskip -3em
\tag1.11
$$
For $t=0$ vectors \thetag{1.10} coincide with coordinate tangent
vectors to $S$, while vector \thetag{1.11} coincides with normal
vector of $S$. Therefore vectors $\bold K_1,\,\ldots,\,\bold K_{n-1},
\,\bold N$ are linearly independent for $t=0$. If we denote $t=u^n$,
and if we build Jacoby matrix
$$
J=\Vmatrix
\dsize\frac{\partial x^1}{\partial u^1} &\hdots
&\dsize\frac{\partial x^1}{\partial u^n}\\
\vspace{1ex}
\vdots & \ddots & \vdots\\
\vspace{2ex}
\dsize\frac{\partial x^n}{\partial u^1} &\hdots
&\dsize\frac{\partial x^n}{\partial u^n}
\endVmatrix,
$$
it will be non-degenerate at the point $p_0$. This means, that the
variables $u^1,\,\ldots,\,u^n$ (where $u^n=t$) could be used as local
coordinates on $M$ in some neighborhood of the point $p_0$. These
coordinates are called {\bf semigeodesic coordinates} (see \cite{**}).
Hypersurface $S$ in these coordinates is described by the equation
$u^n=0$.\par
    Let's consider the equation $u^n=t=\const$ in semigeodesic coordinates.
It determines a piece of some hypersurface. Due to $S\Subset S'$ we can
find $\varepsilon>0$ such that for $t\in I_\varepsilon$ various pieces
of hypersurfaces $u^n=t$ are glued into an open hypersurface $S_t$
diffeomorphic to $S$. At the expense of choosing sufficiently small value
of $\varepsilon$ we can reach the situation, when for $t\in I_\varepsilon$
none of hypersurfaces $S_t$ will intersect another, nor will it have
self-intersection. Therefore $S$ has some neighborhood in $M$ diffeomorphic
to $S\times I_\varepsilon$, and for each $t\in I_\varepsilon$ there is
a diffeomorphism $f_t\!:\,S\to S_t$. Such diffeomorphism built by geodesic
lines as described above is called {\bf geodesic shift} of hypersurface
$S$ to the distance $t$. It's important  to  emphasize  that  any 
hypersurface
$S_t$ has compact closure, and there exists some open hypersurface $S'_t$
such that $S_t\Subset S'_t$.\par
    Let's study the geodesic shift of hypersurface $S$ in semigeodesic
coordinates $x^1=u^1,\,\ldots,\,x^n=u^n$ associated with it. In this case
$x^n=t$ is a parameter of shift, and vectors \thetag{1.10} are coordinate
tangent vectors to hypersurfaces $S_t$ described by the equations $x^n=t
=\const$. Due to $x^j=u^j$ they are the followings:
$$
\bold K_1=\frac{\partial}{\partial x^1},\ \ldots,\ \bold K_{n-1}
=\frac{\partial}{\partial x^{n-1}}.\hskip -3em
\tag1.12
$$
Geodesic lines implementing geodesic shift of hypersurface $x^n=0$
in semigeodesic coordinates are described by functions
$$
x^1(t)=\const,\ \ldots,\ x^{n-1}(t)=\const,\ x^n(t)=t.\hskip -3em
\tag1.13
$$
Let's substitute \thetag{1.13} into the equations of geodesic lines
\thetag{1.4}. This yields
$$
\Gamma^k_{nn}=0\text{\ \ for all \ }k=1,\,\ldots,\,n.\hskip -3em
\tag1.14
$$
Having calculated connection components $\Gamma^k_{nn}$ by formula
\thetag{1.2}, due to non-degene\-racy of the matrix of metric tensor
$g^{ks}$ from \thetag{1.14} we derive the relationships 
$$
2\,\frac{\partial g_{sn}}{\partial x^n}-\frac{\partial g_{nn}}
{\partial x^s}=0\text{\ \ for all \ }s=1,\,\ldots,\,n.\hskip -3em
\tag1.15
$$
Particularly for $s=n$ from \thetag{1.15} we extract the following
equation:
$$
\frac{\partial g_{nn}}{\partial x^n}=0.\hskip -3em
\tag1.16
$$
Thus, diagonal component $g_{nn}$ in metric tensor doesn't depend on
shift parameter $x^n=t$; and $g_{nn}$ is the square of the length of
$n$-th coordinate vector, \pagebreak which in semigeodesic coordinates
coincides with vector \thetag{1.11}. The latter one plays the role of
velocity vector on geodesic lines used to shift $S$:
$$
\bold K_n=\bold N=\bold v(t)=\frac{\partial}{\partial x^n}.
\hskip -3em
\tag1.17
$$
For $x^n=t=0$ velocity vector $\bold v(t)$ coincides with unitary
normal vector to $S$ (see \thetag{1.8}). Hence $g_{nn}=1$ for
$x^n=0$. From this equality combined with \thetag{1.16} we derive
$$
g_{nn}\equiv 1.\hskip -3em
\tag1.18
$$
In particular, \thetag{1.18} could be the proof for the equality
\thetag{1.7} for the velocity vector on geodesic lines. Substituting
\thetag{1.18} into \thetag{1.15} for $s\neq n$, we
immediately get
$$
\frac{\partial g_{sn}}{\partial x^n}=0\text{\ \ for \ }s=1,\,\ldots,
\,n-1.\hskip -3em
\tag1.19
$$
Components $g_{sn}$ in metric tensor are the scalar products of vectors
\thetag{1.12} and \thetag{1.17}:
$$
g_{sn}=(\bold K_s\,|\,\bold K_n).\hskip -3em
\tag1.20
$$
At the initial instant, when $x^n=t=0$, scalar products \thetag{1.20}
are equal to zero, since $\bold K_n$ coincides with unitary normal
vector on $S$, while vectors \thetag{1.12} are tangent to $S$. From
this fact combined with the equations \thetag{1.19} we get
$$
g_{sn}\equiv 0\text{\ \ for all \ }s=1,\,\ldots,
\,n-1.\hskip -3em
\tag1.21
$$
The equalities \thetag{1.21} reflect very important geometric property
of geodesic shift. They mean that not only the initial
hypersurface $S$ is perpendicular to geodesic lines used to shift it,
but all other hypersurfaces $S_t$ generated by shift procedure are
perpendicular as well. Therefore geodesic shift is a {\bf normal shift}
of hypersurfaces.
\head
\S\,2. Normal shift along trajectories of dynamical system.
\endhead
     Geodesic normal shift described in \S\,1 is a well-known classical
construction. The {\bf main idea} that brought about the advent of the
theory of {\bf dynamical systems admitting the normal shift} consists in
replacing geodesic lines by {\bf wider class of parametric curves}.
As a first pretender to this role we considered the trajectories of
Newtonian dynamical systems, since
\roster
\rosteritemwd=5pt
\item they possess prescribed parametrization with the parameter having
      physical meaning of time;
\item they are associated with positively definite quadratic form of
      kinetic energy that equips configuration space with the structure
      of Riemannian manifold;
\item in local coordinates they are described by systems of differential
      equations compatible with Cauchy problems of the form \thetag{1.6}.
\endroster
     Let's consider Newtonian dynamical system with force field $\bold F$,
configuration space of which is a Riemannian manifold $M$ with metric
$\bold g$. Let's choose some local map $(U,x^1,\ldots,x^n)$ on $M$ and
canonically associated local coordinates on tangent bundle $TM$
(see \S\,6 in Chapter \uppercase\expandafter{\romannumeral 2}). Then
trajectories of Newtonian dynamical system with the field $\bold F$
can be constructed by solving the system of differential equations
\thetag{4.1} from Chapter \uppercase\expandafter{\romannumeral 2}.
By analogy with \thetag{1.5} these equations could be written in terms
of covariant derivative with respect to parameter $t$:
$$
\xalignat 2
&\dot x^k=v^k,&&\nabla_tv^k=F^k.\hskip -2em
\tag2.1
\endxalignat
$$
\definition{Definition 2.1} We say that hypersurface $S$ in $M$
belongs to {\bf transformation class}, if it is connected and simply
connected open hypersurface with compact closure such that
$S\Subset S'$ for some other open hypersurface $S'$.
\enddefinition
    Note that closed hypersurface without boundary belongs to
transformation class, if it is connected, simply connected, and
compact, since for closed hypersurface $\overline{S}=S$. Here
we can take $S'=S$.\par
    Suppose that $S\subset M$ is a hypersurface from transformation class.
Let's fix the field of unitary normal vectors $\bold n(p)$ on $S$, and
let's set up the following initial data for the system of differential
equations \thetag{2.1} at the points of $S$:
$$
\xalignat 2
&\quad x^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=x^k(p),
&&v^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=
\nu(p)\cdot n^k(p).\hskip -2em
\tag2.2
\endxalignat
$$
Solution of Cauchy problem with initial data \thetag{2.2} determines
the family of trajectories of Newtonian dynamical system \thetag{2.1}
beginning at the points of $S$ and outgoing from $S$ along normal
vector of this hypersurface (see Fig\. 1.1). The value of $\nu(p)$
determines modulus of velocity vector at the initial point $p$ on such
trajectory. In the case of geodesic normal shift the value of $\nu(p)$
is chosen to be equal to unity everywhere on $S$. Here we shall not
keep this restriction, we shall require only that $\nu(p)\neq 0$ for
all points $p\in S$.
\definition{Definition 2.2} Let $S\subset M$ be a hypersurface from
transformation class. We say that smooth function $\nu(p)$ on $S$
belongs to {\bf transformation class}, if it has smooth expansion
to some bigger open hypersurface $S'$ such that $S\Subset S'$, and if
it is nonzero everywhere on the closure $\overline{S}$.
\enddefinition
    Suppose that on the hypersurface $S$ belonging to the transformation
class on $M$ we defined some function $\nu(p)$ belonging to the
transformation class on $S$. For each point $p=p_0$ on $S$ we draw the
trajectory of dynamical system \thetag{2.1} passing through $p_0$ and
satisfying initial data \thetag{2.2} at this point. Then consider the
point $p(t)=p(p_0,t)$ on such trajectory corresponding to the value $t$
of time variable, and associate it with $p_0$. So we have constructed
a transformation of the hypersurface $S$ that maps $p_0$ to $p(t)$. It
is called the {\bf shift along trajectories of dynamical system} for
the time $t$.\par
     In the case of shift along trajectories of Newtonian dynamical
system, as well as in the case of geodesic shift, we can point out
some interval $I_\varepsilon$ for time variable $t$ such that the
map $S\times I_\varepsilon\to M$ is injective and has no singular
points. This map stratifies some full-dimensional neighborhood of $S$
into a family of hypersurfaces $S_t$ diffeomorphic to $S$: for any
$t\in I_\varepsilon$ we have diffeomorphism $f_t\!:\,S\to S_t$.
For $t\in I_\varepsilon$ hypersurfaces $S_t$ all are in transformation
class, they have no self-intersection points and they do not intersect
each other.
\definition{Definition  2.3}  Shift  of  hypersurface  $S$  along 
trajectories
of Newtonian dynamical system \thetag{2.1} initiated by the function
$\nu(p)$ in Cauchy problem \thetag{2.2} is called {\bf normal shift}, if
for parameter $t$ in some interval $I_\varepsilon=(-\varepsilon,
+\varepsilon)$ all hypersurfaces $S_t$ are orthogonal to the trajectories
used to shift $S$.
\enddefinition
\noindent The definition~2.3 is quite necessary, since, in contrast to
geodesic normal shift, here the orthogonality of $S_t$ and trajectories
is not an unconditional property of shift. Presence (or absence) of this
property essentially depends on how we deal with arbitrariness in the
choice of the function $\nu(p)$ on $S$ and in the choice of force field
$\bold F$ for dynamical system.
\head
\S\, 3. Differential equations for the vector of variation.
\endhead
     Let's choose some local map $(U,x^1,\ldots,x^n)$ on $M$ and consider
the shift of some hypersurface $S$ along trajectories of Newtonian
dynamical system \thetag{2.1}. If $u^1,\,\ldots,\,u^{n-1}$ are local
coordinates on $S$, then trajectories of dynamical system \thetag{2.1}
beginning at the points of this hypersurface are expressed by functions
$$
\align
x^1&=x^1(u^1,\ldots,u^{n-1},t),\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .
\hskip -2em\tag3.1\\
x^n&=x^n(u^1,\ldots,u^{n-1},t).\hskip -2em
\endalign
$$
Parameter $t$ in \thetag{3.1} is the main parameter, it has physical
meaning of time, other parameters $u^1,\,\ldots,\,u^{n-1}$ are auxiliary
ones. If we choose one of auxiliary parameters $u^i$ and fix all others,
then we get the situation considered in \S\,5 of Chapter
\uppercase\expandafter{\romannumeral 4}. In this situation partial
derivatives
$$
\frac{\partial x^1}{\partial u^i},\ \ldots,\ \frac{\partial x^n}
{\partial u^i}\hskip -2em
\tag3.2
$$
are the components of the vector of variation. We denote it by $\boldsymbol
\tau_i$. On the other hand, these derivatives are the components of the
vector $\bold K_i$ from \thetag{1.10} as well. Thus, on the trajectories
of dynamical system \thetag{2.1} used in the construction of shift we have
several vectors of variation simultaneously:
$$
\boldsymbol\tau_1=\bold K_1,\ \ldots,\ \boldsymbol\tau_{n-1}=\bold
K_{n-1}.\hskip -2em
\tag3.3
$$
From geometrical point of view vectors \thetag{3.3} are
interpreted as coordinate tangent vectors on hypersurface $S_t$ in
local coordinates $u^1,\,\ldots,\,u^{n-1}$ induced to $S_t$ from $S$
by means of shift diffeomorphism $f_t\!:\,S\to S_t$.\par
     Let's study time dependence of the vectors \thetag{3.3}, i\.~e\.
their evolution as we move along trajectories of the system
\thetag{2.1}. Components of any of them appear to satisfy the same
differential equations, which can be obtained by linearization of
the equations \thetag{2.1}. In deriving these equations we denote
$u^i=u$ and $\bold K_i=\boldsymbol\tau_i=\boldsymbol\tau$ for to
simplify further calculations. The equations \thetag{2.1} determine
not only the functions $x^i(u^1,\ldots,u^{n-1},t)$ in \thetag{3.1},
but the functions
$$
\align
v^1&=v^1(u^1,\ldots,u^{n-1},t),\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .
\hskip -2em\tag3.4\\
v^n&=v^n(u^1,\ldots,u^{n-1},t)\hskip -2em
\endalign
$$
as well. Therefore we have natural lift of trajectories to the
tangent bundle $TM$ (see definition~2.1 in Chapter
\uppercase\expandafter{\romannumeral 4}). Let's apply the operator
$\nabla_u$ of covariant differentiation with respect to parameter
$u$ (see \S\,5 in Chapter \uppercase\expandafter{\romannumeral 4})
to both sides of second part of the equations \thetag{2.1}. The
result can be written as a vectorial equality
$$
\nabla_u\nabla_t\bold v=\nabla_u(\bold F).\hskip -2em
\tag3.5
$$
In order to transform left hand side of \thetag{3.5} we transpose
covariant differentiations $\nabla_u$ and $\nabla_t$ by means of
formula \thetag{5.14} from Chapter \uppercase\expandafter{\romannumeral
4}:
$$
\nabla_u\nabla_t\bold v=[\nabla_u,\nabla_t]\bold v+\nabla_t\nabla_u
\bold v=\bold S\bold v+\nabla_t\nabla_u\bold v.\hskip -2em
\tag3.6
$$
Operator field $\bold S$ in \thetag{3.6} is determined by formula
\thetag{5.1} from Chapter \uppercase\expandafter{\romannumeral 4}.
Applying this formula, we take into account that dynamic curvature
field $\bold D$ and torsion field $T$ for metric connection
\thetag{1.2} are zero. Therefore
$$
\nabla_u\nabla_t\bold v=\nabla_t\nabla_u\bold v+\bold R(\boldsymbol
\tau,\bold v)\bold v.\hskip -2em
\tag3.7
$$
In order to calculate $\nabla_u\bold v$ we use formula \thetag{5.13}
from Chapter \uppercase\expandafter{\romannumeral 4} and remember that
torsion is zero. Then from \thetag{3.7} we derive
$$
\nabla_u\nabla_t\bold v=\nabla_{tt}\boldsymbol\tau+\bold R(\boldsymbol
\tau,\bold v)\bold v.\hskip -2em
\tag3.8
$$
In order to transform right hand side of equality \thetag{3.5} we use
formula \thetag{5.19} or formula \thetag{5.20} from Chapter
\uppercase\expandafter{\romannumeral 4} and remember that $\bold T=0$:
$$
\nabla_u(\bold F)=C(\boldsymbol\tau\otimes\nabla\bold F)+
C(\nabla_t\boldsymbol\tau\otimes\tilde\nabla\bold F).
\tag3.9
$$
Now, equating the expressions \thetag{3.8} and \thetag{3.9}, we
obtain vectorial differential equation for the vector of variation
$\boldsymbol\tau$:
$$
\pagebreak
\nabla_{tt}\boldsymbol\tau=C(\nabla_t\boldsymbol\tau\otimes
\tilde\nabla\bold F)+C(\boldsymbol\tau\otimes\nabla\bold F)
-\bold R(\boldsymbol\tau,\bold v)\bold v.\hskip -3em
\tag3.10
$$
Written in terms of components in some local map, \thetag{3.10}
looks like 
$$
\nabla_{tt}\tau^k=-\sum^n_{m=1}\sum^n_{i=1}\sum^n_{j=1}R^k_{mij}\,
\tau^i\,v^j\,v^m+\sum^n_{m=1}\nabla_t\tau^m\,\tilde\nabla_mF^k
+\sum^n_{m=1}\tau^m\,\nabla_mF^k.
\hskip -2em
\tag3.11
$$
From \thetag{3.11} we conclude that components of variation vector
$\boldsymbol\tau$ satisfy the system of $n$ linear ordinary equation
of second order respective to time variable $t$. Coefficients in these
linear equations are depending on $t$. They are determined by the force
field $\bold F$ of dynamical system, and they depend on the choice of its
particular trajectory, i\.~e\. they depend on the functions \thetag{3.1}
and \thetag{3.4}.
\head
\S\,4. Function of deviation and its derivatives.
\endhead
     As a vector of variation $\boldsymbol\tau$ in the equations
\thetag{3.10} we can choose any one of vectors $\boldsymbol\tau_1,\,
\ldots,\,\boldsymbol\tau_{n-1}$ from \thetag{3.3}. In the case, when
$f_t\!:\,S\to S_t$ is a normal shift, they all are perpendicular to
the vector $\bold v(t)$, which is tangent to the trajectory. Therefore
as a measure of deviation from normality we use the following scalar
products:
$$
\varphi_i=(\boldsymbol\tau_i\,|\,\bold v)\text{, \ \ where \ \ \ }
i=1,\,\ldots,\,n-1.\hskip -2em
\tag4.1
$$
In the situation of normal shift all these functions are identically
zero for all trajectories of shift.
\definition{Definition 4.1} Scalar product of velocity vector
$\bold v(t)$ with the vector of variation $\boldsymbol\tau(t)$
is called the function of {\bf deviation} on the trajectory of
Newtonian dynamical system.
\enddefinition
    Suppose that $\varphi(t)$ is a function of deviation on some
particular trajectory of Newtonian dynamical system:
$$
\varphi=(\bold v\,|\,\boldsymbol\tau).\hskip -2em
\tag4.2
$$
Let's calculate two time derivatives of this function $\dot\varphi(t)$
and $\ddot\varphi(t)$. For the first derivative of the function
\thetag{4.2} we have
$$
\dot\varphi=\nabla_t\varphi=\nabla_t(\bold v\,|\,\boldsymbol\tau)=
(\nabla_t\bold v\,|\,\boldsymbol\tau)+(\bold v\,|\,\nabla_t\boldsymbol
\tau)\hskip -2em
\tag4.3
$$
Differentiating scalar product $(\bold v\,|\,\boldsymbol\tau)$ in
\thetag{4.3}, we used Leibniz rule. This is correct, since we
can do the following more detailed calculations
$$
\gathered
\nabla_t(\bold v\,|\,\boldsymbol\tau)=\nabla_tC(\bold g\otimes\bold v
\otimes\boldsymbol\tau)=C(\nabla_t\bold g\otimes\bold v\otimes\boldsymbol
\tau)+C(\bold g\otimes\nabla_t\bold v\otimes\boldsymbol\tau)\,+\\
\vspace{1ex}
+\,C(\bold g\otimes\bold v\otimes\nabla_t\boldsymbol\tau)=(\nabla_t\bold
v\,|\,\boldsymbol\tau)+(\bold v\,|\,\nabla_t\boldsymbol\tau)
+C(\nabla_t\bold g\otimes\bold v\otimes\boldsymbol\tau).
\endgathered\hskip -2em
\tag4.4
$$
Covariant derivative $\nabla_t\bold g$ in \thetag{4.4} is equal to zero.
\pagebreak
This follows from the formula \thetag{4.3} in Chapter
\uppercase\expandafter{\romannumeral 4} and the relationships \thetag{1.3}
that express concordance of metric and connection. Indeed, we have
$$
\nabla_t\bold g=C(\bold v\otimes\nabla\bold g)+C(\nabla_t\bold v\otimes
\tilde\nabla\bold g)=0.\hskip -2em
\tag4.5
$$
For the further transformation of \thetag{4.3} we use the equations
of dynamics \thetag{2.1}, second part of which can be written in vectorial
form:
$$
\nabla_t\bold v=\bold F.\hskip -2em
\tag4.6
$$
Substituting \thetag{4.6} into the formula \thetag{4.3}, for the
derivative $\dot\varphi$ we get
$$
\dot\varphi=(\bold F\,|\,\boldsymbol\tau)+(\bold v\,|\,
\nabla_t\boldsymbol\tau).\hskip -2em
\tag4.7
$$\par
    Now let's calculate second derivative $\ddot\varphi$. In order to
do it let's differentiate \thetag{4.7}, taking into account the
equation \thetag{4.6} thereby:
$$
\ddot\varphi=\nabla_t\dot\varphi=(\nabla_t\bold F\,|\,\boldsymbol\tau)
+2\,(\bold F\,|\,\nabla_t\boldsymbol\tau)+(\bold v\,|\,\nabla_{tt}
\boldsymbol\tau).\hskip -2em
\tag4.8
$$
For to calculate $\nabla_t\bold F$ we apply formula \thetag{4.3}
from Chapter \uppercase\expandafter{\romannumeral 4}:
$$
\nabla_t\bold F=C(\bold v\otimes\nabla\bold F)+
C(\bold F\otimes\tilde\nabla\bold F).\hskip -2em
\tag4.9
$$
Second derivative of variation vector $\nabla_{tt}\boldsymbol\tau$
is expressed through $\boldsymbol\tau$ and through first derivative
$\nabla_t\boldsymbol\tau$ by means of the equation \thetag{3.10}.
If additionally we take into account \thetag{4.9}, then for the
derivative $\ddot\varphi$ we get
$$
\gathered
\ddot\varphi=(C(\bold v\otimes\nabla\bold F)\,|\,\boldsymbol\tau)+
(C(\bold F\otimes\tilde\nabla\bold F)\,|\,\boldsymbol\tau)
+2\,(\bold F\,|\,\nabla_t\boldsymbol\tau)\,+\\
\vspace{1ex}
+\,(\bold v\,|\,C(\boldsymbol\tau\otimes\nabla\bold F))
+(\bold v\,|\,C(\nabla_t\boldsymbol\tau\otimes\tilde\nabla\bold F))
-(\bold v\,|\,\bold R(\boldsymbol\tau,\bold v)\bold v).
\endgathered\hskip -2em
\tag4.10
$$
Last term in \thetag{4.10} is equal to zero. This follows from some
properties of curvature tensor of metric connection \thetag{1.2}.
Indeed, let's write the expression $(\bold v\,|\,\bold R(\boldsymbol
\tau,\bold v)\bold v)$ in local coordinates. Here we have
$$
(\bold v\,|\,\bold R(\boldsymbol\tau,\bold v)\bold v)=
\sum^n_{k=1}\sum^n_{m=1}\sum^n_{i=1}\sum^n_{j=1}R_{kmij}\,
v^k\,\tau^i\,v^j\,v^m=0.\hskip -3em
\tag4.11
$$
Sum in \thetag{4.11} vanishes due to skew symmetry of $R_{kmij}$
with respect to first two indices $k$ and $m$ (see \cite{12}, \cite{32},
\cite{76}, or \cite{77}). Therefore
$$
\gathered
\ddot\varphi=(C(\bold v\otimes\nabla\bold F)\,|\,\boldsymbol\tau)+
(C(\bold F\otimes\tilde\nabla\bold F)\,|\,\boldsymbol\tau)\,+\\
\vspace{1ex}
+\,2\,(\bold F\,|\,\nabla_t\boldsymbol\tau)
+(\bold v\,|\,C(\boldsymbol\tau\otimes\nabla\bold F))+
(\bold v\,|\,C(\nabla_t\boldsymbol\tau\otimes\tilde\nabla\bold F)).
\endgathered\hskip -2em
\tag4.12
$$\par
    Let's write the equalities \thetag{4.2}, \thetag{4.7}, and
\thetag{4.12} in local coordinates. For the function of deviation
$\varphi$ and for its first derivative $\dot\varphi$ we have
$$
\align
\varphi&=\sum^n_{i=1}v_i\,\tau^i,\hskip -3em
\tag4.13\\
\dot\varphi&=\sum^n_{i=1}F_i\,\tau^i+\sum^n_{i=1}v_i\,\nabla_t\tau^i
\hskip -3em
\tag4.14
\endalign
$$
For the sake of brevity in \thetag{4.13} and \thetag{4.14} we used
covariant components of vectors  $\bold v$ and $\bold F$, which are
obtained by lowering the indices:
$$
\xalignat 2
&v_i=\sum^n_{j=1}g_{ij}\,v^j,
&&F_i=\sum^n_{j=1}g_{ij}\,F^j.
\hskip -3em
\tag4.15
\endxalignat
$$
Procedure of lowering indices \thetag{4.15} is commutating with
various covariant differentiations due to the concordance of metric
and connection expressed by \thetag{1.3} (see more details in
\cite{77}). This procedure is invertible, therefore we can make
almost no difference between covariant and contravariant components
of tensorial objects in Riemannian geometry.\par
    Now let's write in local coordinates the expression \thetag{4.12}
for the second derivative of the function of variation:
$$
\gathered
\ddot\varphi=\sum^n_{i=1}\left(2\,F_i+\shave{\sum^n_{j=1}} v^j\,
\tilde\nabla_iF_j\right)\nabla_t\tau^i\,+\hskip -2em\\
+\,\sum^n_{i=1}\left(\,\shave{\sum^n_{j=1}}v^j\left(\nabla_jF_i+
\nabla_iF_j\right)+\shave{\sum^n_{j=1}} F^j\,\tilde\nabla_jF_i
\right)\tau^i.\hskip -2em
\endgathered
\tag4.16
$$
In this formula \thetag{4.16} we use covariant and contravariant
components of force field $\bold F$ simultaneously.
\head
\S\,5. Projectors defined by the vector of velocity.
\endhead
    Function of deviation $\varphi$ depends only on orthogonal
projection of variation vector $\boldsymbol\tau$ to the direction
of velocity vector. This follows from \thetag{4.13} or from the
initial formula \thetag{4.2} for the function $\varphi$. Similarly,
first derivative $\dot\varphi$ also depends only on projection of
variation vector $\boldsymbol\tau$ to the direction of velocity
vector. In what follows it will be important to subdivide all entries
of $\boldsymbol\tau$ and $\nabla_t\boldsymbol\tau$ in formulas for
$\varphi$, $\dot\varphi$, and $\ddot\varphi$ into two parts, one
directed along the velocity vector $\bold v$, and the second directed
perpendicular to $\bold v$. With this aim in mind we define a few
auxiliary tensor fields from extended algebra $\bold T(M)$.\par
     Scalar field of modulus of velocity vector $v=|\bold v|$ is an
extended tensor field of type $(0,0)$. \pagebreak Its domain is the
whole tangent bundle $TM$, but it is smooth only at that points
$q=(p,\bold v)$, where $\bold v\neq 0$. In local coordinates for the
field $v$ we have
$$
v=\sqrt{\shave{\sum^n_{i=1}}\shave{\sum^n_{j=1}}g_{ij}\,v^i\,v^j}.
\hskip -2em
\tag5.1
$$
Denote by $\bold N$ the field of unit vectors directed along the velocity
vector $\bold v$. Then
$$
\bold N=\frac{\bold v}{v}.\hskip -2em
\tag5.2
$$
From \thetag{5.2} we see that $\bold N$ is an extended vector field,
the domain of which is the whole tangent bundle $TM$ except for those
points $q=(p,\bold v)$, where $\bold v=0$. The same condition $\bold v
\neq 0$ restricts domain of the following two extended operator fields:
$\bold Q$ and $\bold P$. $\bold Q$ is an operator of orthogonal projection
to the direction of the vector $\bold v$. Operator $\bold P$ is a projector
onto the hyperplane perpendicular to $\bold v$. Then
$$
\bold P+\bold Q=\bold 1.
\tag5.3
$$
Here $\bold 1$ is the field of identical operators. In local coordinates
operator fields $\bold P$ and $\bold Q$ have the following components:
$$
\xalignat 2
&\quad P^i_j=\delta^i_j-N^i\,N_j,
&&Q^i_j=N^i\,N_j.\hskip -2em
\tag5.4
\endxalignat
$$
Here $N^i$ and $N_j$ are the covariant and contravariant components
of the field \thetag{5.2}.\par
    From \thetag{5.1}, \thetag{5.2}, and \thetag{5.4} one can easily
derive the formulas for differentiating all the above defined
auxiliary fields. Let's write these formulas in local coordinates.
In case of vector field of velocity $\bold v$ and scalar field
$v=|\bold v|$ we have
$$
\xalignat 2
\nabla_kv^i&=0,
&\tilde\nabla_kv^i&=\delta^i_k,\hskip -2em
\tag5.5\\
\nabla_kv&=0,
&\tilde\nabla_kv&=N_k.\hskip -2em
\tag5.6
\endxalignat
$$
In case of vector field $\bold N$  from the relationships \thetag{5.5} and
\thetag{5.6} we derive
$$
\xalignat 2
\nabla_kN^i&=0,&\tilde\nabla_kN^i&=\frac{1}{v}\,P^k_i.\hskip -2em
\tag5.7
\endxalignat
$$
Spatial gradients of projector fields $\bold P$ and $\bold Q$ are
identically zero:
$$
\xalignat 2
\nabla_kP^i_j&=0,&\nabla_kQ^i_j&=0.\hskip -2em
\tag5.8
\endxalignat
$$
For velocity gradients of these two fields we have the formulas:
$$
\aligned
\tilde\nabla_kP^i_j=-&\frac{N_j\,P^i_k}{v}
-\sum^n_{r=1}\frac{g_{jr}\,P^r_k\,N^i}{v},\\
\vspace{0.5ex}
\tilde\nabla_kQ^i_j=&\frac{N_j\,P^i_k}{v}+\sum^n_{r=1}
\frac{g_{jr}\,P^r_k\,N^i}{v}.
\endaligned\hskip -2em
\tag5.9
$$
Spatial gradients are identically zero not only for the fields
$\bold P$ and $\bold Q$; for the fields $v$ and $\bold N$ they vanish as
well. This reflects the fact that they all are produced by the velocity
field $\bold v$ and Riemannian metric field $\bold g$, which satisfy the
relationships \thetag{1.3}. Derivation of formulas \thetag{5.5},
\thetag{5.6}, \thetag{5.7}, \thetag{5.8}, and \thetag{5.9} is rather
simple. It is based on formulas \thetag{7.3} and \thetag{7.4} from 
Chapter \uppercase\expandafter{\romannumeral 3}. We shall not give it
here.\par
    Let's use the relationship \thetag{5.3} in order to transform
first summand in formula \thetag{4.14} for the derivative $\dot\varphi$.
In local coordinates \thetag{5.3} is written as $\delta^i_j=P^i_j+Q^i_j$.
We can insert $\delta^i_j$ into \thetag{4.14} and add one more summation
there:
$$
\dot\varphi=\sum^n_{i=1}\sum^n_{j=1}F_i\,\delta^i_j\tau^j
+\sum^n_{i=1}v_i\,\nabla_t\tau^i\hskip -3em
\tag5.10
$$
Now, substituting $P^i_j+Q^i_j$ for $\delta^i_j$ in \thetag{5.10},
we get the formula
$$
\dot\varphi=\sum^n_{i=1}\sum^n_{j=1}F_i\,P^i_j\tau^j+
\sum^n_{i=1}\sum^n_{j=1}\frac{F_i\,N^i}{v}\,v_j\tau^j
+\sum^n_{i=1}v_i\,\nabla_t\tau^i.\hskip -3em
\tag5.11
$$
Formula \thetag{5.11} can be written in coordinate-free form as follows:
$$
\dot\varphi=(\bold F\,|\,\bold P\,\boldsymbol\tau)+
\frac{(\bold F\,|\,\bold N)}{v}\,(\bold v\,|\,\boldsymbol\tau)+
(\bold v\,|\,\nabla_t\boldsymbol\tau).\hskip -3em
\tag5.12
$$
Last two terms in formula \thetag{5.12} contain scalar products of
$\boldsymbol\tau$ and $\nabla_t\boldsymbol\tau$ with the vector of
velocity, while the first term does not depend on the projection of
these two vectors to the direction of $\bold v$.\par
    Now let's perform some similar transformations with the formula
\thetag{4.16} for the second derivative $\ddot\varphi$. Here it's
convenient to introduce auxiliary covectorial fields $\boldsymbol\alpha$
and $\boldsymbol\beta$ with the following components
$$
\aligned
\alpha_i&=2\,F_i+\sum^n_{j=1} v^j\,\tilde\nabla_iF_j,\hskip -2em\\
\beta_i&=\sum^n_{j=1}v^j\left(\nabla_jF_i+
\nabla_iF_j\right)+\sum^n_{j=1} F^j\,\tilde\nabla_jF_i.\hskip -2em
\endaligned
\tag5.13
$$
Then for the second derivative of the function of deviation we get
$$
\aligned
\ddot\varphi&=\boldsymbol\alpha(\bold P\nabla_t\boldsymbol\tau)
+\boldsymbol\beta(\bold P\boldsymbol\tau)
+\frac{\boldsymbol\alpha(\bold N)}{v}\,(\bold v\,|\,\nabla_t\boldsymbol\tau)
+\frac{\boldsymbol\beta(\bold N)}{v}\,(\bold v\,|\,\boldsymbol\tau).
\endaligned\hskip -2em
\tag5.14
$$
Last two terms in \thetag{5.14} contain scalar product of $\boldsymbol\tau$
and $\nabla_t\boldsymbol\tau$ with the vector of velocity $\bold v$. First
two terms do not depend oh the projections of $\boldsymbol\tau$ and
$\nabla_t\boldsymbol\tau$ to the direction of the vector $\bold v$.
\head
\S\,6. The condition of weak normality.
\endhead
     Let's look on the equations \thetag{3.11} and on the formulas
\thetag{4.13}, \thetag{4.14}, and \thetag{4.16} from the point of view
of differential equations theorist. When we fix some trajectory of
dynamical system \thetag{2.1}, the equations \thetag{3.11} form the
system of linear ordinary differential equations, overall order of
which is equal to $2n$. Denote by $\goth T$ the space of solutions of
this system of equations. Then $\dim\goth T=2n$. As the coordinates in
this space we can use components of two vectors $\boldsymbol\tau(t)$
and $\nabla_t\boldsymbol\tau(t)$ taken for some fixed value of
parameter $t=t_0$. Comparing formulas \thetag{4.13}, \thetag{4.14},
and \thetag{4.16}, we see that the function of deviation $\varphi$
and its derivatives $\dot\varphi$ and $\ddot\varphi$ depend linearly
on $\boldsymbol\tau$ and $\nabla_t\boldsymbol\tau$. This is true
for all time derivatives of the function of deviation, it is proved
by applying repeatedly $\nabla_t$ to the equality \thetag{4.12}.
So the value of $\varphi$ by itself, and values of all its
derivatives (corresponding to some fixed value of $t=t_0$)
$$
\varphi(t_0),\ \dot\varphi(t_0),\ \ddot\varphi(t_0),\
\varphi^{(3)}(t_0),\ \ldots,\ \varphi^{(2n)}(t_0),
\hskip -2em
\tag6.1
$$
can be considered as {\bf linear functionals} on the space
$\goth T$. The number of functionals \thetag{6.1} is equal to
$2n+1$, it is one as more than the dimension of dual space
$\goth T^*$. Therefore these functionals are linearly dependent.
Hence 
$$
\sum^{2n}_{i=0}C_i(t_0)\,\varphi^{(i)}(t_0)=0.\hskip -2em
\tag6.2
$$
Coefficients of linear combination \thetag{6.2} certainly depend on
$t_0$. For each particular $n$ they can be calculated explicitly.
These considerations lead to the following theorem.
\proclaim{Theorem 6.1} Suppose that Newtonian dynamical system on
$n$-dimensional Riemannian manifold $M$ is given. For any its
trajectory and for any choice of variation vector $\boldsymbol\tau$
on this trajectory corresponding function of deviation $\varphi(t)$
satisfies some linear homogeneous ordinary differential equation of
the order not greater than $2n$.
\endproclaim
\definition{Definition 6.1} We say that Newtonian dynamical system
on Riemannian manifold $M$ of the dimension $n\geqslant 2$ satisfies
{\bf weak normality} condition, if for each its trajectory there exists
some ordinary differential equation
$$
\ddot\varphi=\Cal A(t)\,\dot\varphi+\Cal B(t)\,\varphi\hskip -2em
\tag6.3
$$
such that any function of deviation $\varphi(t)$ corresponding to any
choice of variation vector $\boldsymbol\tau$ on that trajectory is
the solution of this equation.
\enddefinition
    Definition~6.1 is one of the main definitions in the theory of
dynamical systems admitting the normal shift. The definition of this
kind was first stated in \cite{38} (see also preprint \cite{34}).
It was found that weak normality condition leads to the system of
partial differential equations for the force field $\bold F$ of
Newtonian dynamical system. Derivation of these equations is our
main purpose in this section.\par
     Thus suppose, that Newtonian dynamical system with force field
$\bold F$ satisfies weak normality condition. Let's fix some trajectory
$p(t)$ and write corresponding differential equation \thetag{6.2} in
the following form:
$$
\ddot\varphi-\Cal A\,\dot\varphi-\Cal B\,\varphi=0.\hskip -2em
\tag6.4
$$
Let $p=p(t_0)$ be a point on the fixed trajectory, where velocity
vector doesn't vanish, i\.~e\. $\bold v(t_0)\neq 0$. In left hand side
of \thetag{6.3} we substitute the expression \thetag{4.2} for the
function of deviation $\varphi$, and substitute  the above expressions
\thetag{5.12} and \thetag{5.14} for its derivatives. After some
reductions we get
$$
\gathered
\left(\frac{\boldsymbol\alpha(\bold N)}{v}-\Cal A\right)\,
(\bold v\,|\,\nabla_t\boldsymbol\tau)+
\boldsymbol\alpha(\bold P\nabla_t\boldsymbol\tau)
+\boldsymbol\beta(\bold P\boldsymbol\tau)\,-\\
\vspace{1ex}
-\,\Cal A\,(\bold F\,|\,\bold P\,\boldsymbol\tau)
+\left(\frac{\boldsymbol\beta(\bold N)}{v}-\Cal A\,
\frac{(\bold F\,|\,\bold N)}{v}-\Cal B\right)\,
(\bold v\,|\,\boldsymbol\tau)=0.
\endgathered\hskip -2em
\tag6.5
$$
Let's consider the value of left hand side of \thetag{6.5} for
$t=t_0$. After fixing $t=t_0$ we have only one arbitrariness rest,
it consists in the choice of solution of the equation \thetag{3.11}.
This choice determines the value of $\boldsymbol\tau(t_0)$ and the
value of its derivative $\nabla_t\boldsymbol\tau(t_0)$. Left hand
side of \thetag{6.5} is linear in $\boldsymbol\tau$ and
$\nabla_t\boldsymbol\tau$. Therefore, considered for some fixed
$t=t_0$, \thetag{6.5} is an equality of linear functionals in
$\goth T^*$. The space of solutions of the equations \thetag{3.11}
on a given trajectory $p(t)$ is isomorphic to direct sum of two
copies of tangent space $T_p(M)$ at the point $p=p(t_0)$:
$$
\goth T\cong T_p(M)\oplus T_p(M).\hskip -2em
\tag6.6
$$
This is true, since $\goth T$ is parameterized by components of two
vectors $\boldsymbol\tau(t_0)\in T_p(M)$ and $\nabla_t\boldsymbol
\tau(t_0)\in T_p(M)$. Velocity vector $\bold v(t_0)\in T_p(M)$
subdivides each summand in formula \thetag{6.6} into direct sum of two
subspaces:
$$
T_p(M)=\langle\bold v\rangle\oplus\langle\bold v\rangle_{\sssize
\perp}.\hskip -2em
\tag6.7
$$
Here subspace $\langle\bold v\rangle$ is a linear span of velocity
vector $\bold v(t_0)\neq 0$, and $\langle\bold v\rangle_{\sssize\perp}$
is an orthogonal complement of $\langle\bold v\rangle$ to a whole
space $T_p(M)$ in the metric $g$. From \thetag{6.6} and \thetag{6.7}
we derive the expansion of $\goth T$ into a direct sum of four subspaces:
$$
\goth T\cong \langle\bold v\rangle\oplus\langle\bold v\rangle_{\sssize
\perp}\oplus\langle\bold v\rangle\oplus\langle\bold v\rangle_{\sssize
\perp}.\hskip -2em
\tag6.8
$$
The expansion \thetag{6.8} generates dual expansion in dual space
$\goth T^*$:
$$
\goth T^*\cong \langle\bold v\rangle^*\oplus\langle\bold v\rangle_{\sssize
\perp}^*\oplus\langle\bold v\rangle^*\oplus\langle\bold v\rangle_{\sssize
\perp}^*.\hskip -2em
\tag6.9
$$
Functionals $(\bold v\,|\,\nabla_t\boldsymbol\tau)$, \ $\boldsymbol
\alpha(\bold P\nabla_t\boldsymbol\tau)$, \ $\boldsymbol\beta(\bold P
\boldsymbol\tau)-\Cal A\,(\bold F\,|\,\bold P\boldsymbol\tau)$ and
$(\bold v\,|\,\boldsymbol\tau)$ in \thetag{6.5} belong to four
different subspaces in the expansion \thetag{6.9}. Therefore the
equation \thetag{6.5} is reduced to four separate equations. Two
of them determine $\Cal A$ and $\Cal B$ in \thetag{6.4}:
$$
\xalignat 2
&\quad\Cal A=\frac{\boldsymbol\alpha(\bold N)}{v},
&&\Cal B=\frac{\boldsymbol\beta(\bold N)}{v}-\Cal A\,
\frac{(\bold F\,|\,\bold N)}{v}.\hskip -3em
\tag6.10
\endxalignat
$$
Second pair of equation derived from \thetag{6.5} has the
following form:
$$
\xalignat 2
&\quad\boldsymbol\alpha(\bold P\nabla_t\boldsymbol\tau)=0,
&&\boldsymbol\beta(\bold P\boldsymbol\tau)=\Cal A\,(\bold F\,|\,
\bold P\,\boldsymbol\tau).\hskip -3em
\tag6.11
\endxalignat
$$
Let's calculate coefficients $\Cal A$ and $\Cal B$ in the equation
\thetag{6.3} by means of above formulas \thetag{6.10} and by means of
formulas \thetag{5.13} for the components of convectors
$\boldsymbol\alpha$ and $\boldsymbol\beta$:
$$
\align
&\Cal A=\sum^n_{i=1} \frac{2\,F_i\,N^i}{v}+
\sum^n_{i=1}\sum^n_{j=1} N^i\,N^j\,\tilde\nabla_iF_j,
\tag6.12\\
\vspace{1ex}
&\aligned
\Cal B&=\sum^n_{i=1}\sum^n_{j=1}\left(N^j\left(\nabla_jF_i+
\nabla_iF_j\right)+\frac{F^j\,\tilde\nabla_jF_i}{v}\right)N^i\,-\\
&-\,\sum^n_{i=1}\sum^n_{k=1}\left(\frac{2\,F_i\,N^i}{v}
+\shave{\sum^n_{j=1}}N^i\,N^j\,\tilde\nabla_iF_j\right)
\frac{F_k\,N^k}{v}.
\endaligned\hskip -3em
\tag6.13
\endalign
$$\par
    Now consider the equations \thetag{6.11}. Using \thetag{5.13},
one can write them in local coordinates. Afterwards one should take
into account arbitrariness of vectors $\boldsymbol\tau$ and
$\nabla_t\boldsymbol\tau$. As a result first equation \thetag{6.11}
is written in the following form:
$$
\sum^n_{i=1}\left(2\,F_i+\shave{\sum^n_{j=1}}v^j\,
\tilde\nabla_iF_j\right)P^i_k=0.\hskip -2em
\tag6.14
$$
In order to transform another equation \thetag{6.11} we use formula
\thetag{6.12} for the coefficient $\Cal A$ in the equation \thetag{6.3}.
This yields
$$
\aligned
\sum^n_{i=1}\left(\shave{\sum^n_{j=1}}v^j\left(\nabla_jF_i+
\nabla_iF_j\right)+\shave{\sum^n_{j=1}} F^j\,\tilde\nabla_jF_i\right)
P^i_k=\\
\vspace{1ex}
=\sum^n_{r=1}\left(\shave{\sum^n_{i=1}}\frac{2\,F_i\,N^i}{v}+
\shave{\sum^n_{i=1}}\shave{\sum^n_{j=1}}N^i\,N^j\,\tilde\nabla_iF_j
\right)F_r\,P^r_k.
\endaligned\hskip -3em
\tag6.15
$$
All done. We are only to do some slight cosmetic transformations in the
equation \thetag{6.14} and \thetag{6.15}. The equation \thetag{6.14} is
written as
$$
\sum^n_{i=1}\left(v^{-1}\,F_i+\shave{\sum^n_{j=1}}
\tilde\nabla_i\left(N^j\,F_j\right)\right)P^i_k=0.\hskip -2em
\tag6.16
$$
In deriving \thetag{6.16} we took into account $v^j=v\,N^j$, and used
second relationship \thetag{5.7} together with well-known property
$\bold P^2=\bold P$ of projector $\bold P$. In equation \thetag{6.15}
we shall only transpose some terms:
$$
\pagebreak
\aligned
&\sum^n_{i=1}\sum^n_{j=1}\left(\nabla_iF_j+\nabla_jF_i-2\,v^{-1}
\,F_i\,F_j\right)N^j\,P^i_k\,+\\
\vspace{1ex}
&+\sum^n_{i=1}\sum^n_{j=1}\left(\frac{F^j\,\tilde\nabla_jF_i}{v}
-\sum^n_{r=1}\frac{N^r\,N^j\,\tilde\nabla_jF_r}{v}\,F_i\right)P^i_k=0.
\endaligned\hskip -3em
\tag6.17
$$
The equations \thetag{6.16} and \thetag{6.17}, which we have derived just
now, are called {\bf weak normality} equations. In this form they were
first obtained in \cite{58} (see also \cite{75}). The above derivation of
these equations proves the following theorem.
\proclaim{Theorem 6.2} Newtonian dynamical system on Riemannian manifold
of the dimension $n\geqslant 2$ satisfies weak normality condition if and
only if its force field satisfies the differential equations \thetag{6.16}
and \thetag{6.17} at all points $q=(p,\bold v)$ of tangent bundle $TM$,
except for those, where $\bold v=0$.
\endproclaim
    The equations of weak normality \thetag{6.16} and \thetag{6.17}
form the system of partial differential equations for the force field
$\bold F$. The number of these equations is $2n$. However, projector
matrix $P^i_k$ is degenerate, its rank is $n-1$. Therefore actual number
of independent equations in \thetag{6.16} and \thetag{6.17} is $2n-2$.
The problem of their compatibility is studied in next two Chapters, where
we construct great many of their solutions. Here we note only that
identically zero force field $\bold F$ is the solution of the equations
\thetag{6.16} and \thetag{6.17}. This proves the following theorem.
\proclaim{Theorem 6.3} Geodesic flow on Riemannian manifold $M$ of the
dimension $n\geqslant 2$ is a Newtonian dynamical system satisfying weak
normality condition.
\endproclaim
\head
\S\,7. Cauchy problem for the function of deviation.
\endhead
    Suppose that $M$ is a Riemannian manifold with Newtonian dynamical
system satisfying weak normality condition on it. Let's use this system
to define the shift $f_t\!:\,S\to S_t$ for some hypersurface $S$ from
transformation class on $M$ (see definition~2.1). Functions of deviation
on the trajectories of shift satisfy the differential equations
\thetag{6.3}. These are linear homogeneous differential equations of the
second order:
$$
\ddot\varphi=\Cal A(t)\,\dot\varphi+\Cal B(t)\,\varphi.\hskip -2em
\tag7.1
$$
Coefficients $\Cal A(t)$ and $\Cal B(t)$ can be different for different
trajectories of shift. Upon choosing local coordinates $u^1,\,\ldots,\,
u^{n-1}$ on $S$ and local coordinates $x^1,\,\ldots,\,x^n$ in $M$, we can
define shift functions \thetag{3.1}. Their derivatives determine vectors
of variation \thetag{3.3}, which coincide with coordinate tangent vectors
to hypersurfaces $S_t$:
$$
\boldsymbol\tau_i=\sum^n_{j=1}\frac{\partial x^j}{\partial u^i}\,
\frac{\partial}{\partial x^j}\text{, \ where \ }i=1,\,\ldots,\,n-1.
\hskip -2em
\tag7.2
$$
By differentiating shift functions \thetag{3.1} with respect to time
variable $t$ for the fixed values of  $u^1,\,\ldots,\,u^{n-1}$ we
obtain the velocity vector on the trajectories of shift:
$$
\bold v=\sum^n_{j=1}\dot x^j\,\frac{\partial}{\partial x^j}.
\hskip -2em
\tag7.3
$$
Scalar products of velocity vector $\bold v$ from \thetag{7.3}
with variation vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol
\tau_{n-1}$ from \thetag{7.2} define the set functions of deviations
$$
\varphi_1,\ \ldots,\ \varphi_{n-1}.\hskip -2em
\tag7.4
$$
This is in complete agreement with formulas \thetag{4.1}. Each function
\thetag{7.4} satisfies the equations \thetag{7.1} for fixed values of
variables $u^1,\,\ldots,\,u^{n-1}$. Normality condition for the shift
$f_t\!:\,S\to S_t$ consists in identical vanishing of all functions
\thetag{7.4} (see definition~2.3). In order to satisfy this condition
it is sufficient to provide vanishing of functions $\varphi_k$ and their
first derivatives $\dot\varphi_k$ at the initial instant of time $t=0$,
i\.~e\. at the points of initial hypersurface $S$. Thus, we are to set up
the following Cauchy problem for differential equations \thetag{7.1}
on trajectories of shift:
$$
\xalignat 2
&\quad\varphi_k(t)\,\hbox{\vrule height 8pt depth 8pt width
0.5pt}_{\,t=0}=0,
&&\dot\varphi_k(t)\,\hbox{\vrule height 8pt depth 8pt width
0.5pt}_{\,t=0}=0.\hskip -2em
\tag7.5
\endxalignat
$$
First part of initial conditions \thetag{7.5} appears to be fulfilled
unconditionally due to the initial data \thetag{2.2} that define
trajectories of shift $f_t\!:\,S\to S_t$. Indeed, according to
\thetag{2.2} vector of initial velocity $\bold v(0)$ is directed
along the normal vector of $S$. Therefore it is perpendicular to the
vectors $\boldsymbol\tau_1(0),\,\ldots,\,\boldsymbol\tau_{n-1}(0)$,
which are tangent to $S$.\par
    Let's study in more details the second pert of initial conditions
\thetag{7.5}. In order to do it we use formula \thetag{4.7}. Here this
formula is written as
$$
\dot\varphi_k=(\bold F\,|\,\boldsymbol\tau_k)+(\bold v\,|\,
\nabla_t\boldsymbol\tau_k).\hskip -2em
\tag7.6
$$
In order to calculate vector  $\nabla_t\boldsymbol\tau_k$ in \thetag{7.6}
we apply formula \thetag{5.13} from Chapter
\uppercase\expandafter{\romannumeral 4} and take unto account that here
$u=u^k$. Tensor field of torsion $\bold T$ for Riemannian connection
\thetag{1.2} is equal to zero, therefore
$$
\nabla_t\boldsymbol\tau_k=\nabla_{u^k}\bold v.\hskip -2em
\tag7.7
$$
Let's write the equality \thetag{7.7} for $t=0$, taking into
account initial data \thetag{2.2}. In order to calculate covariant
derivative $\nabla_{u^k}\bold u$ in right hand side of \thetag{7.7}
one should only know the values of velocity vector on initial
hypersurface $S$. This is clear, for instance, due to formula
\thetag{5.8} from Chapter \uppercase\expandafter{\romannumeral 4}.
Then
$$
\nabla_t\boldsymbol\tau_k\,\hbox{\vrule height 8pt depth 8pt width
0.5pt}_{\,t=0}=\nabla_{u^k}(\nu\cdot\bold n)=\frac{\partial\nu}
{\partial u^k}\cdot\bold n+\nu\cdot\nabla_{u^k}\bold n.\hskip -2em
\tag7.8
$$
Covariant derivatives $\nabla_{u^k}\bold n$ are natural in the
theory of hypersurfaces embedded into Riemannian manifold. They
are encountered in derivational formulas of Weingarten (see \cite{16},
\cite{32}, \cite{84}, or \cite{77}). In our case for the hypersurface
$S$ in $M$ derivational formulas of Weingarten are written as follows:
$$
\align
\nabla_{u^k}\boldsymbol\tau_r&=\sum^{n-1}_{m=1}\theta^m_{kr}\,
\boldsymbol\tau_m+b_{kr}\,\bold n,\hskip -3em
\tag7.9\\
\nabla_{u^k}\bold n&=-\sum^{n-1}_{m=1}b^m_k\,\boldsymbol\tau_m.
\hskip -3em
\tag7.10
\endalign
$$
If we denote by $\rho_{ij}$ \pagebreak components of metric tensor 
$\boldsymbol\rho$
for induced Riemannian metric $\boldsymbol\rho$ in local coordinates $u^1,
\,\ldots,\,u^{n-1}$ on $S$, then quantities $\theta^m_{kr}$ in \thetag{7.9}
are the components of corresponding metric connection for the metric
$\boldsymbol\rho$. They are calculated by formula analogous to formula
\thetag{1.2} for $\Gamma^k_{ij}$:
$$
\theta^k_{ij}=\frac{1}{2}\sum^{n-1}_{s=1}\rho^{ks}\left(\frac{\partial
\rho_{sj}}{\partial u^i}+\frac{\partial\rho_{is}}{\partial u^j}-\frac{
\partial\rho_{ij}}{\partial u^s}\right).\hskip -3em
\tag7.11
$$
Quantities $b_{kr}$ in \thetag{7.9} are the components of inner tensor
field $\bold b$ of type $(0,2)$ in $S$ which is called {\bf second
fundamental form} of hypersurface $S$. They are symmetric, i\.~e\.
they do not change in transposing indices:
$$
b_{kr}=b_{rk}.\hskip -3em
\tag7.12
$$
Quantities $b^m_k$ in formula \thetag{7.10} are derived from $b_{kr}$
by index raising procedure:
$$
b^m_k=\sum^{n-1}_{r=1}b_{kr}\,\rho^{rm}.\hskip -3em
\tag7.13
$$
They are the components of inner tensor field of type $(1,1)$ in
submanifold $S$. Such field is  an  operator  version  of  second 
fundamental
form $\bold b$ in $S$.\par
     Let's apply derivational formula of Weingarten \thetag{7.10} in
order to express covariant derivative $\nabla_{u^k}\bold n$ in formula
\thetag{7.8}. This yields
$$
\nabla_t\boldsymbol\tau_k\,\hbox{\vrule height 8pt depth 8pt width
0.5pt}_{\,t=0}=\frac{\partial\nu}{\partial u^k}\cdot\bold n-\nu\cdot
\sum^{n-1}_{m=1}b^m_k\,\boldsymbol\tau_m.\hskip -3em
\tag7.14
$$
Now we substitute \thetag{7.14} into \thetag{7.6} and calculate
derivative $\dot\varphi_k$ for initial instant of time $t=t_0$
on initial hypersurface $S$:
$$
\dot\varphi_k\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=(\bold F\,|\,\boldsymbol\tau_k)+\nu\,\frac{\partial\nu}{\partial u^k}.
\hskip -3em
\tag7.15
$$
In deriving \thetag{7.15} we took into account second part of the
relationships \thetag{2.2} that determine initial value of velocity
vector on $S$.\par
    Let's substitute the above formula \thetag{7.15} expressing initial
value of derivative $\dot\varphi$ into the equations \thetag{7.5}. This
yields the equations
$$
\frac{\partial\nu}{\partial u^k}=-\nu^{-1}\,
(\bold F\,|\,\boldsymbol\tau_k)
\hskip -3em
\tag7.16
$$
for the function $\nu(u^1,\ldots,u^{n-1})$. If $\nu(u^1,\ldots,u^{n-1})$
is the solution of equations \thetag{7.16}, then for all function of
deviation \thetag{7.4} initial conditions \thetag{7.5} are fulfilled.
In the case of dynamical system satisfying weak normality condition
(see definition~6.1) it is sufficient to assure the normality of
shift $f_t\!:\,S\to S_t$.
\head
\S\,8. Pfaff equations for the modulus of initial velocity.
\endhead
     The equations \thetag{7.16} determine two essential different
cases: {\bf two dimensional case}, when $\dim M=n=2$, when we have
only one equation \thetag{7.16}, and {\bf multidimensional case},
when $\dim M=n\geqslant 3$, when the number of equations is greater
than $1$. We intentionally omit here two dimensional case, since it
is considered in {\bf other thesis} \cite{36}.\par
     Let's consider the equations \thetag{7.16} in multidimensional
case. All quantities in right hand side of thetag{7.16} are localized
on initial hypersurface $S$. Force vector $\bold F$ depends on the
point of tangent bundle $q=(p,\bold v)$. Substituting $\bold F$ into
\thetag{7.16} we choose $p\in S$ and $\bold v=\nu\cdot\bold n(p)$ in
accordance with initial data \thetag{2.2}. In local coordinates
$u^1,\,\ldots,\,u^{n-1}$ the point $p\in S$ and coordinate tangent
vectors  $\boldsymbol\tau_k$ are parameterized by these coordinates:
$p=p(u^1,\ldots,u^{n-1})$ and $\boldsymbol\tau_k=\boldsymbol\tau_k(u^1,
\ldots,u^{n-1})$. Therefore the equations \thetag{7.16} has the
following structure:
$$
\frac{\partial\nu}{\partial u^k}=\psi_k(\nu,u^1,\ldots,u^{n-1})
\text{, \ where \ }k=1,\,\ldots,\,n-1.
\hskip -3em
\tag8.1
$$
For $n\geqslant 3$ the equations \thetag{8.1} form {\bf complete
system of Pfaff equations}. Central point of the theory of Pfaff
equations is the {\bf problem of compatibility}. Suppose that
$\nu=\nu(u^1,\ldots,u^{n-1})$ is a solution of the equations
\thetag{8.1}. Let's calculate second order partial derivative
$\nu$ with respect to variables $u^k$ and $u^r$ by virtue of
the equations \thetag{8.1}. This can be done in two ways:
$$
\align
\frac{\partial^2\nu}{\partial u^k\,\partial u^r}&=
\frac{\partial\psi_k}{\partial u^r}+\frac{\partial\psi_k}{\partial\nu}
\,\psi_r=\vartheta_{kr}(\nu,u^1,\ldots,u^{n-1}),\hskip -2em
\tag8.2\\
\vspace{1ex}
\frac{\partial^2\nu}{\partial u^r\,\partial u^k}&=
\frac{\partial\psi_r}{\partial u^k}+\frac{\partial\psi_r}{\partial\nu}
\,\psi_k=\vartheta_{rk}(\nu,u^1,\ldots,u^{n-1}).\hskip -2em
\tag8.3
\endalign
$$
Functions $\vartheta_{kr}(\nu,u^1,\ldots,u^{n-1})$ and $\vartheta_{kr}
(\nu,u^1,\ldots,u^{n-1})$ in right hand sides of \thetag{8.2} and \thetag{8.3}
can be calculated regardless to whether has the system of Pfaff equations
thetag{8.1} some solution or it hasn't. In the case, when it has, the
derivatives \thetag{8.2} and \thetag{8.3} are equal to each other. Equating
\thetag{8.2} and \thetag{8.3}, we get
$$
\aligned
&\vartheta_{kr}(\nu(u^1,\ldots,u^{n-1}),u^1,\ldots,u^{n-1})=\\
\vspace{1ex}
&=\vartheta_{rk}(\nu(u^1,\ldots,u^{n-1}),u^1,\ldots,u^{n-1}).
\endaligned\hskip -2em
\tag8.4
$$
In general, the equality \thetag{8.4} doesn't  provide  identical 
coincidence
of the functions $\vartheta_{kr}$ and $\vartheta_{rk}$. In order to test
whether it is fulfilled or not we are to have some solution $\nu(u^1,
\ldots,u^{n-1})$ of the equations \thetag{8.1}. But there is some situation
when we can escape this difficulty.
\definition{Definition 8.1} System of Pfaff equations \thetag{8.1} is
called {\bf compatible}, if for any fixed point $(\nu_0,\,u^1_0,\,\ldots,
\,u^{n-1}_0)$ from the domain of the functions $\psi_k$ \pagebreak there
is a solution $\nu=\nu(u^1,\ldots,u^{n-1})$ of the equations \thetag{8.1}
defined in some neighborhood of the point $(u^1_0,\ldots,u^{n-1}_0)$ and
normalized by the condition 
$$
\nu(u^1_0,\ldots,u^{n-1}_0)=\nu_0.\hskip -2em
\tag8.5
$$
\enddefinition
     If the system of Pfaff equations \thetag{8.1} is compatible, then
from \thetag{8.5} and \thetag{8.4} we get that functions $\vartheta_{kr}$
and $\vartheta_{rk}$ do identically coincide
$$
\vartheta_{kr}(\nu,u^1,\ldots,u^{n-1})=\vartheta_{rk}(\nu,u^1,\ldots,
u^{n-1}).\hskip -2em
\tag8.6
$$
The relationships \thetag{8.6} are called {\bf compatibility conditions}
for complete system of Pfaff equations \thetag{8.1}. Let's write them
in more explicit form:
$$
\frac{\partial\psi_k}{\partial u^r}+\frac{\partial\psi_k}{\partial\nu}
\,\psi_r=\frac{\partial\psi_r}{\partial u^k}+\frac{\partial\psi_r}
{\partial\nu}\,\psi_k.\hskip -2em
\tag8.7
$$
\proclaim{Theorem 8.1} Compatibility conditions expressed by relationships
\thetag{8.7}, where $k=1,\,\ldots,\,n-1$ and $r=1,\,\ldots,\,n-1$, are
necessary and sufficient for the system of Pfaff equations \thetag{8.1} to
be compatible in the sense of definition~8.1.
\endproclaim
\demo{Proof} The necessity was proved by derivations of relationships
\thetag{8.7}. We are to prove sufficiency. Let's do it by induction in
$n$. For $n=2$ we have one equation \thetag{8.1} which is ordinary
differential equation with independent variable $u^1$. Normalization
\thetag{8.5} is a Cauchy problem for this equation. Compatibility
conditions \thetag{8.7} in this case are identically fulfilled, since
$k=r=1$. Thus, for $n=2$ the proposition of theorem follows from
local solvability of Cauchy problem for ordinary differential equation
(see ``{existence and uniqueness}'' theorem in \cite{82} or \cite{83}).
This is the base of induction.\par
    Let's do the step of induction from $n-1$ to $n$. If we exclude the
last equation from the system \thetag{8.1} and if we fix the last
variable $u^{n-1}=u^{n-1}_0$, then we get into the situation when
inductive hypothesis is applicable. This yields the function
$$
\nu=\nu(u^1,\ldots,u^{n-2})\hskip -2em
\tag8.8
$$
that satisfies the system of first $(n-2)$ Pfaff equations in
\thetag{8.1}, and that satisfies normalizing condition \thetag{8.5}
on the hyperplane $u^{n-1}=u^{n-1}_0$. Now let's consider again the
last equation in \thetag{8.1}, treating it as an ordinary differential
equation with independent variable $u^{n-1}$, and let's set up the
Cauchy problem
$$
\nu\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,u^{n-1}=
u^{n-1}_0}=\nu(u^1,\ldots,u^{n-2}),\hskip -2em
\tag8.9
$$
using the function \thetag{8.8} as initial data. We get the required
function $\nu(u^1,\ldots,u^{n-1})$ as a solution of the above Cauchy
problem with initial data \thetag{8.9}.\par
    Function $\nu(u^1,\ldots,u^{n-1})$ satisfies the last equation
in the system \thetag{8.1} by construction. Moreover, it satisfies
other equations \thetag{8.1} on the hyperplane $u^{n-1}=u^{n-1}_0$
by inductive hypothesis. Suppose that  $1\leqslant k\leqslant n-2$.
Let's show that $k$-th equation \thetag{8.1} is fulfilled for
all values of arguments in $\nu(u^1,\ldots,u^{n-1})$ where this
function is defined. Consider the following functions:
$$
\align
x_k(u^1,\ldots,u^{n-1})&=\frac{\partial\nu}{\partial u^k},\hskip -3em
\tag8.10\\
\vspace{1.5ex}
y_k(u^1,\ldots,u^{n-1})&=\psi_k(\nu,u^1,\ldots,u^{n-1}).\hskip -3em
\tag8.11
\endalign
$$
Let's calculate the derivatives of \thetag{8.10} and \thetag{8.11}
with respect to the variable $u^{n-1}$. For the first of these two
functions we have 
$$
\frac{\partial x_k}{\partial u^{n-1}}=\frac{\partial^2\nu}
{\partial u^k\,\partial u^{n-1}}=\frac{\partial}{\partial u^k}
\left(\psi_{n-1}(\nu,u^1,\ldots,u^{n-1})\right).
$$
By differentiating composite function in the above expression we get
$$
\frac{\partial x_k}{\partial u^{n-1}}=\frac{\partial\psi_{n-1}}
{\partial u^k}+\frac{\partial\psi_{n-1}}{\partial\nu}\,x^k.
\hskip -3em
\tag8.12
$$
By means of analogous differentiation for the function \thetag{8.11} we
obtain
$$
\frac{\partial y_k}{\partial u^{n-1}}=\frac{\partial\psi_k}
{\partial u^{n-1}}+\frac{\partial\psi_k}{\partial\nu}\,\psi_{n-1}.
\hskip -3em
\tag8.13
$$
Let's transform \thetag{8.13} with the help of compatibility condition
\thetag{8.7}, taking $r=n-1$ in it. This yields the relationship
$$
\frac{\partial y_k}{\partial u^{n-1}}=\frac{\partial\psi_{n-1}}
{\partial u^k}+\frac{\partial\psi_{n-1}}{\partial\nu}\,y^k.
\hskip -3em
\tag8.14
$$
Comparing \thetag{8.12} and \thetag{8.14}, we see that functions
$x_k$ and $y_k$ satisfy identical linear ordinary differential equations
of the first order with to independent variable $u^{n-1}$. For
$u^{n-1}=u^{n-1}_0$ their values are equal to each other by
inductive hypothesis. Therefore $x_k=y_k$ for all $u^1,\,\ldots,\,u^{n-1}$.
This means that function $\nu(u^1,\ldots,u^{n-1})$ constructed above
is the solution of whole system of Pfaff equations \thetag{8.1}. And
it satisfies normalizing condition \thetag{8.5} as well. Theorem is
proved.\qed\enddemo
    Let's apply the theorem~8.1 to the equations \thetag{7.16}, and
let's find compatibility condition for them. First calculate partial
derivatives \thetag{8.2} and \thetag{8.3} for the function
$\nu(u^1,\ldots,u^{n-1})$ by virtue of the equations \thetag{7.16}:
$$
\align
&\aligned
&\frac{\partial^2\nu}{\partial u^k\,\partial u^r}=-\nabla_{u^r}
\left(\nu^{-1}\,(\bold F\,|\,\boldsymbol\tau_k)\right)
=\frac{(\bold F\,|\,\boldsymbol\tau_k)}{\nu^2}\,\frac{\partial\nu}
{\partial u^r}-\frac{(\nabla_{u^r}\bold F\,|\,\boldsymbol\tau_k)}{v}\,-\\
\vspace{2ex}
&-\,\frac{(\bold F\,|\,\nabla_{u^r}\boldsymbol\tau_k)}{v}=
-\frac{(\bold F\,|\,\boldsymbol\tau_k)\,(\bold F\,|\,
\boldsymbol\tau_r)}{\nu^3}-\frac{(\nabla_{u^r}\bold F\,|\,
\boldsymbol\tau_k)}{\nu}-\frac{(\bold F\,|\,\nabla_{u^r}
\boldsymbol\tau_k)}{\nu},
\endaligned\hskip -3em
\tag8.15\\
\displaybreak
&\aligned
&\frac{\partial^2\nu}{\partial u^r\,\partial u^k}=-\nabla_{u^k}
\left(\nu^{-1}\,(\bold F\,|\,\boldsymbol\tau_r)\right)
=\frac{(\bold F\,|\,\boldsymbol\tau_r)}{\nu^2}\,\frac{\partial\nu}
{\partial u^k}-\frac{(\nabla_{u^k}\bold F\,|\,\boldsymbol\tau_r)}{v}\,-\\
\vspace{2ex}
&-\frac{(\bold F\,|\,\nabla_{u^k}\boldsymbol\tau_r)}{v}=
-\frac{(\bold F\,|\,\boldsymbol\tau_r)\,(\bold F\,|\,
\boldsymbol\tau_k)}{\nu^3}-\frac{(\nabla_{u^k}\bold F\,|\,
\boldsymbol\tau_r)}{\nu}-\frac{(\bold F\,|\,\nabla_{u^k}
\boldsymbol\tau_r)}{\nu}.
\endaligned\hskip -3em
\tag8.16
\endalign
$$
Let's subtract \thetag{8.16} from \thetag{8.15}, taking into account
that $\nabla_{u^r}\boldsymbol\tau_k=\nabla_{u^k}\boldsymbol\tau_r$.
The latter equality is the consequence of derivational formulas
\thetag{7.9} due to symmetry of components of metric connection
\thetag{7.11}, and due to the relationship \thetag{7.12} expressing
symmetry of second fundamental form. Taking into account all above
facts, we get
$$
(\nabla_{u^r}\bold F\,|\,\boldsymbol\tau_k)=(\nabla_{u^k}\bold F\,|\,
\boldsymbol\tau_r).\hskip -3em
\tag8.17
$$
In order to calculate covariant derivative $\nabla_{u^r}\bold F$
in the equality \thetag{8.17} we apply the formula \thetag{5.19}
from Chapter \uppercase\expandafter{\romannumeral 4}:
$$
\nabla_{u^r}\bold F=C(\nabla_{u^r}\bold v\otimes\tilde\nabla\bold F)
+C(\boldsymbol\tau_r\otimes\nabla\bold F).\hskip -3em
\tag8.18
$$
In calculating $\nabla_{u^r}\bold v$ we take into account the
equality $\bold v=\nu\cdot\bold n$ for the initial velocity on
the initial hypersurface $S$. Then we get 
$$
\nabla_{u^r}\bold v=\nabla_{u^r}(\nu\cdot\bold n)=-\frac{(\bold F\,|\,
\boldsymbol\tau_r)}{\nu}\cdot\bold n-\nu\cdot\sum^{n-1}_{m=1}b^m_r\,
\boldsymbol\tau_m.\hskip -3em
\tag8.19
$$
In obtaining \thetag{8.19} we used derivational formulas \thetag{7.10}
and the equation \thetag{7.16} for $\nu$ by itself. Now let's substitute
\thetag{8.19} into \thetag{8.18}, and let's use the resulting expression
for to transform the equation \thetag{8.17}:
$$
\gather
\frac{(\bold F\,|\,\boldsymbol\tau_k)(C(\bold n\otimes\tilde\nabla\bold F)
\,|\,\boldsymbol\tau_r)}{\nu}-\frac{(\bold F\,|\,\boldsymbol\tau_r)(C(\bold
n\otimes\tilde\nabla\bold F)\,|\,\boldsymbol\tau_k)}{\nu}\,+\hskip -3em\\
\vspace{2ex}
+\,(C(\boldsymbol\tau_r\otimes\nabla\bold F)\,|\,\boldsymbol\tau_k)
-(C(\boldsymbol\tau_k\otimes\nabla\bold F)\,|\,\boldsymbol\tau_r)=
\hskip -3em
\tag8.20\\
\vspace{1ex}
=\sum^{n-1}_{m=1}\nu\left(b^m_r\,(C(\boldsymbol\tau_m\otimes\tilde\nabla
\bold F)\,|\,\boldsymbol\tau_k)-b^m_k\,(C(\boldsymbol\tau_m\otimes\tilde
\nabla\bold F)\,|\,\boldsymbol\tau_r)\right).\hskip -3em
\endgather
$$
The relationship \thetag{8.20} is the compatibility condition \thetag{8.7}
applied to the Pfaff equations \thetag{7.16}. Let's write it in local
coordinates on $M$:
$$
\gathered
\sum^n_{i=1}\sum^n_{j=1}\tau^i_k\,\tau^j_r\left(\,\shave{\sum^n_{m=1}}
n^m\,\frac{F_i\,\tilde\nabla_mF_j-F_j\,\tilde\nabla_mF_i}{\nu}+
\nabla_jF_i-\nabla_iF_j\right)=\\
\vspace{2ex}
=\sum^n_{i=1}\sum^n_{j=1}\sum^{n-1}_{m=1}\nu\left(b^m_r\,
\tau^j_m\tilde\nabla_jF_i\,\tau^i_k-b^m_k\,\tau^j_m\tilde\nabla_jF_i
\,\tau^i_r\right).
\endgathered\hskip -1.5em
\tag8.21
$$
Here $n^m$ are the components of unitary normal vector to $S$, and
$\tau^i_k$ are components of the vector $\boldsymbol\tau_k$ in local
map $(U,x^1,\ldots,x^n)$ on $M$:
$$
\boldsymbol\tau_k=\sum^n_{i=1}\tau^i_k\,
\frac{\partial}{\partial x^i}\text{, \ where \ }k=1,\,\ldots,\,n-1.
\hskip -3em
\tag8.22
$$
Variation vectors \thetag{8.22} tangent to $S$ and the components
$b^m_k$ of inner operator field $\bold b$ from \thetag{7.10} are
determined by the choice of local coordinates $u^1,\,\ldots,\,u^{n-1}$
on the hypersurface $S$. The equations \thetag{8.21} expressing
compatibility condition \thetag{8.20} in local coordinates appear
to be more convenient for the further analysis.
\head
\S\,9. The additional normality condition.
\endhead
    Normalization condition \thetag{8.5} in the theory of Pfaff
equations is similar to initial data in cauchy problem for ordinary
differential equations. Theorem~8.1 is analog of the theorem on
local solvability of Cauchy problem for ordinary differential equations.
The condition \thetag{8.5} prompt us normalization condition for
the function $\nu(p)$ that determines modulus of initial velocity in
\thetag{2.2}. Let $p_0$ be some fixed point on hypersurface $S$ in $M$,
and let $\nu_0\neq 0$ be some fixed real number. Then we set
$$
\nu(p)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,p=p_0}=
\nu_0\hskip -2em
\tag9.1
$$
This equality \thetag{9.1} is called {\bf normalization condition}
for the function $\nu(p)$ on $S$.
\definition{Definition 9.1} Suppose that Riemannian manifold $M$ of
the dimension $n\geqslant 2$ is equipped with some Newtonian dynamical
system, which is used to arrange the shift of hypersurfaces along its
trajectories according to the initial data \thetag{2.2}. We say that
this dynamical system satisfies {\bf additional normality} condition,
if for any hypersurface $S$ in $M$, for any point $p_0$ on $S$, and
for any real number $\nu_0\neq 0$ there exists some smaller piece $S'$
of hypersurface $S$ belonging to transformation class and containing
the point $p_0$, and there exists some function $\nu(p)$ from
transformation class on $S'$ normalized by the condition \thetag{9.1}
and such that, when substituted to \thetag{2.2}, it defines the
shift $f_t\!:\,S'\to S'_t$ such that 
$$
\dot\varphi_k(t)\,\hbox{\vrule height 8pt depth 8pt width
0.5pt}_{\,t=0}=0\text{, \ \ where \ \ \ }k=1,\,\ldots,\,n-1.
\hskip -2em
\tag9.2
$$
Here $\varphi_k=(\boldsymbol\tau_k\,|\,\bold v)$ are the functions of
deviation defined by some choice of local coordinates $u^1,\,\ldots,
\,u^{n-1}$ on $S$ and corresponding vectors of variation $\boldsymbol
\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$.
\enddefinition
     Note that \thetag{9.2} is a second part of the equalities
\thetag{7.5}. If additional normality condition is fulfilled, then
equalities \thetag{7.5} are also fulfilled, since first part of
these equalities holds unconditionally due to the initial data
\thetag{2.2}.\par
     Functions of deviations $\varphi_k$ in \thetag{9.2} depend
on the choice of local coordinates $u^1,\,\ldots,\,u^{n-1}$ on
$S$. But additional normality condition from definition~9.1
is fulfilled or not fulfilled regardless to any particular choice
of local coordinates. Indeed, \pagebreak if we change local coordinates
$u^1,\,\ldots,\,u^{n-1}$ on $S$ for other local coordinates $\tilde u^1,
\,\ldots,\,\tilde u^{n-1}$, then corresponding vectors of variation
are transformed as follows
$$
\tilde{\boldsymbol\tau}_i=\sum^{n-1}_{k=1}\sigma^k_i\,\boldsymbol\tau_k,
\hskip -2em
\tag9.3
$$
Here $\sigma^k_i$ are components of transition matrix given by
$$
\sigma^k_i=\frac{\partial u^k}{\partial\tilde u^i}.
$$
They do not depend on $t$. The matter is that local coordinates $u^1,\,
\ldots,\,u^{n-1}$ and $\tilde u^1,\,\ldots,\,\tilde u^{n-1}$ are transferred
from $S'$ to $S'_t$ by bijective map of shift $f_t\!:\,S'\to S'_t$.
Transition functions 
$$
\xalignat 2
&\tilde u^1=\tilde u^1(u^1,\ldots,u^{n-1}), &&u^1=u^1(\tilde u^1,
\ldots,\tilde u^{n-1}),\\
&.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ &&.\ .\ .\ .\
.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ \\
&\tilde u^{n-1}=\tilde u^{n-1}(u^1,\ldots,u^{n-1}),\quad &&u^{n-1}=
u^{n-1}(\tilde u^1,\ldots,\tilde u^{n-1})
\endxalignat
$$
thereby remain same as on initial hypersurface $S'$. Transformation rule
\thetag{9.3} then is the direct consequence of formula \thetag{7.2}, which,
in turn, is the definition of variation vectors $\boldsymbol\tau_1,\,
\ldots,\,\boldsymbol\tau_{n-1}$.\par
    Let's substitute \thetag{9.3} into \thetag{4.1}. As a result we obtain
the transformation rule for the functions of deviation relating $\varphi_1,
\,\ldots,\,\varphi_{n-1}$ and $\tilde\varphi_1,\,\ldots,\,\tilde
\varphi_{n-1}$:
$$
\tilde\varphi_i=\sum^{n-1}_{k=1}\sigma^k_i\,\varphi_k.\hskip -2em
\tag9.4
$$
Differentiating \thetag{9.4} with respect to $t$ and taking into account
$\dot\sigma^k_i=0$, we get
$$
\DotTildePhi_i=\sum^{n-1}_{k=1}\sigma^k_i\,\dot\varphi_k.
\hskip -2em
\tag9.5
$$
The relationships \thetag{9.5} shows that, once fulfilled for some
particular choice of local coordinates on $S$, the relationships
\thetag{9.2} will remain true for any other choice of local
coordinates on $S$.\par
     The statement of additional normality condition in definition~9.1
exploits some hypersurface $S$ and some local coordinates on it. But,
as we have seen above, these objects are auxiliary tools only.
Additional normality condition, when it is fulfilled, is a property
of dynamical system by itself. Similar to weak normality condition
from definition~6.1, it can be written in form of partial differential
equations for the force field $\bold F$ of Newtonian dynamical system.
Exception is two-dimensional case $\dim M=n=2$.
\proclaim{Theorem 9.1}In two-dimensional case $\dim M=2$ additional
normality condition is fulfilled unconditionally for arbitrary
\pagebreak Newtonian dynamical system on $M$.
\endproclaim
    We shall not prove and comment this theorem, since two-dimensional
case is considered in {\bf other thesis} \cite{36}.\par
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    Suppose that $\dim M\geqslant 3$. Let's consider the Newtonian
dynamical system with force field $\bold F$ on $M$. Suppose that
this system satisfies additional normality condition. Let's choose
some fixed point $p_0$ in $M$ and some nonzero vector $\bold v_0$ at
this point. \vadjust{\vskip -0.1cm\special{em:graph edis-05b.gif}
\vskip 0.1cm}Denote by $\alpha$ the hyperplane perpendicular to
vector $\bold v_0$, and denote by $\bold P$ orthogonal projector
to this hyperplane (see Fig\. 9.1). Then $\bold P$ is a value of
projector field $\bold P$ from \S\,5 at the point $q=(p_0,\bold v_0)$
of tangent bundle $TM$. One can draw infinitely many surfaces passing
through the point $p_0$ and being perpendicular to $\bold v_0$ at
this point. Let's choose one of them belonging to transformation
class and denote it by $S$. Then choose local coordinates $u^1,\,\ldots,
\,u^{n-1}$ in the neighborhood of the point $p_0$ and use this point
to write normalization condition \thetag{9.1}. According to the results
of \S\,7 for $n\geqslant 3$ weak the equalities \thetag{9.2} lead to
the system of Pfaff equations \thetag{7.16} for the function $\nu(p)$,
while arbitrariness of $\nu_0=|\bold v_0|$ in thetag{9.1} means that
these Pfaff equations are compatible in the sense of definition~8.1.
Hence the relationships \thetag{8.20} are fulfilled. In local
coordinates they are written as \thetag{8.21}.  We shall consider
them at the point $p_0$ on $S$.
\proclaim{Theorem 9.2} Let $S$ be some hypersurface in Riemannian manifold
$M$ and let $\bold P$ be operator of orthogonal projection onto the
hyperplane tangent to $S$ at the point $p_0$ on $S$. Then for the matrix
components of operator $\bold P$ in local coordinates $x^1,\,\ldots,\,
x^n$ on $M$ there exists the formula 
$$
P^i_\varepsilon=\sum^n_{s=1}\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}
g_{\varepsilon s}\,\tau^s_a\,\rho^{ak}\,\tau^i_k,\hskip -2em
\tag9.6
$$
where $\tau^s_a$ and $\tau^i_k$ are components of coordinate tangent vectors
$\boldsymbol\tau_a$ and $\boldsymbol\tau_k$ that are defined by some local
coordinates $u^1,\,\ldots,\,u^{n-1}$ on $S$; $\rho^{ak}$ are components of
dual metric tensor for induced Riemannian metric $\bold\rho$ in coordinates
$u^1,\,\ldots,\,u^{n-1}$ on $S$; and finally, $g_{\varepsilon s}$ are
components of metric $\bold g$ in coordinates $x^1,\,\ldots,\,x^n$ on $M$.
\endproclaim
\demo{Proof} Let $\bold y$ be some arbitrary vector at the point $p_0$
on $M$. Its projection $\bold P\bold y$ can be expanded in the base of
vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$:
$$
\bold P\bold y=\sum^{n-1}_{k=1}\beta^k\,\boldsymbol\tau_k.\hskip -2em
\tag9.7
$$
In order to find numeric coefficients $\beta^k$ in \thetag{9.7} \pagebreak
we consider scalar products of the vector $\boldsymbol\tau_a$ with both
sides of the equality \thetag{9.7}. This yields
$$
(\boldsymbol\tau_a\,|\,\bold P\bold y)=\sum^{n-1}_{k=1}\beta^k\,
(\boldsymbol\tau_a\,|\,\boldsymbol\tau_k).\hskip -2em
\tag9.8
$$
In left hand side of \thetag{9.8} we obtain $(\boldsymbol\tau_a\,|\,
\bold P\bold y)=(\bold P\boldsymbol\tau_a\,|\,\bold y)=(\boldsymbol
\tau_a\,|\,\bold y)$, while in right hand side the quantities 
$(\boldsymbol\tau_a\,|\,\boldsymbol\tau_k)$ coincide with components
of metric tensor for induced Riemannian metric on $S$: $\rho_{ak}=
(\boldsymbol\tau_a\,|\,\boldsymbol\tau_k)$. Therefore we can bring
\thetag{9.8} to 
$$
(\boldsymbol\tau_a\,|\,\bold y)=\sum^{n-1}_{k=1}\beta^k\,\rho_{ak},
$$
and we can calculate $\beta^k$ with the use of components $\rho^{ak}$
for dual metric tensor:
$$
\beta^k=\sum^{n-1}_{a=1}(\boldsymbol\tau_a\,|\,\bold y)\,\rho^{ak}.
\hskip -2em
\tag9.9
$$
Let's substitute \thetag{9.9} into the formula \thetag{9.7} for
the vector $\bold P\bold y$. As a result we get
$$
\bold P\bold y=\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}(\boldsymbol\tau_a\,|\,
\bold y)\,\rho^{ak}\,\boldsymbol\tau_k.\hskip -3em
\tag9.10
$$
Now we are only to write vectorial equality \thetag{9.10} in local
coordinates $x^1,\,\ldots,\,x^n$ on the manifold $M$. This yields
the following relationship:
$$
\sum^n_{\varepsilon=1}P^i_\varepsilon\,y^\varepsilon=\sum^n_{\varepsilon=1}
\sum^n_{s=1}\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}g_{\varepsilon s}\,\tau^s_a\,
y^\varepsilon \,\,\rho^{ak}\,\tau^i_k.
$$
Components of vector $\bold y$ in both sides of the obtained equality
are arbitrary real numbers. Therefore it is reduced to the equality
\thetag{9.6} that was to be proved.\qed\enddemo
    We shall use theorem~9.2 to make further changes in \thetag{8.21}.
In order to do it let's take two copies of the equality \thetag{9.6},
redesignating indices in the second copy:
$$
\align
P^i_\varepsilon &=\sum^n_{s=1}\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}
g_{\varepsilon s}\,\tau^s_a\,\rho^{ak}\,\tau^i_k,\\
P^j_\sigma &=\sum^n_{w=1}\sum^{n-1}_{b=1}\sum^{n-1}_{r=1}
g_{\sigma w}\,\tau^w_b\,\rho^{br}\,\tau^j_r.
\endalign
$$
By multiplying these two equalities, we get the relationship
$$
P^i_\varepsilon\,P^j_\sigma=\sum^n_{s=1}\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}
\sum^n_{w=1}\sum^{n-1}_{b=1}\sum^{n-1}_{r=1}g_{\varepsilon s}\,\tau^s_a\,
\rho^{ak}\,g_{\sigma w}\,\tau^w_b\,\rho^{br}\,(\tau^i_k\,\tau^j_r).
$$
Here we intentionally enclosed into brackets last two terms in right hand
side of this relationships. Exactly the same terms $\tau^i_k\,\tau^j_r$
are in left hand side of \thetag{8.21}. Let's multiply \thetag{8.21}
by $g_{\varepsilon s}\,\tau^s_a\,\rho^{ak}\,g_{\sigma w}\,\tau^w_b\,
\rho^{br}$ and let's sum the resulting equality in $s$, $a$, $k$,
$w$, $b$, and $r$. Then left hand side of \thetag{8.21} takes the form
$$
\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(\,\shave{
\sum^n_{m=1}}n^m\,\frac{F_i\,\tilde\nabla_mF_j-F_j\,\tilde\nabla_mF_i}{\nu}+
\nabla_jF_i-\nabla_iF_j\right).
$$
Now let's introduce the following notations
$$
H^j_\varepsilon=\sum^n_{s=1}\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}
\sum^{n-1}_{m=1}g_{\varepsilon s}\,\tau^s_a\,\rho^{ak}\,b^m_k\,
\tau^j_m.\hskip -3em
\tag9.11
$$
Then in right hand side of transformed equality \thetag{8.21} we can
detect the quantities $H^j_\varepsilon$ and $H^j_\sigma$ that were
formed as a result of the above transformations. I terms of notations
\thetag{9.11} right hand side of \thetag{8.21} is written as
$$
\sum^n_{i=1}\sum^n_{j=1}\nu\,(P^i_\varepsilon\,H^j_\sigma-
P^i_\sigma\,H^j_\varepsilon)\,\tilde\nabla_jF_i.
$$
Now, equating transformed expressions for left and right hand sides
of \thetag{8.21}, we complete first step in transforming this equality:
$$
\gathered
\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(\,\shave{
\sum^n_{m=1}}n^m\,\frac{F_i\,\tilde\nabla_mF_j-F_j\,\tilde\nabla_mF_i}
{\nu^2}\right.\,+\\
\vspace{2ex}
+\,\left.\frac{\nabla_jF_i-\nabla_iF_j}{\nu}\vphantom{\shave{\sum^n_{m=1}}
n^m\,\frac{F_i\,\tilde\nabla_mF_j-F_j\,\tilde\nabla_mF_i}{\nu}}\right)=
\sum^n_{i=1}\sum^n_{j=1}(P^i_\varepsilon\,H^j_\sigma-
P^i_\sigma\,H^j_\varepsilon)\,\tilde\nabla_jF_i.
\endgathered\hskip -2em
\tag9.12
$$\par
     Let's postpone for a while further transformation of \thetag{9.12}.
Instead, let's study the quantities $H^j_\varepsilon$ introduced by
\thetag{9.11}. They have transparent geometric interpretation. In
\thetag{9.11} we find the quantities $b^m_k$ that were derived from
$b_{kr}$ by index raising procedure (see formula \thetag{7.13}). They
are the components of inner tensor field $\bold B$ of type $(1,1)$
in $S$. The value of such field at the point $p_0\in S$ can be
interpreted as linear operator in the tangent space $T_{p_0}(S)$ to $S$.
Tangent space $T_{p_0}(S)$ is naturally embedded into the tangent space
$T_{p_0}(M)$ as $(n-1)$-dimensional hyperplane $\alpha$ (see Fig\. 9.1).
We have orthogonal projector $\bold P$ onto this hyperplane. Therefore
composition $\bold H=\bold B\compos\bold P$ can be interpreted as
linear operator in $T_{p_0}(M)$. Let's calculate the matrix of this
operator in local coordinates $x^1,\,\ldots,\,x^n$. Suppose that
$\bold y$ is some arbitrary tangent vector at the point $p_0$ on $M$.
Then
$$
\bold H(\bold y)=\bold B\compos\bold P(\bold y)=\bold B(\bold P\bold y).
\hskip -3em
\tag9.13
$$
In order to calculate projection $\bold P\bold y$ in \thetag{9.13} we
use formula \thetag{9.10}. The result of applying $\bold B$ to base
vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ in hyperplane
$\alpha$ is defined by matrix of $\bold B$:
$$
\bold B(\boldsymbol\tau_k)=\sum^{n-1}_{m=1}b^m_k\,\boldsymbol\tau_m.
\hskip -3em
\tag9.14
$$
Combining \thetag{9.13}, \thetag{9.14}, and \thetag{9.10}, for the
vector $\bold H(\bold y)$ in \thetag{9.13} we get 
$$
\bold H(\bold y)=\sum^{n-1}_{m=1}\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}
(\boldsymbol\tau_a\,|\,\bold y)\,\rho^{ak}\,b^m_k\,\boldsymbol\tau_m.
\hskip -3em
\tag9.15
$$
Now let's bring vectorial equality \thetag{9.15} to coordinate form:
$$
\sum^n_{\varepsilon=1}H^j_\varepsilon\,y^\varepsilon=
\sum^n_{\varepsilon=1}\sum^n_{s=1}\sum^{n-1}_{m=1}\sum^{n-1}_{a=1}
\sum^{n-1}_{k=1}g_{\varepsilon s}\,\tau^s_a\,y^\varepsilon\,
\rho^{ak}\,b^m_k\,\tau^j_m.
$$
Components of vector $\bold y$ are arbitrary real numbers. Taking into
account this fact, we find that components of operator $\bold H$ in local
coordinates $x^1,\,\ldots,\,x^n$ are given exactly by formula \thetag{9.11}.
\proclaim{Theorem 9.3} Operator $\bold H=\bold B\compos\bold P$ in
tangent space $T_{p_0}(M)$ defined by second fundamental form of
hypersurface $S$ according to the formula \thetag{9.11} for its
components is symmetric in Riemannian metric $\bold g$ of the
manifold $M$, i\.~e\.
$$
(\bold x\,|\,\bold H\bold y)=(\bold H\bold x\,|\,\bold y),
\hskip -3em
\tag9.16
$$
where $\bold x$ and $\bold y$ are arbitrary vectors from tangent space
$T_{p_0}(M)$ at the point $p_0\in S$.
\endproclaim
\demo{Proof} In order to prove the relationship \thetag{9.16} we use
the above formula \thetag{9.15} for the vector $\bold H\bold y=
\bold H(\bold y)$. This yields
$$
(\bold x\,|\,\bold H\bold y)=\sum^{n-1}_{m=1}\sum^{n-1}_{a=1}
\sum^{n-1}_{k=1}(\boldsymbol\tau_a\,|\,\bold y)\,\rho^{ak}\,b^m_k\,
(\boldsymbol\tau_m\,|\,\bold x).
$$
Taking into account \thetag{7.13}, this relationship can be transformed
as follows:
$$
(\bold x\,|\,\bold H\bold y)=\sum^{n-1}_{m=1}\sum^{n-1}_{a=1}
\sum^{n-1}_{k=1}\sum^{n-1}_{r=1}(\boldsymbol\tau_a\,|\,\bold y)
\,\rho^{ak}\,b_{kr}\,\rho^{rm}\,(\boldsymbol\tau_m\,|\,\bold x).
\hskip -3em
\tag9.17
$$
Right hand side of the equality \thetag{9.17} is invariant under the
transposition of vectors $\bold x$ and $\bold y$. This is due to the
symmetry in components of second fundamental form of $S$ (see equality
\thetag{7.12} in \S\,7). Therefore we get the required relationship
\thetag{9.16}: \pagebreak $(\bold x\,|\,\bold H\bold y)=(\bold y\,|\,
\bold H\bold x)=(\bold H\bold x\,|\,\bold y)$.\qed\enddemo


    Symmetric in metric $\bold g$ operator $\bold H$ commutate with
operator $\bold P$ of orthogonal projection to hyperplane $\alpha=
T_{p_0}(S)$, i\.~e\. we have:
$$
\bold P\compos\bold H=\bold H\compos\bold P=\bold H.\hskip -3em
\tag9.18
$$
Let $\bold x\in T_{p_0}(M)$. Operator $\bold H$ maps vector $\bold x$
to the vector $\bold H\bold x\in \alpha$. Therefore, applying $\bold P$
to $\bold H\bold x$, we get the same vector $\bold H\bold x$, i\.~e\.
$\bold P(\bold H\bold x)=\bold H\bold x$. This means that $\bold P\compos
\bold H=\bold H$. On the other hand $\bold H\compos\bold P=\bold B\compos
\bold P^2$. If we recall that $\bold P^2=\bold P$, then we get $\bold H
\compos\bold P=\bold B\compos\bold P=\bold H$. This proves the equality
\thetag{9.18}.
\proclaim{Theorem 9.4 (\bf on the second fundamental form)} Let $\bold v_0$
be nonzero vector at some point $p_0$ on Riemannian manifold $M$, and let
$\bold P$ be the operator of orthogonal projection onto the hyperplane
$\alpha$ perpendicular to $\bold v_0$. Then for any linear operator $\bold H$
in $T_{p_0}(M)$ symmetric in metric $\bold g$ of Riemannian manifold $M$
and satisfying the relationships \thetag{9.18} one can find hypersurface
$S$ passing through the point $p_0$ and perpendicular to $\bold v_0$ such
that matrix elements of the operator $\bold H$ are determined by second
fundamental form of $S$ according to the formula \thetag{9.11}.
\endproclaim
    We shall prove this theorem in \S\,10 (see below). Now we use it
to analyze the equations \thetag{9.12}. Identically zero operator
$\bold H=0$ satisfy the hypothesis of theorem~9.4. Therefore its 
matrix $H^i_j=0$ can be present in right hand side of \thetag{9.12}.
Substituting $H^j_\sigma=0$ and $H^j_\varepsilon=0$ into \thetag
{9.12} we obtain
$$
\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\nabla_jF_i-\nabla_iF_j+\shave{\sum^n_{m=1}}n^m\,\frac{F_i\,
\tilde\nabla_mF_j-F_j\,\tilde\nabla_mF_i}{\nu}\right)=0.
$$
Left  hand  side  of  \thetag{9.12}  doesn't  depend  on   second 
fundamental
form of $S$, hence it doesn't depend on $\bold H$. The above equality
means that left hand side of \thetag{9.12} is equal to zero. Therefore
right hand side of \thetag{9.12} is zero too:
$$
\sum^n_{i=1}\sum^n_{j=1}(P^i_\varepsilon\,H^j_\sigma-P^i_\sigma\,
H^j_\varepsilon)\,\tilde\nabla_jF_i=0.\hskip -3em
\tag9.19
$$
First of the above two relationships can be written in the form that
doesn't depend on $S$ at all. In order to do it let's recall that
normal vector to $S$ at the point $p_0$ is directed along the vector
of velocity, while function $\nu$ iz normalized at this point by
$\nu=\nu_0=|\bold v_0|$. Hence $\bold n$ can be replaced by $\bold N$,
and $\nu$ can be replaced by $v=|\bold v|$:
$$
\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\nabla_jF_i-\nabla_iF_j+\shave{\sum^n_{m=1}}N^m\,\frac{F_i\,
\tilde\nabla_mF_j-F_j\,\tilde\nabla_mF_i}{v}\right)=0.
$$
Arbitrariness in the choice of point $p_0$ (see definition~9.1) means
that this equality holds at all points $q=(p,\bold v)$ of tangent
bundle $TM$, where $\bold v\neq 0$. It is the differential equation
for the force field of dynamical system on $M$ that follows from
additional normality condition for it. For the sake of more symmetry
we shall write it as
$$
\gathered
\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_i\,\tilde\nabla_mF_j}{v}
-\nabla_iF_j\right)=\\
\vspace{1ex}
=\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_j\,\tilde\nabla_mF_i}{v}
-\nabla_jF_i\right).
\endgathered\hskip -3em
\tag9.20
$$\par
    The equation \thetag{9.19}, unlike to \thetag{9.20}, require
further transformation, since it contain the quantities $H^j_\sigma$
and $H^j_\varepsilon$, which depend on the choice of hypersurface $S$.
Let's consider a linear operator in $T_{p_0}(M)$ with the following
components:
$$
K^\sigma_\varepsilon=\sum^n_{m=1}\sum^n_{i=1}\sum^n_{j=1}
g^{\sigma m}\,P^j_m\,\tilde\nabla_jF_i\,P^i_\varepsilon.
\hskip -3em
\tag9.21
$$
Operator $\bold K$ with components \thetag{9.21} satisfies the
relationships
$$
\bold K\compos\bold P=\bold P\compos\bold K=\bold K.
\hskip -3em
\tag9.22
$$
The relationships \thetag{9.22} are analogous to the relationships
\thetag{9.18} for the operator $H$. They can be verified by direct
calculations, if we take into account that $\bold P^2=\bold P$ and
take into account the equalities
$$
\sum^n_{m=1}g^{\sigma m}\,P^j_m=\sum^n_{m=1}P^\sigma_m\,g^{mj}
\hskip -3em
\tag9.23
$$
that reflect the symmetry of projector $\bold P$ with respect to
Riemannian metric $\bold g$ on $M$. Further let's consider the
operator $\bold M=\bold H\compos\bold K$, and let's calculate its
components in local coordinates. For the components of $\bold M$
we obtain
$$
M^\gamma_\varepsilon=\sum^n_{\sigma=1}H^\gamma_\sigma\,
K^\sigma_\varepsilon=\sum^n_{\sigma=1}\sum^n_{i=1}\sum^n_{j=1}
g^{\gamma\sigma}\,H^j_\sigma\,\tilde\nabla_jF_i\,P^i_\varepsilon.
$$
In deriving this formula we used \thetag{9.23}, and used one of
the relationships \thetag{9.18}. By means of $\bold M$ we define
bilinear form
$$
\boldsymbol\Theta(\bold x,\bold y)=(\bold x\,|\,\bold M\bold y).
\hskip -3em
\tag9.24
$$
Form \thetag{9.24} is associated with tensor $\boldsymbol\Theta$
of type $(0,2)$. Here are its components:
$$
\theta_{\sigma\varepsilon}=\sum^n_{i=1}\sum^n_{j=1}H^j_\sigma\,
\tilde\nabla_jF_i\,P^i_\varepsilon.\hskip -3em
\tag9.25
$$
Let's compare formulas \thetag{9.25} and \thetag{9.19}. It's easy to
note that \thetag{9.19} is exactly the condition of symmetry for bilinear
form $\boldsymbol\Theta$, \pagebreak i\.~e\. $\theta_{\sigma\varepsilon}=
\theta_{\varepsilon\sigma}$ or $\boldsymbol\Theta(\bold x,\bold y)=
\boldsymbol\Theta(\bold y,\bold x)$. Hence $\bold M$ is a symmetric
operator in Riemannian metric $\bold g$ of the manifold $M$:
$$
(\bold x\,|\,\bold M\bold y)=(\bold M\bold x\,|\,\bold y),
\hskip -3em
\tag9.26
$$
Let's state this result as the following lemma.
\proclaim{Lemma 9.1} The equations \thetag{9.19} are equivalent to
the symmetry of operator $\bold M=\bold H\compos\bold K$ in the metric
$\bold g$ of $M$, operators $\bold H$ and $\bold K$ being defined by
their components \thetag{9.11} and \thetag{9.21}.
\endproclaim
     First let's take $\bold H=\bold P$ and let's apply theorem~9.4
to such operator $\bold H$. This is correct, since operator $\bold H=
\bold P$ is symmetric, and it satisfies the relationships \thetag{9.18}.
From theorem~9.4 we get that operator $\bold H=\bold P$ can be defined
by second fundamental form of some hypersurface $S$ passing through
the point $p_0$. For $\bold H=\bold P$ we obtain $\bold M=\bold K$.
Therefore due to lemma~9.1 we conclude that operator $\bold K$ with
components \thetag{9.21} is symmetric in Riemannian metric $\bold g$
of the manifold $M$.\par
     The equality $\bold H=\bold P$, which holds for some particular
choice of hypersurface $S$, is only some special case, when theorem~9.4
is applicable. In general, operator $\bold H$ is an arbitrary symmetric
operator satisfying the relationships \thetag{9.18} and such that
composition of two symmetric operators $\bold H$ and $\bold K$ is
symmetric operator $\bold M=\bold H\compos\bold K$. This is possible
if and only if operators $\bold H$ and $\bold K$ are commutating.
Indeed, from the equality \thetag{9.26} we derive
$$
(\bold x\,|\,\bold H(\bold K\bold y))=(\bold H(\bold K\bold x)\,|\,
\bold y).\hskip -3em
\tag9.27
$$
On the other hand symmetry of operators $\bold H$ and $\bold K$ yields
$$
(\bold x\,|\,\bold H(\bold K\bold y))=(\bold H\bold x\,|\,\bold K
\bold y)=(\bold K(\bold H\bold x)\,|\,\bold y).\hskip -3em
\tag9.28
$$
Comparing \thetag{9.27} and \thetag{9.28}, and taking into account
the arbitrariness of $\bold x$ and $\bold y$ in these relationships,
we obtain the operator equality
$$
\bold H\compos\bold K=\bold K\compos\bold H.\hskip -3em
\tag9.29
$$
This equality \thetag{9.29} means that $\bold H$ and $\bold K$ are
commutating. We state this result in form of the following lemma.
\proclaim{Lemma 9.2} Suppose that the equations \thetag{9.19} are
fulfilled for any hypersurface $S$ passing through the point $p_0$
and perpendicular to the vector $\bold v_0$ at this point. Then
operator $\bold K$ commutates with any symmetric operator $\bold H$
that satisfies the relationships \thetag{9.18}.
\endproclaim
    Let $\alpha$ be a hyperplane perpendicular to the vector
$\bold v_0$. Tangent space to the manifold $M$ at the point
$p_0$ is represented by a sum of two subspaces
$$
\pagebreak
T_{p_0}(M)=\alpha\oplus\langle\bold v_0\rangle.\hskip -3em
\tag9.30
$$
Here $\langle\bold v_0\rangle$ is a linear span of vector
$\bold v_0$. Each subspace in the expansion \thetag{9.30}
is invariant under the action of operators $\bold H$ and
$\bold K$. This is easily derived from \thetag{9.18} and
\thetag{9.22}. Moreover, the restrictions of $\bold H$ and
$\bold K$ to $\langle\bold v_0\rangle$ are zero. Let's
choose some orthonormal base $\bold E_1,\,\ldots,\,\bold E_{n-1}$
in hyperplane $\alpha$ and let's complete it by unitary vector
$\bold E_n$ directed along the vector $\bold v_0$. Operators
$\bold H$ and $\bold K$ in the base $\bold E_1,\,\ldots,\,\bold E_n$
are defined by the following symmetric matrices:
$$
\gather
H=\Vmatrix
H^1_1 & \hdots & H^1_{n-1} & 0\\
\vspace{1ex}
\vdots &\ddots & \vdots &\vdots\\
\vspace{2ex}
H^{n-1}_1 & \hdots & H^{n-1}_{n-1} & 0\\
\vspace{2ex}
0 &\hdots & 0 & 0
\endVmatrix,\hskip -3em
\tag9.31\\
\vspace{3ex}
K=\Vmatrix
K^1_1 & \hdots & K^1_{n-1} & 0\\
\vspace{1ex}
\vdots &\ddots & \vdots &\vdots\\
\vspace{2ex}
K^{n-1}_1 & \hdots & K^{n-1}_{n-1} & 0\\
\vspace{2ex}
0 &\hdots & 0 & 0
\endVmatrix.\hskip -3em
\tag9.32
\endgather
$$
Matrix of projector $\bold P$ in the base $\bold E_1,\,\ldots,\,
\bold E_n$ has similar structure:
$$
P=\Vmatrix
1 & \hdots & 0 & 0\\
\vspace{0.3ex}
\vdots &\ddots & \vdots &\vdots\\
\vspace{0.3ex}
0 & \hdots & 1 & 0\\
\vspace{0.3ex}
0 &\hdots & 0 & 0
\endVmatrix.\hskip -3em
\tag9.33
$$
Operator $\bold K$ is some fixed operator with matrix \thetag{9.32},
but $\bold H$, as it follows from lemma~9.2, is an arbitrary operator
with matrix of the form \thetag{9.31}. Using this arbitrariness, let's
choose the matrix of operator $\bold H$ to be the diagonal matrix with
distinct elements $H^1_1,\,\ldots,\,H^{n-1}_{n-1},\,H^n_n=0$:
$$
H=\Vmatrix
H^1_1 & \hdots & 0 & 0\\
\vspace{1ex}
\vdots &\ddots & \vdots &\vdots\\
\vspace{2ex}
0 & \hdots & H^{n-1}_{n-1} & 0\\
\vspace{2ex}
0 &\hdots & 0 & 0
\endVmatrix.\hskip -3em
\tag9.34
$$
Commutativity of operators $\bold K$ and $\bold H$ means that their
matrices \thetag{9.32} and \thetag{9.34} are also commutating. This
yields $K^i_j\,H^j_j=H^i_i\,K^i_j$. Diagonal elements in matrix
\thetag{9.34} are distinct: $H^i_i\neq H^j_j$ for $i\neq j$. Therefore
$K^i_j=0$ for $i\neq j$, i\.~e\. matrix of operator $\bold K$ is
diagonal in the base of vectors $\bold E_1,\,\ldots,\,\bold E_n$.
This reduces matrix \thetag{9.32} to the following form:
$$
K=\Vmatrix
K^1_1 & \hdots & 0 & 0\\
\vspace{1ex}
\vdots &\ddots & \vdots &\vdots\\
\vspace{2ex}
0 & \hdots & K^{n-1}_{n-1} & 0\\
\vspace{2ex}
0 &\hdots & 0 & 0
\endVmatrix.\hskip -3em
\tag9.35
$$
Now let's use again the arbitrariness of operator $\bold H$. In this
case we choose it so that its matrix \thetag{9.31} has the form:
$$
H=\Vmatrix
0 & \hdots & 1 &\hdots & 0\\
\vspace{0.3ex}
\vdots &\ddots & \vdots & &\vdots\\
\vspace{0.3ex}
1 & \hdots & 0 &\hdots & 0\\
\vspace{0.3ex}
\vdots & & \vdots &\ddots &\vdots\\
\vspace{0.3ex}
0 &\hdots & 0 &\hdots & 0
\endVmatrix.\hskip -3em
\tag9.36
$$
The only nonzero elements in matrix \thetag{9.36} are $H^1_i=1$ and
$H^i_1=1$, where $1<i<n$. Commutativity of $\bold K$ and $\bold H$
implies commutativity of their matrices \thetag{9.35} and \thetag{9.36}.
This yields $K^1_1\,H^1_i=H^1_i\,K^i_i$ and $K^i_i\,H^i_1=H^i_1\,K^1_1$. 
These relationships are equivalent to $K^i_i=K^1_1$. Due to arbitrariness
in $i$ we get
$$
K^1_1=\ldots=K^{n-1}_{n-1}=\lambda.\hskip -3em
\tag9.37
$$
The relationship \thetag{9.37} reduces matrix \thetag{9.35} to the
following form:
$$
K=\Vmatrix
\lambda & \hdots & 0 & 0\\
\vspace{0.3ex}
\vdots &\ddots & \vdots &\vdots\\
\vspace{0.3ex}
0 & \hdots & \lambda & 0\\
\vspace{0.3ex}
0 &\hdots & 0 & 0
\endVmatrix.
$$
Now it's obvious that such matrix commutates with any matrix of the form
\thetag{9.31}. Comparing it with \thetag{9.33}, we get $\bold K=\lambda
\cdot\bold P$. We state this result as a third lemma. 
\proclaim{Lemma 9.3} Operator $\bold K$ satisfying the relationships
\thetag{9.22} commutates with arbitrary symmetric operator $\bold H$
satisfying the relationships \thetag{9.18} if and only if it differs
from projector $\bold P$ only by some numeric multiple: $\bold K=\lambda
\cdot\bold P$.
\endproclaim
    Thus, by means of theorem~9.4 and by means of three lemmas~9.1, 9.2,
and 9.3 we reduced \thetag{9.19} to the operator equality
$$
\bold K=\lambda\cdot\bold P.\hskip -3em
\tag9.38
$$
Let's write \thetag{9.38} in local coordinates. In order to do it we
use formula \thetag{9.21} that determines components of operator
$\bold K$. As a result we get the following relationship:
$$
\sum^n_{m=1}\sum^n_{i=1}\sum^n_{j=1}g^{\sigma m}\,P^j_m\,
\tilde\nabla_jF_i\,P^i_\varepsilon=\lambda\,
P^\sigma_\varepsilon.\hskip -3em
\tag9.39
$$
In formula \thetag{9.39} one can lower index $\sigma$ and one can
raise index $\varepsilon$. If we take into account symmetry of operator
$\bold P$, we can write 
$$
\sum^n_{i=1}\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF^i\,
P^\varepsilon_i=\lambda\,P^\varepsilon_\sigma.\hskip -3em
\tag9.40
$$
In the equations \thetag{9.40} we have numeric parameter $\lambda$.
Let's find it by calculating traces of operators in both sides of
the equality \thetag{9.38}:
$$
\tr\bold K=\lambda\cdot\tr\bold P.\hskip -3em
\tag9.41
$$
It's easy to calculate trace of projector: $\tr\bold P=n-1$. It is
nonzero, since we consider the case $n\geqslant 3$. Finding $\lambda$
from \thetag{9.41}, we can write \thetag{9.40} in the form that
has no indefinite parameters:
$$
\sum^n_{i=1}\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF^i\,
P^\varepsilon_i=\sum^n_{i=1}\sum^n_{j=1}\sum^n_{m=1}
\frac{P^j_m\,\tilde\nabla_jF^i\,P^m_i}{n-1}\,P^\varepsilon_\sigma.
\hskip -3em
\tag9.42
$$\par
    Similar to \thetag{9.20}, the equations \thetag{9.42} are the
partial differential equations for the force field $\bold F$ of
Newtonian dynamical system. The above calculations can be reverted.
From \thetag{9.42} we could get \thetag{9.38}. Then from \thetag{9.38}
and \thetag{9.22} we could get the equality \thetag{9.29}, which
now is written as
$$
\bold H\compos\bold K=\bold K\compos\bold H=\lambda\cdot\bold H.
$$
Hence operator $\bold M=\bold H\compos\bold K=\lambda\cdot\bold H$ is
symmetric. Due to lemma~9.1 this is equivalent to \thetag{9.19}.
Combining \thetag{9.19} and \thetag{9.20}, we get the equality
\thetag{9.12}, which, in turn, is equivalent to \thetag{8.21}. The
equality \thetag{8.21} is the compatibility condition for the system
of Pfaff equations \thetag{7.16}. It provides solvability of these
equations for arbitrary choice of the point $p_0\in S$, and for
arbitrary choice of numeric parameter $\nu_0\neq 0$ in normalization
condition \thetag{9.1}. As a result we can formulate the main
theorem of this section, which was proved by the above considerations.
\proclaim{Theorem 9.5} Newtonian dynamical system on Riemannian
manifold of the dimension $n\geqslant 3$ satisfies additional
normality condition if and only if its force field satisfies the
equations \thetag{9.20} and \thetag{9.42} at all points $q=(p,\bold v)$
of tangent bundle $TM$, except for those, where $\bold v=0$.
\endproclaim
    The equations \thetag{9.20} and \thetag{9.42} are called
{\i additional normality conditions}. They were first derived
in \cite{38} (see also preprint \cite{34}) for the case
$M=\Bbb R^n$. \pagebreak The above derivation of the equations
\thetag{9.20} and \thetag{9.42} is based on the theorem~9.4,
which is not yet proved.
\head
\S\,10. Proof of the theorem on a second quadratic form.
\endhead
    In theorem~9.4 on the second fundamental form we consider a
point $p_0$ on Riemannian manifold $M$ and nonzero vector $\bold v_0$
at this point. Vector $\bold v_0$ defines a hyperplane $\alpha$ in
$T_{p_0}(M)$ perpendicular to $\bold v_0$ and an operator $\bold P$
of orthogonal projection onto the hyperplane $\alpha$. Then various
hypersurfaces passing through $p_0$ and tangent to $\alpha$ at that
point are considered. Basic tangent vectors $\boldsymbol\tau_1,\,
\ldots,\,\boldsymbol\tau_{n-1}$ at the point $p_0$ for any of such
hypersurfaces are in the hyperplane $\alpha$ and form the base in it.
Theorem~9.4 (which we need to prove) asserts that for any symmetric
linear operator $\bold H$ in $T_{p_0}(M)$ satisfying the relationships
\thetag{9.18} its matrix is defined by vectors $\boldsymbol\tau_1,\,
\ldots,\,\boldsymbol\tau_{n-1}$ and by second fundamental form of
some hypersurface $S$ according to the formula \thetag{9.11}. Vector
$\bold v_0$ and hyperplane $\alpha$ define the expansion of tangent
space $T_{p_0}(M)$ into a direct sum of two subspaces:
$$
T_{p_0}(M)=\alpha\oplus\langle\bold v_0\rangle.\hskip -3em
\tag10.1
$$
WE have already considered this expansion (see \thetag{9.30}).
Let $\bold H$ be some arbitrary symmetric in metric $\bold g$
on $M$ linear operator $\bold H\!:T_{p_0}(M)\to T_{p_0}(M)$
satisfying the relationships \thetag{9.18}. From $\bold H=\bold P
\compos\bold H$ we find that subspace $\alpha$ is invariant under
the action of $\bold H$. From $\bold H=\bold H\compos\bold P$, in
turn, we get that subspace $\langle\bold v_0\rangle$ in \thetag{10.1}
is also invariant under the action of $\bold H$, the restriction
of $\bold H$ to $\langle\bold v_0\rangle$ being zero. Let's denote
by $\bold B$ the restriction of $\bold H$ to hyperplane $\alpha$:
$$
\bold B=\bold H\,\hbox{\vrule height 8pt depth 6pt
width 0.5pt}_{\,\alpha}.
$$
Such restriction is a symmetric linear operator $\bold B\!:\alpha\to\alpha$
such that $\bold H=\bold B\compos\bold P$. And conversely, for any
symmetric linear operator $\bold B\!:\alpha\to\alpha$ the composition
$\bold H=\bold B\compos\bold P$ is symmetric operator $\bold H\!:
T_{p_0}(M)\to T_{p_0}(M)$ satisfying the relationships \thetag{9.18}.
\par
    Let $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ be some
arbitrary base in hyperplane $\alpha$. Consider a matrix with the
following components
$$
b_{ij}=(\boldsymbol\tau_i\,|\,\bold B\boldsymbol\tau_j).
\hskip -3em
\tag10.2
$$
Matrix \thetag{10.2} is symmetric, i\.~e\. $b_{ij}=b_{ji}$, this is
due to the symmetry of operator $\bold B$. Matrix with components
\thetag{10.2} can be obtained from the matrix of operator $\bold B$
in the base $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$
by means of lowering index procedure:
$$
b_{ij}=\sum^{n-1}_{m=1}\rho_{im}\,b^m_j.
\hskip -3em
\tag10.3
$$
Here $\rho_{im}$ \pagebreak are components of metric tensor for the metric
$\bold g$ restricted to the hyperplane $\alpha$. Formula \thetag{10.3}
for $b_{ij}$ can be inverted as follows:
$$
b^m_k=\sum^{n-1}_{i=1}\rho^{mi}\,b_{ik}.
\hskip -3em
\tag10.4
$$
Operator $\bold B$ is reconstructed by matrix \thetag{10.2} according
to the formula
$$
\bold B\bold y=\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}\sum^{n-1}_{m=1}
(\boldsymbol\tau_a\,|\,\bold y)\,\rho^{ak}\,b^m_k\,\boldsymbol
\tau_m.\hskip -3em
\tag10.5
$$
Formula \thetag{10.5} and formula \thetag{10.4} inverting \thetag{10.3}
mean that defining symmetric operator $\bold B$ in hyperplane $\alpha$
is equivalent to fixing some base $\boldsymbol\tau_1,\,\ldots,\,
\boldsymbol\tau_{n-1}$ in $\alpha$ and choosing some symmetric
matrix $b_{ij}$. Substituting \thetag{10.5} into the formula $\bold H=
\bold B\compos\bold P$, for components of operator $\bold H$ we get
the expression, which exactly the same as in \thetag{9.11}. Therefore
theorem~9.4 is the consequence of the following more simple
auxiliary theorem.
\proclaim{Theorem 10.1} Suppose that in tangent space $T_{p_0}(M)$
at some point $p_0$ on Riemannian manifold of the dimension $n\geqslant
3$ we choose some nonzero vector $\bold v_0$, mark hyperplane
$\alpha$ perpendicular to $\bold v_0$, mark some base $\boldsymbol
\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ in $\alpha$, and choose some
symmetric $(n-1)\times (n-1)$ matrix $b$. Then there exists some
hypersurface $S$ passing through the point $p_0$ tangent to $\alpha$
and there exist some local coordinates $u^1,\,\ldots,\,u^{n-1}$ on $S$
such that vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$
are coordinate tangent vectors to $S$ at the point $p_0$, while $b$
coincides with the matrix of second fundamental form for $S$ at this
point.
\endproclaim
    Before proving the theorem let's consider the following lemma that
states the fact, which is well-known in geometry.
\proclaim{Lemma 10.1} For any point $p_0$ on Riemannian manifold $M$
there exist local coordinates $x^1,\,\ldots,\,x^n$ such that component
of metric connection $\Gamma^k_{ij}$ vanishes at the point $p_0$ in
these local coordinates.
\endproclaim
\demo{Proof} Suppose that $\tx^1,\,\ldots,\,\tx^n$ are some arbitrary
local coordinates in some neighborhood of the point $p_0$. Without loss
of generality we can assume that
$$
\tx^1(p_0)=\ldots=\tx^n(p_0)=0.
$$
Let $\tilde\Gamma^k_{ij}(p_0)$ be components of metric connection at the
point $p_0$ in these local coordinates. Let's define new local coordinates
$x^1,\,\ldots,\,x^n$ by the following formula
$$
\pagebreak
x^k=\tx^k+\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1}\tilde\Gamma^k_{ij}(p_0)
\,\tx^i\,\tx^j.\hskip -3em
\tag10.6
$$
For the components of transition matrix $S$ at the point $p_0$ from
\thetag{10.6} we get
$$
S^k_i(p_0)=\frac{\partial x^k}{\partial\tx^i}\,\hbox{\vrule height 12pt
depth 9pt width 0.5pt}_{\,p_0}=\delta^k_i.
$$
This means that $S$ is unitary matrix at the point $p_0$. Matrix $T=S^{-1}$
then is also unitary matrix at the point $p_0$. In order to calculate
components of metric connection $\Gamma^k_{ij}$ in newly defined local
coordinates let's use formula \thetag{4.11} from Chapter
\uppercase\expandafter{\romannumeral 3}: 
$$
\tilde\Gamma^k_{ij}=\sum^n_{m=1}\sum^n_{a=1}\sum^n_{c=1} T^k_m\,S^a_i
\,S^c_j\,\Gamma^m_{ac}+\sum^n_{m=1} T^k_m\,\frac{\partial
S^m_i}{\partial\tx^j}.\hskip -3em
\tag10.7
$$
If we take into account that $S$ and $T$ are unitary matrices at the
point $p_0$, then for components of metric connection $\Gamma^k_{ij}(p_0)$
and $\tilde\Gamma^k_{ij}(p_0)$ from \thetag{10.7} we get
$$
\tilde\Gamma^k_{ij}(p_0)=\Gamma^k_{ij}(p_0)+\frac{\partial^2 x^k}
{\partial\tx^i\,\partial\tx^j}\,\hbox{\vrule height 12pt
depth 9pt width 0.5pt}_{\,p_0}.\hskip -3em
\tag10.8
$$
Differentiating \thetag{10.6}, we calculate second derivative in
\thetag{10.8}:
$$
\frac{\partial^2 x^k}{\partial\tx^i\,\partial\tx^j}\,
\hbox{\vrule height 12pt depth 9pt width 0.5pt}_{\,p_0}
=\tilde\Gamma^k_{ij}(p_0).\hskip -3em
\tag10.9
$$
Now from \thetag{10.8} and \thetag{10.9} we see that components of
metric connection $\Gamma^k_{ij}$ for newly constructed local
coordinates \thetag{10.6} are zero at the point $p_0$, as it was
stated in lemma. Lemma is proved.\qed\enddemo
{\bf Remark}. Suppose that at the point $p_0$ we have $n$ vectors
$\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_n$ forming the base
in tangent space $T_{p_0}(M)$. Local coordinates $x^1,\,\ldots,\,x^n$
with $\Gamma^k_{ij}(p_0)=0$ can be chosen so that corresponding
coordinate tangent vectors
$$
\bold E_1=\frac{\partial}{\partial x^1},\ \ldots,\ \bold E_n=
\frac{\partial}{\partial x^n}\hskip -4em
\tag10.10
$$
at the point $p_0$ will coincide with vectors $\boldsymbol\tau_1,\,
\ldots,\,\boldsymbol\tau_n$. Indeed, the condition of vanishing
$\Gamma^k_{ij}$ at the point $p_0$ is invariant under the linear
change of local coordinates
$$
\xalignat 2
&x^k=\sum^n_{i=1}S^k_i\,\tx^i,
&&\tx^i=\sum^n_{i=1}T^i_k\,x^k,
\endxalignat
$$
where $S$ and $T=S^{-1}$ are some constant matrices. While vectors
\thetag{10.10} under such linear change of local coordinates are
transformed as follows:
$$
\xalignat 2
&\tilde\bold E_i=\sum^n_{i=1}S^k_i\,\bold E_k,
&&\bold E_k=\sum^n_{i=1}T^i_k\,\tilde\bold E_i.
\endxalignat
\pagebreak
$$
Therefore we can bring vectors $\bold E_1,\,\ldots,\,\bold E_n$ to
the coincidence with $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_n$,
keeping condition $\Gamma^k_{ij}(p_0)=0$ fulfilled.
\demo{Proof of theorem 10.1} In theorem~10.1 we consider the point
$p_0$ with nonzero vector $\bold v_0$ at this point and the base 
$\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ in hyperplane
$\alpha\subset T_{p_0}(M)$ perpendicular to the vector $\bold v_0$.
By denoting
$$
\boldsymbol\tau_n=\frac{\bold v_0}{|\bold v_0|}
$$
we complete base $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$
up to the base $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_n$ in
$T_{p_0}(M)$. Let's apply lemma~10.1 and choose local coordinates
$x^1,\,\ldots,\,x^n$ in the neighborhood of $p_0$ on $M$ such that
$\Gamma^k_{ij}(p_0)=0$ and such that
$$
\bold E_1=\boldsymbol\tau_1,\ \ldots,\ \bold E_{n-1}=\boldsymbol
\tau_{n-1},\qquad\bold E_n=\boldsymbol\tau_n=\frac{\bold v_0}
{|\bold v_0|}.
$$
Moreover, without loss of generality we can assume that
$x^1(p_0)=\ldots=x^n(p_0)=0$. Now let's define hypersurface $S$
by the equation
$$
x^n=\frac{1}{2}\sum^{n-1}_{i=1}\sum^{n-1}_{j=1}b_{ij}\,x^i\,x^j
\hskip -4em
\tag10.11
$$
in local coordinates chosen above. Here $b_{ij}$ are components
of constant symmetric matrix $b$ from the statement of theorem~10.1.
We can write \thetag{10.11} in parametric form by introducing local
coordinates $u^1,\,\ldots,\,u^{n-1}$ on $S$:
$$
\align
x^k&=u^k\text{ \ for \ }k=1,\,\ldots,\,n-1,\hskip -4em\\
\vspace{-1.4ex}
&\tag10.12\\
\vspace{-1.4ex}
x^n&=\frac{1}{2}\sum^{n-1}_{i=1}\sum^{n-1}_{j=1}b_{ij}\,u^i\,u^j.
\endalign
$$
Let's differentiate parametric equations \thetag{10.12} with respect to
the variable $u^i$:
$$
\frac{\partial x^k}{\partial u^i}=1\text{ \ for \ }i=k,\quad
\frac{\partial x^k}{\partial u^i}=0\text{ \ for \ }i\neq k<n.
\hskip -4em
\tag10.13
$$
By differentiating $x^n$ with respect to $u^i$ we get the formula
$$
\frac{\partial x^n}{\partial u^i}=\sum^{n-1}_{j=1}b_{ij}\,u^j.
\hskip -4em
\tag10.14
$$
Hypersurface $S$ defined by the equations \thetag{10.12} in parametric
form passes through the point $p_0$; this point has zero local coordinates
on $S$: $u^1(p_0)=\ldots=u^{n-1}(p_0)=0$. Partial derivatives \thetag{10.13}
and \thetag{10.14} form components of $i$-th coordinate tangent vector
$\boldsymbol\tau_i$ of $S$ in local coordinates $x^1,\,\ldots,\,x^n$ on
$M$:
$$
\xalignat 2
&\quad\tau^k_i=
\cases
1 &\text{for \ }i=k,\\
0 &\text{for \ }i\neq k<n,
\endcases
&&\tau^n_i=\sum^{n-1}_{j=1}b_{ij}\,u^j.\hskip -2em
\tag10.15
\endxalignat
$$
Substituting coordinates $u^1(p_0)=\ldots=u^{n-1}(p_0)$ of the point $p_0$
into these formulas, we get $\tau^n_i=0$. This means that $S$ is tangent
to $\alpha$, while coordinate tangent vectors to $S$ at the point $p_0$
coincide with the base $\bold E_1=\boldsymbol\tau_1,\,\ldots,\,\bold
E_{n-1}=\boldsymbol\tau_{n-1}$ in hyperplane $\alpha$. This proves first
proposal in theorem~10.1.\par
    In order to prove second proposal in theorem~10.1 let's calculate
the matrix of second fundamental form of hypersurface $S$ at the point
$p_0$, applying derivational formula of Weingarten \thetag{7.9} for this
purpose:
$$
\nabla_{u^j}\boldsymbol\tau_i=\sum^{n-1}_{m=1}\theta^m_{ij}\,
\boldsymbol\tau_m+\beta_{ij}\,\bold n.\hskip -4em
\tag10.16
$$
Let's calculate components of vector $\nabla_{u^j}\boldsymbol\tau_i$
in left hand side of derivational formula \thetag{10.16} according
to formula \thetag{5.8} from Chapter
\uppercase\expandafter{\romannumeral 4}:
$$
\nabla_{u^j}\tau^k_i=\frac{\partial\tau^k_i}{\partial u^j}+
\sum^n_{m=1}\sum^n_{r=1}\tau^m_j\,\Gamma^k_{mr}\,\tau^r_i.\hskip -4em
\tag10.17
$$
At the point $p=p_0$ we have $\Gamma^k_{mr}(p_0)=0$ due to special choice
of local coordinates $x^1,\,\ldots,\,x^n$ on $M$. For $\tau^k_i$ we apply
formula \thetag{10.15}. Then
$$
\xalignat 2
&\nabla_{u^j}\tau^k_i(p_0)=0\text{ \ for \ }k<n,
&&\nabla_{u^j}\tau^n_i(p_0)=b_{ij}.
\endxalignat
$$
Let's write these relationships in vectorial form, and let's calculate
components of unitary normal vector $\bold n$ to $S$ at the point $p_0$:
$$
\xalignat 2
&\qquad\nabla_{u^j}\boldsymbol\tau_i(p_0)=\Vmatrix 0\\ \vdots\\ 0\\
b_{ij}\endVmatrix,&&\bold n(p_0)=\Vmatrix 0\\ \vdots\\ 0\\ 1\endVmatrix.
\hskip -2em
\tag10.18
\endxalignat
$$
The relationships \thetag{10.15} for the quantities $\tau^i_j$,
written in vectorial form, yield
$$
\boldsymbol\tau_1(p_0)=\Vmatrix 1\\ \vdots\\ 0\\ 0\endVmatrix,\
\ldots,\
\boldsymbol\tau_{n-1}(p_0)=\Vmatrix 0\\ \vdots\\ 1\\ 0\endVmatrix.
\hskip -4em
\tag10.19
$$
Now, substituting \thetag{10.18} and \thetag{10.19} into vectorial
equality \thetag{10.16}, we find the values of parameters $\theta^m_{ij}$
$\beta_{ij}$ at the point $p_0$:
$$
\xalignat 2
&\theta^k_{ij}(p_0)=0,
&&\beta_{ij}=b_{ij}.
\endxalignat
$$
So, we see that matrix of second fundamental form $\beta$ for the
hypersurface $S$ defined by the equation \thetag{10.11} at the point
$p_0$ coincides with matrix $b$. Theorem~10.1 is now proved.\qed\enddemo
    Along with theorem~10.1 we have proved theorem~9.4 on the second
fundamental form. This follows from considerations preceding the
statement of theorem~10.1.
\head
\S\,11. Newtonian dynamical systems admitting the normal shift.
\endhead
\rightheadtext{\S\,11. Newtonian dynamical systems \dots}
     Having proved the theorem on second quadratic form, let's return
to the results of \S\,6 and \S\,9. Central points of these two sections
are definitions~6.1 and 9.1 that introduce weak and additional normality
conditions for Newtonian dynamical systems on Riemannian manifolds.
We managed to write each of these two conditions in form of systems of
partial differential equations for the force field $\bold F$ of the
dynamical system (see equations \thetag{6.16} and \thetag{6.17} for
weak normality condition, and equations \thetag{9.20} and \thetag{9.42}
for additional normality condition).
\definition{Definition 11.1} Weak normality condition from definition~6.1
and additional normality condition from definition~9.1, combined together,
form {\bf complete normality} condition for Newtonian dynamical system with
force field $\bold F$ on Riemannian manifold $M$.
\enddefinition
    The equations \thetag{6.16}, \thetag{6.17}, \thetag{9.20}, and
\thetag{9.42} form complete system of normality equations for the force
field of dynamical system $\bold F$. The whole set of these equations
is equivalent to complete normality condition from definition~11.1. 
Identically zero field $\bold F=0$ satisfies all these equations, therefore
geodesic flow is a simples example of dynamical system that satisfies
complete normality condition.\par
    Let's consider some Newtonian dynamical system with force field
$\bold F$ satisfying complete normality condition. Let $S$ be some
hypersurface in the manifold $M$. Let's mark some point $p_0\in S$ and
choose some local coordinates $u^1,\,\ldots,\,u^{n-1}$ in the
neighborhood of the point $p_0$ on $S$. Suppose $S'$ to be some smaller
piece of hypersurface $S$ containing the point $p_0$. Let's draw the
trajectories of dynamical system
$$
\xalignat 2
&\dot x^k=v^k,&&\nabla_tv^k=F^k,\hskip -3em
\tag11.1
\endxalignat
$$
passing through each point $p$ of $S'$. We define them by
the following initial data:
$$
\xalignat 2
&\qquad x^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=x^k(p),
&&v^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=
\nu(p)\cdot n^k(p).\hskip -3em
\tag11.2
\endxalignat
$$
This arranges the shift $f_t\!:S'\to S'_t$ of $S'\subset S$ along the
trajectories of the dynamical system \thetag{11.1}. Piece $S'$ in $S$
can always be chosen belonging to the transformation class (see
definition~2.1). Due to additional normality condition being the part
of complete normality condition we can choose the function $\nu(p)$ on
$S'$ such that the shift $f_t\!:S'\to S'_t$ approximates normal shift
up to a first order derivatives. This means that all functions of
deviation $\varphi_1,\,\ldots,\,\varphi_{n-1}$ satisfy initial conditions
$$
\xalignat 2
&\qquad \varphi_k\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=0,
&&\dot\varphi_k\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=0
\hskip -3em
\tag11.3
\endxalignat
$$
on initial hypersurface $S'$ (see conditions \thetag{7.5} above).
Function $\nu$ thereby can be chosen belonging to transformation
class (see definition~2.2) and normalized by 
$$
\nu(p_0)=\nu_0,\hskip -3em
\tag11.4
$$
where $\nu_0$ is an arbitrary nonzero real number.\par
   Weak normality condition, which also is a part of complete normality
condition, means that all functions of deviation $\varphi_1,\,\ldots,\,
\varphi_{n-1}$ satisfy linear homogeneous differential equations of
the second order in $t$ on trajectories of shift:
$$
\ddot\varphi_k-\Cal A(t)\,\dot\varphi_k-\Cal B(t)\,\varphi_k=0.
\hskip -3em
\tag11.5
$$
Combining differential equations \thetag{11.5} and initial data
\thetag{11.5}, we find that functions of deviation $\varphi_1,\,
\ldots,\,\varphi_{n-1}$ for the shift $f_t\!:S'\to S'_t$ are
identically zero. This, in turn, means that the shift we are
considering now is a normal shift in the sense of definition~2.3.
\definition{Definition 11.2} Newtonian dynamical system on Riemannian
manifold $M$ is called the system {\bf admitting the normal shift}, if
for any hypersurface $S$ in $M$, for any point $p_0\in S$, and for any
real number $\nu_0\neq 0$ there is a piece $S'$ of hypersurface $S$
belonging to transformation class and containing $p_0$, and there exists
a function $\nu(p)$ belonging to transformation class on $S'$ and
normalized by \thetag{11.4} such that the shift $f_t\!:S'\to S'_t$
defined by this function is a normal shift along the trajectories
of dynamical system.
\enddefinition
    Definition~11.2 states the central concept of this thesis, the
concept of Newtonian dynamical system {\bf admitting the normal shift}.
The condition stated i this definition was first introduced in \cite{39}.
There it was called {\bf strong normality} condition. In earlier papers
\cite{34}, \cite{35}, \cite{38}, and \cite{58} we used more simple version
of this condition, which did not include normalization \thetag{11.4}.
This condition was called {\bf normality condition}. It is more natural
from geometric point of view, but it is not in a good agreement with the
Pfaff equations that arises in our theory. Therefore presently, saying
dynamical system {\bf admitting the normal shift}, we imply the
definition~11.2.
\head
\S\,12. Equivalence of strong and complete normality conditions.
\endhead
\rightheadtext{\S\,12. Equivalence of strong and complete
normality \dots}
     Complete normality condition, which consists of weak normality
condition from definition~6.1 and additional normality condition
from definition~9.1, \pagebreak is sufficient for the strong normality
condition from definition~11.2 to be fulfilled. This follows from above
considerations preceding the statement of definition~11.2. Thus,
Newtonian dynamical system satisfying both weak and additional normality
conditions appears to be admitting the normal shift. It is applicable for
arranging normal shift of any preassigned hypersurface. This result can
be strengthened in form of the following theorem.
\proclaim{Theorem 12.1} {\bf Complete} and {\bf strong} normality
conditions for Newtonian dynamical systems on Riemannian manifolds
are {\bf equivalent} to each other.
\endproclaim
\demo{Proof} As we showed in \S\,11 above, complete normality condition
is {\bf sufficient} for the strong normality condition to be fulfilled.
Now we are to show that it is {\bf necessary} as well. Suppose that strong
normality condition for Newtonian dynamical system \thetag{11.1} is
fulfilled. This means that for any hypersurface $S$, for any point
$p_0\in S$, and for any real number $\nu_0\neq 0$ there is a piece $S'$
of hypersurface $S$ and there exists a function $\nu(p)$ on $S'$
normalized by the condition \thetag{11.4} and such that all functions
of deviations on shift trajectories are identically zero:
$$
\varphi_k(t)=0\text{ \ for all \ }k=1,\,\ldots,\,n-1.
\hskip -3em
\tag12.1
$$
This is the very condition that provides normality of shift $f_t\!:S'\to
S'$. From \thetag{12.2} we get that functions of deviations and their
first derivatives are zero at initial instant of time, i\.~e\. the
initial conditions \thetag{11.3} are fulfilled. Hence strong normality
condition implies additional normality condition to be fulfilled.\par
     Now we shall show that strong normality condition implies weak
normality condition to be fulfilled as well. Let's fix some trajectory
$p(t)$ of Newtonian dynamical system \thetag{11.1} and let's mark
some point $p_0=p(0)$ on it. Suppose that velocity vector $\bold v_0
=\bold v(0)$ at this point is nonzero: $\bold v_0\neq 0$. Then in tangent
space $T_{p_0}(M)$ we can take hyperplane $\alpha$ perpendicular to the
vector $v_0$. Let's consider various hyperplanes $S$ in $M$ passing
through the point $p_0$. For our preassigned trajectory $p(t)$ to be
one of the trajectories of normal shift for $S$ it should be
perpendicular to $S$ at the point $p_0$, i\.~e\. $S$ should be tangent
to hyperplane $\alpha$ at this point. Let's define unitary normal vector
at $p_0$ and a real number $\nu_0\neq 0$:
$$
\xalignat 2
&\bold n=\frac{\bold v_0}{|\bold v_0|},
&&\nu_0=|\bold v_0|.
\endxalignat
$$
This defines smooth field of unitary normal vectors on $S$ in some
neighborhood of $p_0$. Relying upon strong normality condition of
dynamical system, let's choose some piece $S'$ of hypersurface $S$
containing $p_0$, and choose some function $\nu(p)$ on $S'$ that
satisfies the condition \thetag{11.4} and initiates normal shift
$f_t\!:S'\to S'_t$. Let $u^1,\,\ldots,\,u^{n-1}$ be local coordinates
on $S'$. These local coordinates define variation vectors $\boldsymbol
\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ and deviation functions
$$
\pagebreak
\varphi_k=(\boldsymbol\tau_i\,|\,\bold v)\text{, \ \ where\ \ \ }
k=1,\,\ldots,\,n-1.\hskip -3em
\tag12.2
$$
Time derivatives of variation functions \thetag{12.2} are calculated
by formula \thetag{4.7}. Here this formula is written in the following
form:
$$
\dot\varphi_k=(\bold F\,|\,\boldsymbol\tau_k)+(\bold v\,|\,
\nabla_t\boldsymbol\tau_k).\hskip -3em
\tag12.3
$$
Covariant derivatives $\nabla_t\boldsymbol\tau_k$ in \thetag{12.3} at
the initial instant of time $t=0$ are calculated by formula \thetag{7.14}.
Using this formula, we take into account that function $\nu(p)$ on
initial hypersurface $S'$ satisfies the equation \thetag{7.16}. Hence
for covariant derivatives $\nabla_t\boldsymbol\tau_k$
in formula \thetag{12.3} we obtain
$$
\nabla_t\boldsymbol\tau_k\,\hbox{\vrule height 8pt depth 8pt width
0.5pt}_{\,t=0}=-\frac{(\bold F\,|\,\boldsymbol\tau_k)}{\nu}\cdot
\bold n-\nu\cdot\sum^{n-1}_{m=1}b^m_k\,\boldsymbol\tau_m.\hskip -3em
\tag12.4
$$
Normal vector $\bold n$ of hypersurface $S'$ is directed along the
vector of velocity; the function $\nu(p)$ determines the modulus of
initial velocity on $S'$. Therefore 
$$
\nabla_t\boldsymbol\tau_k\,\hbox{\vrule height 8pt depth 8pt width
0.5pt}_{\,t=0}=-\frac{(\bold F\,|\,\boldsymbol\tau_k)}{|\bold v|^2}
\cdot\bold v-|\bold v|\cdot\sum^{n-1}_{m=1}b^m_k\,\boldsymbol\tau_m.
\hskip -3em
\tag12.5
$$
Strong normality condition, that provides normality of shift $f_t\!:S'\to
S'_t$, implies identical in $t$ vanishing of all functions of
deviations. In particular this means 
$$
\xalignat 2
&\qquad \varphi_k\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=0,
&&\dot\varphi_k\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=0.
\hskip -3em
\tag12.6
\endxalignat
$$
Vanishing of $\varphi_k$ and $\dot\varphi_k$ in \thetag{12.6} is
granted by orthogonality $\boldsymbol\tau_k\perp\bold v$ upon
substituting \thetag{12.5} into \thetag{12.3}. Let's consider
second derivatives 
$$
\ddot\varphi_k\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=0.
\hskip -3em
\tag12.7
$$
Their vanishing is also the consequence of $\varphi_k(t)=0$. We 
calculate second order derivatives in the left hand side of
\thetag{12.7} by formula \thetag{5.14}. As a result we get
$$
\boldsymbol\beta(\boldsymbol\tau_k)-\frac{\boldsymbol\alpha(\bold N)}
{|\bold v|}\,(\bold F\,|\,\boldsymbol\tau_k)-|\bold v|\sum^{n-1}_{m=1}
b^m_k\,\boldsymbol\alpha(\boldsymbol\tau_m)=0.
\hskip -3em
\tag12.8
$$
Here covectorial fields $\boldsymbol\alpha$ and $\boldsymbol\beta$ are
defined by their components \thetag{5.13}. When referred to the marked
point $p_0$, the quantities $b^m_k$ can be interpreted as components of
symmetric operator $\bold B$ in hyperplane $\alpha$ orthogonal to the
vector of velocity $\bold v=\bold v_0$ (see formula \thetag{9.14}).
Therefore equations \thetag{12.8} can be written as
$$
\boldsymbol\beta(\boldsymbol\tau_k)-\frac{\boldsymbol\alpha(\bold N)}
{|\bold v|}\,(\bold F\,|\,\boldsymbol\tau_k)-|\bold v|\,\boldsymbol
\alpha(\bold B\,\boldsymbol\tau_k)=0.
\hskip -3em
\tag12.9
$$
For further analysis of the equations \thetag{12.9} we use the
arbitrariness in the choice of hypersurface $S$ passing through
the point $p_0$ and perpendicular to vector $\bold v_0$ at this
point. Let's apply theorem~10.1. According to this theorem vector
$\boldsymbol\tau_k$ can be replaced by arbitrary vector $\boldsymbol
\tau$ from hyperplane $\alpha$, while operator $\bold B$ in \thetag{12.9}
can be understood as arbitrary symmetric operator $\bold B\!:\alpha
\to\alpha$. Hence vector $\bold B\,\boldsymbol\tau_k$ in \thetag{12.9}
can be replaced by some other arbitrary vector from hyperplane $\alpha$,
which do not depend on $\boldsymbol\tau$. Due to these reasons
equation \thetag{12.9} breaks into two parts
$$
\xalignat 2
&\boldsymbol\beta(\boldsymbol\tau)-\frac{\boldsymbol\alpha(\bold N)}
{|\bold v|}\,(\bold F\,|\,\boldsymbol\tau)=0,
&&\boldsymbol\alpha(\boldsymbol\tau)=0,
\endxalignat
$$
where $\boldsymbol\tau\in\alpha$ is arbitrary vector perpendicular to
$\bold v_0$ at the point $p_0$. We can avoid restriction $\boldsymbol\tau
\perp\bold v_0$, if we substitute $\boldsymbol\tau$ by $\bold P\boldsymbol
\tau$:
$$
\xalignat 2
&\boldsymbol\beta(\bold P\boldsymbol\tau)=\frac{\boldsymbol\alpha(\bold N)}
{|\bold v|}\,(\bold F\,|\,\bold P\boldsymbol\tau),
&&\boldsymbol\alpha(\bold P\boldsymbol\tau)=0.\hskip -4em
\tag12.10
\endxalignat
$$
Here $\boldsymbol\tau$ is arbitrary vector at the point $p_0$ on $M$.
Second equation \thetag{12.10} coincides with first equation \thetag{6.11},
since $\boldsymbol\tau$ in \thetag{12.10} and $\nabla_t\boldsymbol\tau$
in \thetag{6.11} stand for arbitrary vectors at marked point $p_0$ on
the trajectory of shift. First equation \thetag{12.10} coincides with
second equation \thetag{6.11}. In order to see it one should only denote
$$
\xalignat 2
&\quad\Cal A=\frac{\boldsymbol\alpha(\bold N)}{|\bold v|},
&&\Cal B=\frac{\boldsymbol\beta(\bold N)}{|\bold v|}-\Cal A\,
\frac{(\bold F\,|\,\bold N)}{|\bold v|}\hskip -4em
\tag12.11
\endxalignat
$$
(compare with \thetag{6.10} in \S\,6). Now let's use arbitrariness in
the choice of point $p_0$ and in the choice of vector $\bold v_0$ at
this point. This means that the equations \thetag{12.10} coinciding
with weak normality equations \thetag{6.17} and \thetag{6.17} are
fulfilled at all points $q=(p,\bold v)$  of tangent bundle $TM$,
where $\bold v\neq 0$.\par
     Now let's take some arbitrary function of deviation $\varphi$
on the trajectory $p(t)$ and form linear combination of its time
derivatives
$$
\ddot\varphi-\Cal A\,\dot\varphi-\Cal B\,\varphi,\hskip -4em
\tag12.12
$$
using parameters $\Cal A$ and $\Cal B$ from \thetag{12.11} as
coefficients in it. Here we do not associate vector of variation
$\boldsymbol\tau$ in $\varphi=(\bold v\,|\,\boldsymbol\tau)$ with
any hypersurface, and therefore we do not restrict it by the
condition $\bold v\perp\boldsymbol\tau$, as it was in the case
of normal shift. Here $\boldsymbol\tau=\boldsymbol\tau(t)$ is
arbitrary vector function that varies in $t$ according to the
differential equation \thetag{3.10}. From \thetag{3.10} we derive
the equalities \thetag{5.12} and \thetag{5.14} for $\dot\varphi$
and $\ddot\varphi$. Substituting them into \thetag{12.12} and taking
into account \thetag{12.10} and \thetag{12.11}, we find that
linear combination is identically zero: $\ddot\varphi-\Cal A\,\dot\varphi
-\Cal B\,\varphi=0$. This means that arbitrary function of deviation
satisfies homogeneous linear ordinary differential equation of the
second order with respect to parameter $t$ on trajectories of dynamical
system. According to definition~6.1 this is exactly the weak normality
condition for that dynamical system. Theorem~12.1 is proved.
\enddemo
\enddocument
\end

ediss-06.tex
\input amstex
\documentstyle{amsppt}
%
% Author:   Ruslan Abdulovich Sharipov
% Title:    Dynamical systems admitting the normal shift
% Comment:  Thesis for the degree of Doctor of Sciences in Russia
% Address:  Rabochaya street 5, Ufa, 450003, Russia
% E-mail:   R_Sharipov@ic.bashedu.ru
%           ruslan-sharipov@usa.net
% Web-page: http://www.geocities.com/CapeCanaveral/Lab/5341
%
\nopagenumbers
\catcode`@=11
\redefine\logo@{\relax}
\catcode`\@=\active
\def\startpage#1{\pageno=#1}
\pagewidth{12.8cm}
\pageheight{20cm}
\def\const{\operatorname{const}}
\accentedsymbol\tx{\tilde x}
\Monograph
\topmatter
\title\chapter{6} Problem of metrizability.
\endtitle
\endtopmatter
\loadbold
\startpage{128}
\document
\head
\S\,1. Geodesic flows of conformally equivalent metrics.
\endhead
     Concept of dynamical system {\bf admitting the normal shift} is
a central concept of this thesis. For Newtonian dynamical system on
Riemannian manifolds of the dimension $n\geqslant 2$ it is expressed
by {\bf strong normality} condition (see definition~11.2 in Chapter
\uppercase\expandafter{\romannumeral 5}). This is purely geometric
condition. Theorem~12.1 in Chapter \uppercase\expandafter{\romannumeral 5}
reduces {\bf strong normality} condition to the couple of other geometric
conditions: {\bf weak normality} condition and {\bf additional normality}
condition. These two conditions, joint together, form {\bf complete
normality} condition. Theorem~6.2 in Chapter
\uppercase\expandafter{\romannumeral 5} transforms {\bf weak normality}
condition to the system of partial differential equations for the
force field $\bold F$ of Newtonian dynamical system:
$$
\align
&\sum^n_{i=1}\left(v^{-1}\,F_i+\shave{\sum^n_{j=1}}
\tilde\nabla_i\left(N^j\,F_j\right)\right)P^i_k=0.\hskip -2em
\tag1.1\\
\vspace{2ex}
&\aligned
&\sum^n_{i=1}\sum^n_{j=1}\left(\nabla_iF_j+\nabla_jF_i-2\,v^{-2}
\,F_i\,F_j\right)N^j\,P^i_k\,+\\
\vspace{0.4ex}
&+\sum^n_{i=1}\sum^n_{j=1}\left(\frac{F^j\,\tilde\nabla_jF_i}{v}
-\sum^n_{r=1}\frac{N^r\,N^j\,\tilde\nabla_jF_r}{v}\,F_i\right)P^i_k=0.
\endaligned\hskip -3em
\tag1.2
\endalign
$$
{\bf Additional normality} condition separates two quite different
subcases: {\bf two-dimensional case} $\dim M=n=2$ and
{\bf multidimensional case} $\dim M=n\geqslant 3$. In two
dimensional case additional normality condition is fulfilled
unconditionally for any Newtonian dynamical system (see theorem~9.1
in Chapter \uppercase\expandafter{\romannumeral 5}). We shall not
consider two dimensional case at all, it is considered in {\bf other
thesis} \cite{36}. In multidimensional case theorem~9.5 in Chapter
\uppercase\expandafter{\romannumeral 5} transforms {\bf additional
normality} condition to the system of partial differential equations
for the force field $\bold F$:
$$
\align
&\aligned
&\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_i\,\tilde\nabla_mF_j}{v}
-\nabla_iF_j\right)=\\
\vspace{0.4ex}
&\quad\,=\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_j\,\tilde\nabla_mF_i}{v}
-\nabla_jF_i\right).
\endaligned\hskip -3em
\tag1.3\\
\displaybreak
\vspace{1ex}
&\sum^n_{i=1}\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF^i\,
P^\varepsilon_i=\sum^n_{i=1}\sum^n_{j=1}\sum^n_{m=1}
\frac{P^j_m\,\tilde\nabla_jF^i\,P^m_i}{n-1}\,P^\varepsilon_\sigma.
\hskip -3em\tag1.4
\endalign
$$\par
    Once normality equations \thetag{1.1}, \thetag{1.2}, \thetag{1.3},
and \thetag{1.4} are written, the problem of their compatibility arises.
We begin to study this problem by constructing some simplest solutions
for these equations. Existence of such solutions follows from geometric
nature of the equations \thetag{1.1}, \thetag{1.2}, \thetag{1.3},
and \thetag{1.4}. Let $M$ be Riemannian manifold of the dimension
$n\geqslant 3$ and let $\bold g$ be metric tensor of $M$. Geodesic
flow of metric $\bold g$ is a Newtonian dynamical system admitting
the normal shift in metric $\bold g$ (see \S\,1 in Chapter
\uppercase\expandafter{\romannumeral 5}). Let's consider the metric
conformally equivalent to metric $\bold g$:
$$
\tilde\bold g=e^{-2f}\,\bold g.\hskip -2em
\tag1.5
$$
Here $f$ is some smooth function on $M$ (non-extended scalar field).
Geodesic flow of metric \thetag{1.5} is obviously a dynamical system
admitting the normal shift in metric \thetag{1.5} by itself. However,
angles, measured in metric $\bold g$, have the same measure as in metric
$\tilde\bold g$, i\.~e\. normal shift in metric $\tilde\bold g$
remains to be normal shift with respect to basic metric $\bold g$ of
manifold $M$. Hence geodesic flow of  metric  \thetag{1.5}  is  a 
dynamical
system admitting the normal shift in metric $\bold g$.\par
    Components of metric connection $\tilde\Gamma^k_{ij}$ for conformally
equivalent metric \thetag{1.5} are calculated by standard formula
$$
\tilde\Gamma^k_{ij}=\frac{1}{2}\sum^n_{s=1}\tilde g^{ks}\left(\frac{
\partial\tilde g_{sj}}{\partial x^i}+\frac{\partial\tilde g_{is}}{
\partial x^j}-\frac{\partial\tilde g_{ij}}{\partial x^s}\right).
\hskip -2em
\tag1.6
$$
Substituting $\tilde g_{ij}=e^{-2f}\,g_{ij}$ and $\tilde g^{ij}=e^{2f}\,
g^{ij}$ into the formula \thetag{1.6}, we get
$$
\tilde\Gamma^k_{ij}=\Gamma^k_{ij}-\nabla_if\,\delta^k_j-\nabla_jf\,
\delta^k_i+g_{ij}\sum^n_{s=1}g^{ks}\,\nabla_sf.\hskip -2em
\tag1.7
$$
Let's substitute \thetag{1.7} into the equations of geodesic lines
for metric $\tilde\bold g$:
$$
\ddot x^k+\sum^n_{i=1}\sum^n_{j=1}\tilde\Gamma^k_{ij}\,\dot x^i\,
\dot x^j=0.\hskip -2em
\tag1.8
$$
In basic metric $\bold g$ of manifold $M$ the equations \thetag{1.8}
are written as the equations of Newtonian dynamical system
$$
\xalignat 2
&\dot x^k=v^k,&&\nabla_tv^k=F^k\hskip -2em
\tag1.9
\endxalignat
$$
with the force field $\bold F$ given by the following components:
$$
F^k=-|\bold v|^2\shave{\sum^n_{s=1}}g^{ks}\,\nabla_sf+2
\shave{\sum^n_{s=1}}\nabla_sf
\,v^s\,v^k.\hskip -3em
\tag1.10
$$
Force field \thetag{1.10} satisfies all normality equations \thetag{1.1},
\thetag{1.2}, \thetag{1.3}, and \thetag{1.4}. This fact follows from
geometric nature of the field \thetag{1.10} as described above. \pagebreak
But one
can check it by direct substitution of \thetag{1.10} into the equations
\thetag{1.1}, \thetag{1.2}, \thetag{1.3}, and \thetag{1.4}. Functions
\thetag{1.10} form parametric family of solutions of these normality
equations. This family is parameterized by one scalar function on the
manifold $M$.
\head
\S\,2. Inheriting the trajectories and trajectory equivalence.
\endhead
    Let $f_t\!:S\to S_t$ be the shift of hypersurface $S$ along
trajectories of Newtonian dynamical system \thetag{1.9} with force
field $\bold F$. The same shift can be implemented by Newtonian
dynamical system with some other force field $\tilde\bold F$,
provided the trajectories of first system (those which are used
in shift procedure) are among trajectories of the second one.
Let's generalize this situation by the following definition.
\definition{Definition 2.1} Suppose that we have two Newtonian dynamical
systems with force fields $\bold F$ and $\tilde\bold F$ on Riemannian
manifold $M$. We say that second system {\bf inherits} trajectories of
first system, if each trajectory of second system as line (up to a
regular reparametrization) coincides with some trajectory of first
dynamical system.
\enddefinition
\definition{Definition 2.2} Two Newtonian dynamical systems on
Riemannian manifold $M$ inheriting trajectories of each other
are called {\bf trajectory equivalent}
\enddefinition
    The concepts of {\bf trajectory inheriting} and {\bf trajectory
equivalence} can be traced back to the classical paper \cite{50}.
The concept of {\bf geodesic equivalence} for Riemannian and affine
manifolds is nearly related concept (see \cite{55--57}). Fore general
definition of trajectory equivalence were stated in \cite{43--48},
where it was used for topological classification of dynamical systems.
\definition{Definition 2.3} Newtonian dynamical system on Riemannian
manifold $M$ is called {\bf metrizable}, if it inherits trajectories
of dynamical system with force field \thetag{1.10} defined by conformally
equivalent metric  $\tilde\bold g=e^{-2f}\,\bold g$.
\enddefinition
    Suppose that we have chosen some point $p_0$ of Riemannian manifold $M$
and some vector $\tilde\bold v_0\neq 0$ at this point. Then we have a
trajectory $\tilde p(t)$ of second dynamical system such that $\tilde p(0)
=p_0$ and $\tilde\bold v(0)=\bold v_0$, where $\tilde\bold v(t)$ is a
velocity vector on this trajectory. If second dynamical system inherits
trajectories of first system, then there exists the trajectory $p(\tau)$
of the first system coinciding with $\tilde p(t)$. This means that there
exists the regular reparametrization $\tau=T(t)$ such that $p(T(t)=
\tilde p(t)$. Without loss of generality we can assume that $T(0)=0$.
Velocity vectors on common trajectory of these two dynamical systems are
bound by relationship
$$
\tilde\bold v(t)=\frac{dT}{dt}\,\bold v(T(t)).\hskip -2em
\tag2.1
$$
Let's apply covariant differentiation $\nabla_t$ to both sides of the
equality \thetag{2.1} considered as vector fields on the curve $p(t)$.
In order to apply such covariant differentiation we should choose local
coordinates $x^1,\,\ldots,\,x^n$ in the neighborhood of marked point
$p_0$ and should use formula \thetag{4.1} from Chapter
\uppercase\expandafter{\romannumeral 4}. We should also take into account
that $\nabla_\tau\bold v=\bold F$ and $\nabla_t\tilde\bold v
=\tilde\bold F$. Upon completing these calculations we get 
$$
\tilde\bold F(p,\tilde\bold v)=\left(\frac{dT}{dt}\right)^2\,
\bold F(p,\bold v)+\frac{d^2T}{dt^2}\,\bold v.\hskip -2em
\tag2.2
$$
The relationship \thetag{2.2} binds force fields of two dynamical
systems. It is non-local, since $\bold F$ and $\tilde\bold F$ are
taken in two different points $q=(p,\bold v)$ and
$\tilde q=(p,\tilde\bold v)$ on tangent bundle $TM$. But there are
some cases when it reduces to the local relationship. Let $\bold F$
be homogeneous function in its vectorial argument $\bold v$, and let
$\gamma$ be degree of homogeneity. Then from \thetag{2.1} we derive
$$
\bold F(p,\bold v)=\left(\frac{dT}{dt}\right)^{-\gamma}\,
\bold F(p,\tilde\bold v).\hskip -2em
\tag2.3
$$
Since $\tau=T(t)$ is regular reparametrization, we have
$$
\frac{dT}{dt}\neq 0.\hskip -2em
\tag2.4
$$
Inequality \thetag{2.4} provides correctness of \thetag{2.3}. Suppose
that $\gamma=2$. This is the very case applicable to the force
field \thetag{1.10} that corresponds to geodesic flow of conformally
equivalent metric $\tilde\bold g=e^{-2f}\,\bold g$. Other cases
$\gamma\neq 2$ were considered in \cite{41}, they are not interesting
for us now.\par
    Let's substitute \thetag{2.3} with $\gamma=2$ into the formula
\thetag{2.2} and let's  express  vector  $\bold  v$  through  the 
vector
$\tilde\bold v$ in it. Then \thetag{2.2} is written as
$$
\tilde\bold F(p,\tilde\bold v)=\bold F(p,\tilde\bold v)+\frac{d^2T}
{dt^2}\,\left(\frac{dT}{dt}\right)^{-1}\tilde\bold v.\hskip -2em
\tag2.5
$$
Due to the arbitrariness in choice of point $p$ and in choice of
vector $\tilde\bold v\neq 0$ at this point the relationship \thetag{2.5}
mean that the field $\tilde\bold F$ differs from $\bold F$ by a vector
directed along the vector of velocity. Let's express this fact as
follows:
$$
\tilde\bold F(p,\bold v)=\bold F(p,\bold v)+\frac{H(p,\bold v)}
{|\bold v|}\,\bold v.\hskip -2em
\tag2.6
$$
Here $H=H(p,\bold v)$ is some arbitrary scalar field from extended
algebra of tensor fields on $M$. We replaced $\tilde\bold v$ by
$\bold v$ for to simplify the appearance of formula.
\proclaim{Theorem 2.1} Consider some two Newtonian dynamical 
systems
on Riemannian manifold $M$. Suppose that the force field of first
system is homogeneous function of degree $2$ with respect to velocity
vector $\bold v$ in fibers of tangent bundle $TM$. Second dynamical
system inherits trajectories of the first system if and only if its
force field $\tilde\bold F$ is given by formula \thetag{2.6}.
\endproclaim
\demo{Proof} If second dynamical system inherits trajectories of
the first one, then its force field is given by \thetag{2.6}. This
was shown above. Conversely, suppose that force field of the second
dynamical system is defined by formula \thetag{2.6}. Let's take
the trajectory $\tilde p(t)$ of the second system, and ret's arrange
reparametrization $\tau=T(t)$ of this curve defined by the solution of
the following ordinary differential equation:
$$
\ddot T=\frac{H(\tilde p(t),\tilde\bold v(t))}{|\tilde\bold v(t)|}
\,\dot T.\hskip -2em
\tag2.7
$$
Solution of the equation \thetag{2.7} can be normalized by the condition
$$
\dot T\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=a\neq 0
$$
that provides regularity of reparametrization $T\!:t\to\tau$. Denote
by $\theta\!:\tau\to t$ the inverse reparametrization and take $p(\tau)
=\tilde p(\theta(\tau))$. Then $\tilde p(t)=p(T(t))$, and the
relationship \thetag{2.1} is fulfilled. Differentiating \thetag{2.1},
we get
$$
\tilde\bold F(\tilde p,\tilde\bold v)=\dot T^2\,\nabla_\tau\bold v+
\ddot T\,\bold v.\hskip -2em
\tag2.8
$$
Now substitute $\ddot T$, obtained from \thetag{2.7}, into the
formula \thetag{2.8}. Then take into account the relationship
\thetag{2.1} by itself and formula \thetag{2.6} for $\tilde\bold F$.
As a result we get
$$
\bold F(\tilde p,\tilde\bold v)=\dot T^2\,\nabla_\tau\bold v.
\hskip -2em
\tag2.9
$$
Let's apply formula \thetag{2.1} written as $\tilde\bold v=\dot T\,
\bold v$ and take into account homogeneity of force field $\bold F(\tilde p,
\tilde\bold  v)$  in  its  vectorial  argument.  The  degree   of 
homogeneity $\gamma
=2$. Then the equality \thetag{2.9} is brought to
$$
\nabla_\tau\bold v=\bold F(\tilde p,\bold v).\hskip -3em
\tag2.10
$$
This relationship, which we have just derived, means that curve $\tilde
p(t)$ after reparamet\-rization $p(\tau)=\tilde p(\theta(\tau))$ becomes
the trajectory of Newtonian dynamical system with force field $\bold F$.
\qed\enddemo
\proclaim{Theorem 2.2} Newtonian dynamical system on Riemannian
manifold $\bold M$ is met\-ri\-zable if and only if its force field
has the components given by formula
$$
F^k=-|\bold v|^2\shave{\sum^n_{s=1}}g^{ks}\,\nabla_sf+2\shave{\sum^n_{s=1}}
\nabla_sf\,v^s\,v^k+\frac{H}{|\bold v|}\,v^k,\hskip -3em
\tag2.11
$$
where $f$ is an ordinary (not extended) scalar field, while $H$ is a
scalar field from extended algebra of tensor fields on $M$.
\endproclaim
    Theorem~2.2 is a simple consequence of previous theorem~2.1. It
doesn't require the separate proof.
\head
\S\,3. Problem of metrizability.
\endhead
     {\bf Problem of metrizability} considered in \cite{41} and \cite{59}
is stated as follows: {\bf Which} of Newtonian dynamical systems admitting
the normal shift {\bf are metrizable}, i\.~e\. which of them have
trajectories coinciding with trajectories of geodesic flow for
conformally equivalent metric $\tilde\bold g=e^{-2f}\,\bold g$.
Theorem~2.2 gives easy way for solving this problem. Instead of looking
for metrizable dynamical systems among those admitting the normal shift,
we can look for systems admitting the normal shift in the set of
metrizable systems. This is easier, since according to the theorem~2.2
force field of metrizable dynamical systems has quite definite form
\thetag{2.11}.\par
    Let's substitute \thetag{2.11} into the normality equations
\thetag{1.1}, \thetag{1.2}, \thetag{1.3}, and \thetag{1.4}. This will
require substantial amount of calculations. Let's begin with \thetag{1.1}.
Here
$$
\sum^n_{j=1}N^j\,F_j=H+|\bold v|\sum^n_{s=1}v^s\,\nabla_sf.
$$
Applying covariant derivative $\tilde\nabla_i$ to this equality,
we use formulas \thetag{5.5} and \thetag{5.6} from Chapter
\uppercase\expandafter{\romannumeral 5}. As a result we obtain
$$
\sum^n_{j=1}\tilde\nabla_i(N^j\,F_j)=\tilde\nabla_iH+
N_i\sum^n_{s=1}v^s\,\nabla_sf+|\bold v|\,\nabla_if.
$$
Upon contracting with the operator of projection $\bold P$ we get
$$
\sum^n_{i=1}\sum^n_{j=1}\tilde\nabla_i(N^j\,F_j)\,P^i_k=
\sum^n_{i=1}\tilde\nabla_iH\,P^i_k+|\bold v|\sum^n_{i=1}
\nabla_if\,P^i_k.\hskip -2em
\tag3.1
$$
Moreover, we have the following relationship being immediate consequence
of \thetag{2.11}:
$$
\sum^n_{i=1}|\bold v|^{-1}\,F_i\,P^i_k=-|\bold v|\sum^n_{i=1}
\nabla_if\,P^i_k.\hskip -2em
\tag3.2
$$
If we add left hand sides of \thetag{3.2} and \thetag{3.1}, we obtain
the expression coinciding with left hand side of normality equation
\thetag{1.1}. Therefore, when applied to the force field \thetag{2.11},
the normality equation \thetag{1.1} gives
$$
\sum^n_{i=1}\tilde\nabla_iH\,P^i_k=0.\hskip -2em
\tag3.3
$$\par
    As a second step, let's substitute \thetag{2.11} into the equation
\thetag{1.2}. Upon opening brackets in left hand side of \thetag{1.2}
we get five terms. Let's calculate third term:
$$
\gathered
-\sum^n_{i=1}\sum^n_{j=1}\frac{2\,F_i\,F_j\,N^j\,P^i_k}{|\bold v|^2}=
2\,H\sum^n_{i=1}\nabla_if\,\,P^i_k\,+\\
+\,2\,|\bold v|\sum^n_{i=1}\sum^n_{s=1}v^s\,\nabla_sf\,
\nabla_if\,P^i_k.
\endgathered\hskip -2em
\tag3.4
$$
Then find derivatives $\nabla_iF_j$ for the force field \thetag{2.11}:
$$
\nabla_iF_j=-|\bold v|^2\,\nabla_i\nabla_jf+2\sum^n_{s=1}
\nabla_i\nabla_sf\,v^s\,v_j+\nabla_iH\,N_j.\hskip -2em
\tag3.5
$$
For first and second term in left hand side of the equation
\thetag{1.2} from \thetag{3.5} we get
$$
\align
&\sum^n_{i=1}\sum^n_{j=1}\nabla_iF_j\,N^j\,P^i_k=|\bold v|^2
\sum^n_{i=1}\sum^n_{j=1}\nabla_i\nabla_jf\,N^j\,P^i_k+
\sum^n_{i=1}\nabla_iH\,P^i_k,\\
\vspace{0.4ex}
&\sum^n_{i=1}\sum^n_{j=1}\nabla_jF_i\,N^j\,P^i_k=-|\bold v|^2
\sum^n_{i=1}\sum^n_{j=1}\nabla_j\nabla_if\,N^j\,P^i_k.
\endalign
$$
Let's add \thetag{3.4} and two above equalities. The result is
as follows:
$$
\gathered
\sum^n_{i=1}\sum^n_{j=1}\left(\nabla_iF_j+\nabla_jF_i-\frac{2\,F_i
\,F_j}{|\bold v|}\right)N^j\,P^i_k=\sum^n_{i=1}\nabla_iH\,P^i_k\,+\\
\vspace{0.4ex}
+\,2\,H\sum^n_{i=1}\nabla_if\,\,P^i_k+2\,|\bold v|
\sum^n_{i=1}\sum^n_{s=1}v^s\,\nabla_sf\,\nabla_if\,P^i_k.
\endgathered\hskip -3em
\tag3.6
$$
In deriving \thetag{3.6} we took into account that $\nabla_i\nabla_jf
=\nabla_j\nabla_if$. This follows from formula \thetag{7.10} in Chapter
\uppercase\expandafter{\romannumeral 3} because tensor of torsion here
is equal to zero and because $f$ is ordinary (not extended) scalar field
on the manifold $M$.\par
    Further let's calculate covariant derivative $\tilde\nabla_jF_r$
for the force field of metrizable dynamical system given by formula
\thetag{2.11}:
$$
\gathered
\tilde\nabla_jF_r=2\,\nabla_jf\,v_r-2\,\nabla_rf\,v_j
+\tilde\nabla_jH\,N_r\,+\\
\vspace{0.4ex}
+\,2\,g_{rj}\sum^n_{s=1}\nabla_sf\,v^s+\frac{H}{|\bold v|}
\sum^n_{s=1}g_{sj}\,P^s_r.
\endgathered\hskip -2em
\tag3.7
$$
Contracting this expression with $N^r\,N^j$ and with $P^r_k$, we derive
$$
\gather
\sum^n_{j=1}\sum^n_{r=1}N^r\,N^j\,\tilde\nabla_jF_r=
\sum^n_{j=1}\tilde\nabla_jH\,N^j+2\sum^n_{s=1}\nabla_sf\,v^s,\\
\vspace{0.4em}
\sum^n_{i=1}\tilde\nabla_jF_i\,P^i_k=
\sum^n_{i=1}\left(\left(2\shave{\sum^n_{s=1}}\nabla_sf\,v^s+\frac{H}
{|\bold v|}\right)g_{ij}-2\,v_j\,\nabla_if\right)P^i_k.
\endgather
$$
Let's combine these expressions with \thetag{3.2} and \thetag{2.11}.
As a result we get
$$
\gathered
\sum^n_{i=1}\sum^n_{j=1}\left(\frac{F^j\,\tilde\nabla_jF_i}{|\bold v|}
-\sum^n_{r=1}\frac{N^r\,N^j\,\tilde\nabla_jF_r}{|\bold v|}\,F_i\right)
P^i_k=\\
\vspace{0.4ex}
=\sum^n_{i=1}\nabla_if\,P^i_k\left(\,\shave{\sum^n_{j=1}}\tilde\nabla_jH
\,v^j-3\,H-2\,|\bold v|\shave{\sum^n_{s=1}}\nabla_sf\,v^s\right).
\endgathered\hskip -2em
\tag3.8
$$
And finally, let's add the equalities \thetag{3.5} and \thetag{3.8}.
This yields the equality
$$
\sum^n_{i=1}\left(\nabla_iH+\left(\,\shave{\sum^n_{j=1}}v^j\,
\tilde\nabla_jH-H\right)\nabla_if\right)P^i_k=0.\hskip -2em
\tag3.9
$$
The equality \thetag{3.9} is an ultimate result of substituting
\thetag{2.11} into \thetag{1.2}. It should be understood as a system
of differential equations for $H$.\par
    Let's study the equations \thetag{3.3} and \thetag{3.9} for
extended scalar field $H$. In order to do it  we choose some local
map $(U,x^1,\ldots,x^n)$ on $M$ and associated map $(\pi^{-1}(U),x^1,
\ldots,x^n,v^1,\ldots,v^n)$ on tangent bundle. Then scalar field
$H$ is expressed by function $H(x^1,\ldots,x^n,v^1,\ldots,v^n)$,
while equation \thetag{3.3} takes the form
$$
\sum^n_{i=1}\frac{\partial H}{\partial v^i}\,P^i_k=0.\hskip -3em
\tag3.10
$$
Recall that $P^i_k$ are components of projector onto the hyperplane
perpendicular to velocity vector. Therefore the equation \thetag{3.10}
means that velocity gradient of $H$ is projected to zero. This mean
that $\tilde\nabla H$ is directed along the vector $\bold N$, which,
in turn, is unit vector directed along $\bold v$. Covariant components
of the vector $\bold N$ can be represented by velocity gradient
of scalar field $v=|\bold v|$:
$$
N_i=\tilde\nabla_iv=\frac{\partial v}{\partial v^i}\hskip -3em
\tag3.11
$$
(see formulas \thetag{5.6} in Chapter
\uppercase\expandafter{\romannumeral 5}). Now we use the following
well-known fact.
\proclaim{Lemma 3.1} If velocity gradients of two scalar fields $H$ and
$v=|\bold v|$ are proportional at all point of tangent bundle $TM$,
except for those where $\bold v=0$, then $H$ depends only on modulus
of velocity vector, i\.~e\. $H=H(x^1,\ldots,x^n,v)$.
\endproclaim
\demo{Proof} Let's fix some point $p\in M$ and consider the fiber
of tangent bundle over this point. In such fiber lets consider the
point where velocity gradient of $v=|\bold v|$ iz nonzero. According
to \thetag{3.11} this is equivalent to $\bold v\neq 0$. In the
neighborhood of such point $q=(p,\bold v)$ cartesian coordinates
$v^1,\ldots,v^n$ in this fiber can be replaced by local coordinates
$u^1,\ldots,u^n$ such that
$$
\left\{
\aligned
u^1&=v(v^1,\ldots,v^n),\\
u^2&=u^2(v^1,\ldots,v^n),\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ \\
u^n&=u^n(v^1,\ldots,v^n).
\endaligned\right.
$$
In these coordinates $v=v(u^1,\ldots,u^n)=u^1$, therefore velocity
gradient of $v$ is represented by unitary vector:
$\tilde\nabla v=(1,0,\ldots,0)$. Condition of proportionality
for velocity gradients of $H$ and $v$, \pagebreak which we have in the
statement of lemma, is written as a condition of collinearity
for two vector-columns, one being unitary vector-column:
$$
\xalignat 2
&\qquad\Vmatrix
\partial H/\partial u^1\\
\partial H/\partial u^2 \\
\vdots\\
\partial H/\partial u^n
\endVmatrix,
&&\Vmatrix
1\vphantom{\partial H/\partial u^1}\\
0\vphantom{\partial H/\partial u^2}\\
\vdots\\
0\vphantom{\partial H/\partial u^n}
\endVmatrix.\hskip -2em
\tag3.12
\endxalignat
$$
From collinearity of vector-columns \thetag{3.12} we derive
$$
\frac{\partial H}{\partial u^2}=0, \ \ldots, \
\frac{\partial H}{\partial u^n}=0.
$$
This means that $H$ depend only on the variable $u^1=v$ within the
fiber over fixed point $p\in M$. If we recall that $H$ can depend
on $p$, then we get $H=H(x^1,\ldots,x^n,v)$. Lemma is proved.
\qed\enddemo
    Let's proceed with studying the equations \thetag{3.3} and \thetag{3.9}
for $H$ in local map $(\pi^{-1}(U),x^1,\ldots,x^n,v^1,\ldots,v^n)$ on
tangent bundle $TM$. According to lemma~3.1 the equation \thetag{3.3}
written in form of \thetag{3.10} determines the dependence of $H$ on
vector $\bold v$ within fibers of tangent bundle: $H$ depends on $\bold v$
only through its dependence on $v=|\bold v|$. Taking into account this
circumstance, we reduce the equation \thetag{3.9} for $H=H(x^1,\ldots,
x^n,v)$ to the following form:
$$
\sum^n_{i=1}\left(\nabla_iH+\left(v\,\frac{\partial H}{\partial v}
-H\right)\nabla_if\right)P^i_k=0.\hskip -3em
\tag3.13
$$
Covariant derivatives $\nabla_if$ are easily calculated, they coincide
with partial derivatives since $f$ doesn't depend on vector of velocity
$\bold v$:
$$
\nabla_if=\frac{\partial f}{\partial x^i}.\hskip -3em
\tag3.14
$$
In order to calculate $\nabla_iH$ we apply formula \thetag{7.3} from
Chapter \uppercase\expandafter{\romannumeral 3}. When applied to
scalar field $H$ this formula is written as follows:
$$
\nabla_iH=\frac{\partial H}{\partial x^i}-\sum^n_{j=1}\sum^n_{k=1}
\Gamma^k_{ij}\,v^j\,\frac{\partial H}{\partial v^k}.\hskip -3em
\tag3.15
$$
Note that, calculating partial derivatives in \thetag{3.15}, we
should treat $H(x^1,\ldots,x^n,v)$ as the function of $2n$ variables
$x^1,\,\ldots,\,x^n,\,v^1,\,\ldots,\,v^n$, where dependence of $v$
upon these variables is given by the following expression:
$$
v=\sqrt{\shave{\sum^n_{i=1}\sum^n_{j=1}}g_{ij}(x^1,\ldots,x^n)
\,v^i\,v^j\,}.
$$
Therefore formula \thetag{3.15} for covariant derivatives $\nabla_iH$
is brought to the form:
$$
\nabla_iH=\frac{\partial H}{\partial x^i}+\frac{\partial H}{\partial v}
\sum^n_{j=1}\sum^n_{k=1}\left(\frac{1}{2v}\,\frac{\partial g_{jk}}
{\partial x^i}\,v^j\,v^k-\Gamma^k_{ij}\,v^j\,N_k\right).\hskip -3em
\tag3.16
$$
Here partial derivatives of the function $H=H(x^1,\ldots,x^n,v)$
are calculated respective to its natural set of arguments $x^1,
\,\ldots,\,x^n$ and $v$. In order to simplify \thetag{3.16} we
use formula \thetag{1.2} from Chapter 
\uppercase\expandafter{\romannumeral 5}, that expresses $\Gamma^k_{ij}$.
Then we get the following result, which could seem striking at first
sight:
$$
\nabla_iH=\frac{\partial H}{\partial x^i}.\hskip -3em
\tag3.17
$$
Components of spatial gradient for extended scalar field $H$ depending
{\bf only} on {\bf modulus} of velocity vector are calculated as
{\bf partial derivatives} of its representation in local map
$H=H(x^1,\ldots,x^n,v)$ with respect to spatial variables $x^1,\,\ldots,
\,x^n$.\par
   Now let's substitute \thetag{3.14} and \thetag{3.17} into the equation
\thetag{3.13}. As a result of this substitution we bring this equation to
the form
$$
\sum^n_{i=1}\left(\frac{\partial H}{\partial x^i}+\left(v\,
\frac{\partial H}{\partial v}-H\right)\frac{\partial f}
{\partial x^i}\right)P^i_k=0.\hskip -3em
\tag3.18
$$
Note that in equation \thetag{3.18} components of projection operator
$\bold P$ depend on the direction of velocity vector. Other terms 
$$
L_i=\frac{\partial H}{\partial x^i}+\left(v\,\frac{\partial H}
{\partial v}-H\right)\frac{\partial f}{\partial x^i}\hskip -3em
\tag3.19
$$
depend only on modulus of velocity vector $v=|\bold v|$. Keeping the
value of $|\bold v|$ unchanged, we can rotate vector $\bold v$ so
that it will become perpendicular to the vector $\bold L$, whose
covariant components are given by formula \thetag{3.19}. Then we
shall get the equality $\bold P(\bold L)=\bold L$ that reduces
\thetag{3.18} to the following form 
$$
\frac{\partial H}{\partial x^i}+\left(v\,\frac{\partial H}{\partial v}
-H\right)\frac{\partial f}{\partial x^i}=0.\hskip -3em
\tag3.20
$$
Now let's change variables in \thetag{3.20} replacing
$x^1,\,\ldots,\,x^n,v$ by $\tx^1,\,\ldots,\,\tx^n,\tilde v$:
$$
\left\{\aligned &\tx^1=x^1,\\ \vspace{-0.8ex} &.\ .\ .\ .\ .\ .\ \\
&\tx^n=x^n,\\ &\tilde v=v\,\exp(-f(x^1,\ldots,x^n)).
\endaligned\right.\hskip -3em
\tag3.21
$$
Upon making change of variables \thetag{3.21} the equations
\thetag{3.18} look like
$$
\frac{\partial H}{\partial\tx^i}-H\,\frac{\partial f}{\partial\tx^i}=0.
\hskip -3em
\tag3.22
$$
We can make these equations more simple by writing them as follows:
$$
\frac{\partial}{\partial\tx^i}\left(H\,e^{-f}\right)=0.\hskip -3em
\tag3.23
$$
From \thetag{3.23} now it's clear that these equations are compatible
and general form of their solution is determined by arbitrary
function of one variable $H(\tilde v)$:
$$
H(\tx^1,\ldots,\tx^n,\tilde v)=H(\tilde v)\,e^f.\hskip -3em
\tag3.24
$$
By inverting \thetag{3.21} we can return to initial variables
$x^1,\,\ldots,\,x^2$:
$$
H(x^1,\ldots,x^n,v)=H(v\,e^{-f})\,e^f.\hskip -3em
\tag3.25
$$
Formula \thetag{3.25} determines general solution for the system
of differential equations \thetag{3.20}. We shall formulate this
result in form of the following theorem.
\proclaim{Theorem 3.1} Newtonian dynamical system on Riemannian
manifold $M$ with metric $\bold g$ satisfying weak normality condition
is metrizable by conformally equivalent metric $\tilde\bold g=e^{-2f}\,
\bold g$ if and only if its force field $\bold F$ is given by
formula 
$$
F_k=-|\bold v|^2\,\nabla_kf+2\sum^n_{s=1}
\nabla_sf\,v^s\,v_k+\frac{H(v\,e^{-f})\,e^f}{|\bold v|}\,v_k.\hskip -3em
\tag3.26
$$
\endproclaim
   Theorem~3.1 is not complete solution for problem of metrizability
formulated in the beginning of this section. In order to have complete
solution we should substitute \thetag{3.26} into additional normality
equations \thetag{1.3} and \thetag{1.4}. From \thetag{3.5} we derive
$$
\sum^n_{i=1}\sum^n_{j=1}\nabla_iF_j\,P^i_\varepsilon\,
P^j_\sigma=-|\bold v|^2\sum^n_{i=1}\sum^n_{j=1}\nabla_i\nabla_jf
\,P^i_\varepsilon\,P^j_\sigma.\hskip -3em
\tag3.27
$$
In a similar way from formula \thetag{3.7} we derive
$$
\sum^n_{m=1}\sum^n_{j=1}N^m\,\tilde\nabla_mF_j\,P^j_\sigma=
-2\,|\bold v|\sum^n_{j=1}\nabla_jf\,P^j_\sigma.
$$
Let's combine this relationship with \thetag{3.2}. As a result we obtain
$$
\sum^n_{i=1}\sum^n_{j=1}\sum^n_{m=1}P^i_\varepsilon\,P^j_\sigma\,
N^m\frac{F_i\,\tilde\nabla_mF_j}{|\bold v|}=2\,|\bold v|^2
\sum^n_{i=1}\sum^n_{j=1}\nabla_if\,\nabla_jf\,P^i_\varepsilon\,
P^j_\sigma.
$$
Right hand size of this equality do not change if we transpose indices
$\varepsilon$ and $\sigma$. Similar property we observe in right hand
size of \thetag{3.27}, it's the consequence of $\nabla_i\nabla_jf
=\nabla_j\nabla_if$ (see comment to formula \thetag{3.6} above). From
these two observations we easily derive that the equations \thetag{1.3}
are fulfilled by force field \thetag{2.11} for arbitrary choice of
function $H$ in it.\par
    Now let's consider the equation \thetag{1.4}. Substituting
\thetag{2.11} into the equation \thetag{1.4}, due to \thetag{3.7}
we derive the following relationship:
$$
\sum^n_{i=1}\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF_i\,
P^i_\varepsilon=\left(2\shave{\sum^n_{i=1}}\nabla_sf\,v^s+
\frac{H}{|\bold v|}\right)\sum^n_{j=1}P^j_\sigma\,
g_{j\varepsilon}.
$$
Upon raising index $\varepsilon$ in this relationship we obtain
$$
\sum^n_{i=1}\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF^i\,
P^\varepsilon_i=\left(2\shave{\sum^n_{i=1}}\nabla_sf\,v^s+
\frac{H}{|\bold v|}\right)P^\varepsilon_\sigma.
\hskip -3em
\tag3.28
$$
Right hand size of the equality \thetag{3.28} differs from
components of the projection operator $\bold P$ only by scalar
multiple
$$
\lambda= 2\sum^n_{i=1}\nabla_sf\,v^s+\frac{H}{|\bold v|}.
$$
This circumstance mean that the equations \thetag{1.4} are
fulfilled by force field \thetag{2.11} unconditionally for
arbitrary function $H$ in it. Summarizing this result with
previous one we formulate the following theorem.
\proclaim{Theorem 3.2} Arbitrary Newtonian dynamical system
with force field \thetag{2.11} satisfies additional normality
equations \thetag{1.3} and \thetag{1.4}.
\endproclaim
   Combining theorems~3.1 and 3.2 we obtain complete solution for
the problem of metrizability formulated in the beginning of this
section. 
\proclaim{Theorem 3.3} Newtonian dynamical system admitting the normal
shift on Riemannian manifold $M$ with metric $\bold g$ is metrizable
by conformally equivalent metric $\tilde\bold g=e^{-2f}\,\bold g$ if
and only if its force field $\bold F$ is given by formula \thetag{3.26}
\endproclaim
\head
\S\,4. Wider treatment of the problem of metrizability.
\endhead
    Theorem~3.3 proved above solves the problem of metrizability
of dynamical systems by conformally equivalent metric. In wider
treatment of this problem we should give up this preassigned requirement
of conformal equivalence of metrics $\bold g$ and $\tilde\bold g$.
Problem of metrizability should sound as follows: under which condition
Newtonian dynamical system admitting normal shift in metric $\bold g$
inherits trajectories of geodesic flow of another metric
$\tilde\bold g$\,? This statement is natural generalization of the
problem of metrizability as it was stated in \S\,3. However, in the
sense of normal shift in metric $\bold g$ it is less motivated. The
matter is that normal shift of hypersurface in metric $\bold g$ shouldn't
be the normal shift in metric $\tilde\bold g$ (unless these metrics are
conformally equivalent). This explains our previous choice of conformally
equivalent metrics.\par
     Let $\Gamma^k_{ij}$ and $\tilde\Gamma^k_{ij}$ be components of metric
connections corresponding to metrics $\bold g$ and $\tilde\bold g$
respectively. Denote by $M^k_{ij}$ their difference:
$$
M^k_{ij}=\tilde\Gamma^k_{ij}-\Gamma^k_{ij}.\hskip -2em
\tag4.1
$$
Quantities \thetag{4.1} form the set of components for some tensor
field $\bold M$ of type $(1,2)$ on the manifold $M$. Metric $\bold g$
will be understood as basic metric on $M$. Therefore geodesic flow
of second metric $\tilde\bold g$ can be written as Newtonian dynamical
system on Riemannian manifold $M$ with metric $\bold g$. Force field
of such dynamical system is defined by the following formula for
its components:
$$
F^k=-\sum^n_{i=1}\sum^n_{j=1}M^k_{ij}\,v^i\,v^j.\hskip -2em
\tag4.2
$$
\definition{Definition 4.1} We say that Newtonian dynamical system on
Riemannian manifold $M$ is {\bf metrizable} by geodesic flow of some
affine connection on $M$, if it inherits  trajectories of this geodesic
flow.
\enddefinition
    The origin of affine connection defining geodesic flow in the
definition~4.1 is absolutely unessential. In case when this connection
is metrical, i\.~e\. when its components $\tilde\Gamma^k_{ij}$ are
given by formula \thetag{1.6}, we say that dynamical system is
metrizable by metric $\tilde\bold g$. If metric $\tilde\bold g$ is
conformally equivalent to basic metric of Riemannian manifold $M$,
i\.~e\. if $\tilde\bold g=e^{-2f}\,\bold g$, then we return to the
situation described by definition~2.3.\par
    In general case described by definition~4.1 tensor field $M$ with
components $M^k_{ij}$ in \thetag{4.2} is an arbitrary symmetric
(non-extended) tensor field of type $(1,2)$ on the manifold $M$.
Components of force \thetag{4.2} are homogeneous functions of degree
$\gamma=2$ respective to velocity vector $\bold v$. Therefore we
can apply results of \S\,2. For the force field of metrizable dynamical
system this gives
$$
F^k=-\sum^n_{i=1}\sum^n_{j=1}M^k_{ij}\,v^i\,v^j+\frac{H}{|\bold v|}\,v^k.
\hskip -2em
\tag4.3
$$
Now we are to study, which of Newtonian dynamical systems, metrizable in
the sense of definition~4.1, admit the normal shift\,? In order to
explore this question we substitute \thetag{4.3} into the equations
of normality \thetag{1.1}, \thetag{1.2}, \thetag{1.3}, \thetag{1.4}.
Let's begin with the last equation. Therefore let's calculate
$\tilde\nabla_jF^i$:
$$
\tilde\nabla_jF^i=-2\sum^n_{s=1}M^i_{js}\,v^s+\tilde\nabla_jH\,N^i+
\frac{H}{|\bold v|}\,P^i_j.\hskip -2em
\tag4.4
$$
Contracting this expression with $P^j_\sigma$ and $P^\varepsilon_i$
we get
$$
\sum^n_{i=1}\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF^i\,P^\varepsilon_i=
-2\sum^n_{i=1}\sum^n_{j=1}\sum^n_{s=1}P^j_\sigma\,M^i_{js}\,v^s\,
P^\varepsilon_i+\frac{H}{|\bold v|}\,P^\varepsilon_\sigma.
$$
Last term in right hand side of this equality differs from
$P^\varepsilon_\sigma$ by scalar multiple. Hence for the force field
\thetag{4.2} to satisfy the equation {1.4} we should provide the equality
$$
\sum^n_{i=1}\sum^n_{j=1}\sum^n_{s=1}P^j_\sigma\,M^i_{js}\,v^s\,
P^\varepsilon_i=\lambda\,P^\varepsilon_\sigma\hskip -2em
\tag4.5
$$
to be fulfilled. Here $\lambda$ is a scalar parameter. The value of this
parameter is easily calculated by contracting \thetag{4.5} with respect
to indices $\varepsilon$ and $\sigma$:
$$
\lambda=\sum^n_{i=1}\sum^n_{j=1}\sum^n_{s=1}\frac{P^j_i\,M^i_{js}\,
v^s}{n-1}.\hskip -2em
\tag4.6
$$
The relationships \thetag{4.5} with parameter given by \thetag{4.6}
form the set of equations for the components of tensor $\bold M$ at
each $p$ on the manifold $M$. They are purely algebraic equations.
These equations contain components of velocity vector $\bold v$,
though quantities $M^i_{jk}$ do not depend on $\bold v$. In order
to solve the equations \thetag{4.5} let's introduce two operators
$\bold X$ and $\bold Y$ with the following components:
$$
\xalignat 2
&\quad X^i_j=\sum^n_{s=1}M^i_{js}\,v^s,
&&Y^\varepsilon_j=\sum^n_{i=1}P^\varepsilon_i\,X^i_j.
\hskip -2em
\tag4.7
\endxalignat
$$
Then the equation \thetag{4.5} and second relationship in \thetag{4.7}
are equivalent to operator equalities $\bold Y\,\bold P=\lambda\,\bold P$
and $\bold P\,\bold X=\bold Y$ respectively.
\proclaim{Lemma 4.1} The equations \thetag{4.5} are reduced to the
relationships
$$
\sum^n_{s=1}M^i_{js}\,v^s=\lambda\,P^i_j+u^i\,N_j+N^i\,w_j,
\hskip -2em
\tag4.8
$$
where $\lambda$ is a scalar parameter defined by \thetag{4.6}, $w_j$
are components of some covector, and $u^i$ are components of some
vector perpendicular to $\bold v$.
\endproclaim
\demo{Proof} Let's write operator equality $\bold Y\,\bold P=\lambda\,
\bold P$, equivalent to \thetag{4.5}, in local coordinates, and then
use formulas \thetag{5.4} from Chapter
\uppercase\expandafter{\romannumeral 5}:
$$
\sum^n_{j=1}Y^\varepsilon_j\,P^j_\sigma=\sum^n_{j=1}Y^\varepsilon_j
\left(\delta^j_\sigma-N^j\,N_\sigma\right)=Y^\varepsilon_\sigma
-\left(\,\shave{\sum^n_{j=1}}Y^\varepsilon_j\,N^j\right)N_\sigma=
\lambda\,P^\varepsilon_\sigma.
$$
This relationship can be rewritten as follows:
$$
Y^\varepsilon_\sigma=\lambda\,P^\varepsilon_j+\left(\,\shave{\sum^n_{j=1}}
Y^\varepsilon_j\,N^j\right)N_\sigma=\lambda\,P^\varepsilon_\sigma+
u^\varepsilon\,N_\sigma.\hskip -2em
\tag4.9
$$
By $u^\varepsilon$ we denote components of vector $\bold u=\bold Y(\bold N)
=\bold P(\bold X(\bold N))$. Operator $\bold P$ is a projector to hyperplane
perpendicular to the vector of velocity. Therefore $\bold u\perp\bold v$.
\par
    Further let's write in coordinates operator equality $\bold P\,\bold
X=\bold Y$ and let's apply formulas \thetag{5.4} from Chapter
\uppercase\expandafter{\romannumeral 5} again:
$$
\sum^n_{i=1}P^\varepsilon_i\,X^i_\sigma=
\sum^n_{i=1}\left(\delta^\varepsilon_i-N^\varepsilon\,N_i\right)
X^i_\sigma=X^\varepsilon_\sigma-\left(\,\shave{\sum^n_{j=1}}N_i\,
X^i_\sigma\right)N^\varepsilon=Y^\varepsilon_\sigma.
$$
Denote by $w_\sigma$ the coefficients of $N^\varepsilon$ in the above
formula binding $\bold X$ and $\bold Y$. They are components of some
covector $\bold w$:
$$
w_\sigma=\sum^n_{j=1}N_i\,X^i_\sigma.
$$
Now the relationship $\bold P\,\bold X=\bold Y$ for $\bold X$ and $\bold Y$
can be written as
$$
X^\varepsilon_\sigma=Y^\varepsilon_\sigma+w_\sigma\,N^\varepsilon.
\hskip -3em
\tag4.10
$$
Combining \thetag{4.9}, \thetag{4.10}, and first formula in \thetag{4.7},
we get the relationship coinciding with \thetag{4.8}. Lemma is proved
\qed\enddemo
     Components of vector $\bold u$ and components of covector $\bold w$
in \thetag{4.8} can be calculated in explicit form. In order to find
$w_j$ we multiply \thetag{4.8} by $N_i$ and contract with respect to
index $i$. Thereby we take into account the condition $\bold u\perp\bold N$,
which follows from orthogonality of vectors $\bold u$ and $\bold v$:
$$
w_j=\sum^n_{i=1}\sum^n_{s=1}M^i_{js}\,v^s\,N_i.\hskip -3em
\tag4.11
$$
In order to find components of vector $\bold u$ we multiply \thetag{4.8}
by $N^j$ and contract with respect to index $j$. This yields
$$
u^i=\sum^n_{j=1}\sum^n_{s=1}N^j\,M^i_{js}\,v^s-\sum^n_{j=1}w_j\,
N^j\,N^i.
\hskip -3em
\tag4.12
$$
Parameter $\lambda$ is calculated by formula \thetag{4.6}, as it was
above. However, this formula now can be written as follows:
$$
\lambda=\sum^n_{j=1}\sum^n_{s=1}\frac{M^j_{js}\,v^s}{n-1}
-\sum^n_{j=1}\frac{w_j\,N^j}{n-1}.\hskip -3em
\tag4.13
$$\par
     The relationships \thetag{4.8}, \thetag{4.11}, \thetag{4.12}, and
\thetag{4.13}, as well as the initial relationships \thetag{4.5}, are
the equations for the components of tensor $\bold M$ that should be
fulfilled identically for all $\bold v\neq 0$. Components of tensor
$\bold M$by themselves do not depend on $\bold v$. Now we shall use
the arbitrariness in $\bold v$. Let's choose vector $\bold v$ such
that all its components are zero, except for one of them:
$$
v^s=\delta^s_r
=\cases
0 &\text{for \ }s\neq r,\\
1 &\text{for \ }s=r.
\endcases\hskip -3em
\tag4.14
$$
From  \thetag{4.14}  one  easily  calculate   $|\bold   v|$   and 
components
of the vector $\bold N$ as well:
$$
\xalignat 3
&\qquad |\bold v|^2=g_{rr},&&N^s=\frac{\delta^s_r}{\sqrt{g_{rr}}},
&&N_s=\frac{g_{sr}}{\sqrt{g_{rr}}}.\hskip -2em
\tag4.15
\endxalignat
$$
Further let's first calculate parameters $w_j$ and $u^i$ by formulas
\thetag{4.11} and \thetag{4.12}:
$$
\gathered
w_j=\frac{M_{rjr}}{\sqrt{g_{rr}}}=\sum^n_{k=1}\frac{M^k_{jr}\,g_{kr}}
{\sqrt{g_{rr}}},\\
\vspace{1ex}
u^i=\frac{M^i_{rr}}{\sqrt{g_{rr}}}-\frac{M_{rrr}}{g_{rr}\,\sqrt{g_{rr}}}
\,\delta^i_r.
\endgathered\hskip -3em
\tag4.16
$$
In a similar way, let's calculate parameter $\lambda$ by formula
\thetag{4.13}:
$$
\lambda=\frac{1}{n-1}\left(\,\shave{\sum^n_{k=1}}M^k_{kr}-
\frac{M_{rrr}}{g_{rr}}\right).\hskip -3em
\tag4.17
$$
Now let's substitute \thetag{4.15}, \thetag{4.16}, and \thetag{4.17}
into the formulas \thetag{4.8}. This yields 
$$
\gathered
M^i_{jr}=\frac{1}{n-1}\left(\,\shave{\sum^n_{k=1}}M^k_{kr}-
\frac{M_{rrr}}{g_{rr}}\right)\left(\delta^i_j-\frac{\delta^i_r\,
g_{jr}}{g_{rr}}\right)+\\
\vspace{2ex}
+\left(M^i_{rr}-\frac{M_{rrr}}{g_{rr}}\,\delta^i_r\right)
\frac{g_{jr}}{g_{rr}}+M_{rjr}\,\frac{\delta^i_r}{g_{rr}}.
\endgathered\hskip -3em
\tag4.18
$$
Formula \thetag{4.18} gives the expressions for components of
tensor $\bold M$. Let's study the dependence of $M^i_{jr}$
upon indices $i$, $j$, and $r$. For this purpose, first of all,
let's introduce the following notations:
$$
\align
\mu_r&=\frac{1}{n-1}\left(\,\shave{\sum^n_{k=1}}M^k_{kr}-
\frac{M_{rrr}}{g_{rr}}\right),\hskip -3em
\tag4.19\\
\vspace{1ex}
\theta^i_r&=\frac{M^i_{rr}}{g_{rr}},\hskip -3em
\tag4.20\\
\vspace{1ex}
\varkappa_{jr}&=\frac{M_{rjr}}{g_{rr}}-\frac{M_{rrr}\,g_{jr}}
{(g_{rr})^2}-\frac{g_{jr}\,\mu_r}{g_{rr}}.\hskip -3em
\tag4.21
\endalign
$$
Taking into account notations \thetag{4.19}, \thetag{4.20}, and
\thetag{4.21}, we can rewrite formula \thetag{4.18}, bringing
it to the following form:
$$
M^i_{jr}=\mu_r\,\delta^i_j+\varkappa_{jr}\,\delta^i_r+\theta^i_r\,
g_{jr}.\hskip -3em
\tag4.22
$$
Apart from formula \thetag{4.22} we take into account symmetry
of tensor $\bold M$. Let $j\neq r$. Let's calculate the difference
$M^i_{jr}-M^i_{rj}$. Equating it to zero, we get the equation
$$
(\mu_r-\varkappa_{rj})\,\delta^i_j+(\varkappa_{jr}-\mu_j)\,
\delta^i_r+(\theta^i_r-\theta^i_j)\,g_{jr}=0.\hskip -3em
\tag4.23
$$
The equation \thetag{4.23} \pagebreak substantially simplifies, if
we assume that $i\neq j$ and $i\neq r$. In this case we have
$(\theta^i_r-\theta^i_j)\,g_{jr}=0$. Without loss of generality we
can choose local coordinates so that all components of metric tensor
are nonzero: $g_{jr}\neq 0$. Then
$$
\theta^i_r=\theta^i_j\text{ \ for \ }i\neq j\text{ \ and \ }i\neq r.
\hskip -3em
\tag4.24
$$
Remember that we consider multidimensional case $n\geqslant 3$,
two-dimensional case is considered in {\bf other thesis} (see
\cite{36}). In multidimensional case $n\geqslant 3$ the range
of indices is large enough for to satisfy inequalities
$i\neq r$, $i\neq j$, and $j\neq r$ simultaneously. Therefore
the relationships \thetag{4.24} are not trivial. They mean that
the quantities $\theta^i_r$ in \thetag{4.22} do not depend on $r$
for all $r\ne i$. Let's write this fact as 
$$
\theta^i_r=\cases
z^i&\text{for \ }r\neq i,\\
\theta^r_r&\text{for \ }r=i.
\endcases\hskip -3em
\tag4.25
$$
It is convenient to bring \thetag{4.25} to the following form
$$
\theta^i_r=z^i+(\theta^r_r-z^r)\,\delta^i_r.\hskip -3em
\tag4.26
$$\par
    Apart from the case, when $i\neq j$ and $i\neq r$, in \thetag{4.23}
have two other cases, when $i=j$ or $i=r$. In both cases $j\neq r$, since
for $j=r$ the equation \thetag{4.23}  turns  to  identity  $0=0$. 
Considering
the equation \thetag{4.23} in both these cases, we get the same result,
which is written as follows: 
$$
\varkappa_{jr}=\mu_j+(z^r-\theta^r_r)\,g_{jr}\text{ \ for \ }
j\neq r.\hskip -3em
\tag4.27
$$
Let's substitute \thetag{4.26} and \thetag{4.27} into \thetag{4.22}.
As a result we obtain
$$
M^i_{jr}=\mu_r\,\delta^i_j+\mu_j\,\delta^i_r+z^i\,g_{jr}
\text{ \ for \ }j\neq r.\hskip -3em
\tag4.28
$$
Components $M^i_{rr}$ corresponding to the case, when $j=r$, can be
determined from formula \thetag{4.20}. This yields $M^i_{rr}=\theta^i_r\,
g_{rr}$. If we take into account \thetag{4.26}, then we get
$$
M^i_{rr}=(\theta^r_r-z^r)\,g_{rr}\,\delta^i_r+z^i\,g_{rr}.
\hskip -3em
\tag4.29
$$
For the sake of uniformity in the above formulas \thetag{4.28} and
\thetag{4.29} it is convenient to introduce the following notations:
$$
\tilde\mu_r=\frac{(\theta^r_r-z^r)\,g_{rr}}{2}.
$$
Then \thetag{4.28} and \thetag{4.29} can be united into one formula
$$
M^i_{jr}=\cases
\mu_r\,\delta^i_j+\mu_j\,\delta^i_r+z^i\,g_{jr}
&\text{ \ for \ }j\neq r,\\
\tilde\mu_r\,\delta^i_j+\tilde\mu_j\,\delta^i_r+z^i\,g_{jr}
&\text{ \ for \ }j=r.
\endcases\hskip -3em
\tag4.30
$$
\proclaim{Lemma 4.2} Tensor $\bold M$ satisfies the equations
\thetag{4.5} if and only if its components are defined by
formula
$$
M^i_{jr}=\mu_r\,\delta^i_j+\mu_j\,\delta^i_r+z^i\,g_{jr}.\hskip -3em
\tag4.31
$$
\endproclaim
\demo{Proof} By deriving formula \thetag{4.30} we have already done
the part of proof for lemma~4.2. Now we are to prove that $\mu_i=\tilde
\mu_i$ in this formula. Let's substitute the expression \thetag{4.30}
into \thetag{4.5} and into the formula \thetag{4.6} for $\lambda$.
This yields
$$
\align
&\sum^n_{s=1}\mu_s\,v^s\,P^\varepsilon_\sigma+
\sum^n_{j=1}2\,(\tilde\mu_j-\mu_j)\,v^j\,P^\varepsilon_j\,P^j_\sigma=
\lambda\,P^\varepsilon_\sigma,\hskip -3em
\tag4.32\\
\vspace{1ex}
&\lambda=\sum^n_{s=1}\mu_s\,v^s+\sum^n_{j=1}\frac{2\,(\tilde\mu_j-\mu_j)
\,v^j\,P^j_j}{n-1}.\hskip -3em
\tag4.33
\endalign
$$
By substituting \thetag{4.33} into \thetag{4.32} we bring the equation
\thetag{4.5} to the form
$$
\sum^n_{j=1}(\tilde\mu_j-\mu_j)\,v^j\,P^\varepsilon_j\,P^j_\sigma=
\sum^n_{j=1}\frac{(\tilde\mu_j-\mu_j)\,v^j\,P^j_j\,P^\varepsilon_\sigma}
{n-1}.\hskip -3em
\tag4.34
$$\par
     Similar to \thetag{4.5}, the equations \thetag{4.34} derived from
\thetag{4.5} should be fulfilled identically for all  $\bold  v$. 
Therefore
we can use again the trick with special choice of $\bold v$ in them.
However, the choice of \thetag{4.14} yields nothing new, bringing
\thetag{4.34} to the identities. We are to modify this trick by considering
vector $\bold v$ that slightly differs from \thetag{4.14}:
$$
v^s=\delta^s_r+\xi^s.\hskip -3em
\tag4.35
$$
Assuming $\xi^s$ in \thetag{4.35} to be small parameters, let's consider
Taylor expansions for all terms in \thetag{4.34}.  Thereby we shall hold
only terms linear in $\xi^s$. For $|\bold v|$ we have
$$
|\bold v|^2=g_{rr}+2\sum^n_{k=1}g_{rk}\,\xi^k+\ldots
=g_{rr}+2\,\xi_r+\ldots\,.\hskip -3em
\tag4.36
$$
Further let's calculate covariant and contravariant components of
unitary vector $\bold N$ directed along the velocity vector $\bold v$:
$$
\aligned
N^s&=\frac{\delta^s_r}{\sqrt{g_{rr}}}\left(1-\frac{\xi_r}{g_{rr}}\right)
+\frac{\xi^s}{\sqrt{g_{rr}}}+\ldots\,,\\
\vspace{1ex}
N_s&=\frac{g_{sr}}{\sqrt{g_{rr}}}\left(1-\frac{\xi_r}{g_{rr}}\right)
+\frac{\xi_s}{\sqrt{g_{rr}}}+\ldots\,.
\endaligned\hskip -3em
\tag4.37
$$
Using \thetag{4.37} and using formulas \thetag{5.4} from Chapter
\uppercase\expandafter{\romannumeral 5}, we calculate components
of orthogonal projection operator $\bold P$:
$$
P^i_j=\delta^i_j-\frac{\delta^i_r\,g_{jr}}{g_{rr}}\left(1-2\,\frac{\xi_r}
{g_{rr}}\right)-\frac{\xi^i\,g_{jr}+\xi_j\,\delta^i_r}{g_{rr}}+\ldots\,.
\hskip -3em
\tag4.38
$$
For the sake of brevity in formula \thetag{4.34} we introduce the following
notations:
$$
m_j=\tilde\mu_j-\mu_j.\hskip -3em
\tag4.39
$$
From \thetag{4.35} we get $v^j=\delta^j_r+\xi^j$. Let's substitute
this expression into the equations \thetag{4.34} and take into
account above notations:
$$
P^\varepsilon_r\,m_r\,P^r_\sigma-\frac{P^r_r\,m_r\,P^\varepsilon_\sigma}
{n-1}=\sum^n_{j=1}m_j\,\xi^j\left(\frac{P^j_j\,P^\varepsilon_\sigma}{n-1}-
P^\varepsilon_j\,P^j_\sigma\right).\hskip -3em
\tag4.40
$$
Right hand side of \thetag{4.40} contains small parameters $\xi^j$
as multiplicands. Hence for components of projector $\bold P$ in right
hand side of \thetag{4.40} we can hold only zero order terms from
their Taylor expansions \thetag{4.38}:
$$
\gather
\sum^n_{j=1}m_j\,\xi^j\,\frac{P^j_j\,P^\varepsilon_\sigma}{n-1}
=\sum^n_{j=1}m_j\,\xi^j\left(1-\frac{\delta^j_r\,g_{jr}}{g_{rr}}
\right)\frac{P^\varepsilon_\sigma}{n-1}+\ldots\,,\\
\sum^n_{j=1}m_j\,\xi^j\,P^\varepsilon_j\,P^j_\sigma=
\sum^n_{j=1}m_j\,\xi^j\,P^\varepsilon_j\left(\delta^j_\sigma-
\frac{\delta^j_r\,g_{r\sigma}}{g_{rr}}\right)+\ldots\,.
\endgather
$$
Let's accomplish summation in right hand sides of the obtained
expansions. Then 
$$
\gathered
\sum^n_{j=1}m_j\,\xi^j\,\frac{P^j_j\,P^\varepsilon_\sigma}{n-1}
=\sum_{j\neq r}m_j\,\xi^j\,\frac{P^\varepsilon_\sigma}{n-1}
+\ldots\,,\\
\sum^n_{j=1}m_j\,\xi^j\,P^\varepsilon_j\,P^j_\sigma=
m_\sigma\,\xi^\sigma\,P^\varepsilon_\sigma-m_r\,\xi^r\,
P^\varepsilon_r\frac{g_{r\sigma}}{g_{rr}}+\ldots\,.
\endgathered\hskip -3em
\tag4.41
$$
Now we take into account that zero order terms in the expansions
of $P^\varepsilon_r$ are zero. Indeed, from formula \thetag{4.38}
we find that 
$$
P^\varepsilon_r=\frac{\xi_r}{g_{rr}}\,\delta^\varepsilon_r
-\xi^\varepsilon+\ldots\,.\hskip -3em
\tag4.42
$$
Let's substitute the expansions \thetag{4.41} and \thetag{4.42}
into the equations \thetag{4.40}. Collecting first order terms
in resulting equality, we obtain
$$
\gather
m_r\left(\frac{\delta^\varepsilon_r}{g_{rr}}\,\xi_r-\xi^\varepsilon
\right)\left(\delta^r_\sigma-\frac{g_{\sigma r}}{g_{rr}}\right)+
m_\sigma\,\xi^\sigma\left(\delta^\varepsilon_\sigma-
\frac{\delta^\varepsilon_r\,g_{r\sigma}}{g_{rr}}\right)=\\
\vspace{1ex}
=\left(\,\shave{\sum_{j\neq r}}\frac{m_j\,g_{rr}+m_r\,g_{rj}}
{g_{rr}\,(n-1)}\,\xi^j\right)\left(\delta^\varepsilon_\sigma-
\frac{\delta^\varepsilon_r\,g_{r\sigma}}{g_{rr}}\right).
\endgather
$$
Remember that we consider case $n\geqslant 3$. Two-dimensional
case is considered in {\bf other thesis} (see \cite{36}). In
multidimensional case $n\geqslant 3$ we can choose three
distinct values for $\varepsilon$, $r$, and $\sigma$, i\.~e\.
$\varepsilon\neq r$, $r\neq\sigma$, and $\sigma\neq\varepsilon$.
This choice substantially simplifies the above equalities.
Now they look like 
$$
\frac{m_r\,g_{\sigma r}}{g_{rr}}\,\xi^\varepsilon=0.\hskip -3em
\tag4.43
$$
As we already mentioned above, without loss of generality we
can choose local coordinates so that all components of metric
tensor $g_{\sigma r}$ are nonzero. Parameters $\xi^1,\,\ldots,\,
\xi^n$ are arbitrary real numbers, they can be chosen nonzero
as well. Hence from \thetag{4.43} and \thetag{4.39} we get $m_r=0$
and $\tilde\mu_r=\mu_r$. Thus, we proved that formula \thetag{4.35}
follows from the equations \thetag{4.5}. And conversely, \thetag{4.5}
follows from \thetag{4.31}. This is proved immediately by substituting
\thetag{4.31} into the equations \thetag{4.5}. \qed\enddemo
    Combining the results of lemma~4.1 and lemma~4.2, we get the theorem.
\proclaim{Theorem 4.1} On Riemannian manifold $M$ of the dimension
$n\geqslant 3$ the force field \thetag{4.5} satisfies the equations
of normality \thetag{1.4} if and only if components of tensor $\bold M$
in \thetag{4.5} are defined by formula \thetag{4.31}.
\endproclaim
    Parameters $z^i$ and $\mu_j$ in \thetag{4.31} are the components
of some vector field $\bold z$ and some covector field $\boldsymbol\mu$
respectively. In order to prove it let's consider the following
two expressions in formula \thetag{4.31}:
$$
\gathered
\sum^n_{r=1}\sum^n_{s=1}g^{ir}\,M^s_{sr}=(n+1)\,\mu^i+z^i,\\
\vspace{1ex}
\sum^n_{r=1}\sum^n_{s=1}g^{sr}\,M^i_{sr}=2\mu^i+n\,z^i.
\endgathered\hskip -3em
\tag4.44
$$
By means of \thetag{4.44} we can express parameters $z^i$ and $\mu_j$
explicitly through components of tensor field $\bold M$:
$$
\gathered
z^i=\sum^n_{r=1}\sum^n_{s=1}\frac{(n+1)\,g^{sr}\,M^i_{sr}
-2\,g^{ir}\,M^s_{sr}}{n^2+n-2},\\
\vspace{1ex}
\mu_j=\sum^n_{i=1}\sum^n_{r=1}\sum^n_{s=1}\frac{n\,g^{ir}\,M^s_{sr}
-g^{sr}\,M^i_{sr}}{n^2+n-2}\,g_{ij}.
\endgathered\hskip -3em
\tag4.45
$$
Formulas \thetag{4.45} makes transparent vectorial and covectorial nature
of the quantities $z^i$ and $\mu_j$ in \thetag{4.31}.\par
    Let's substitute \thetag{4.31} into the formula \thetag{4.3} for the
force field $\bold F$. As a result we get the following expression for
covariant components of $\bold F$:
$$
F_i=-2\sum^n_{j=1}\mu_j\,v^j\,v_i-z_i\,|\bold v|^2+H\,N_i.
\hskip -3em
\tag4.46
$$
Next step consist in substituting \thetag{4.46} into the normality
equations \thetag{1.3}. Let's first calculate $\tilde\nabla_mF_j$ and
$\nabla_iF_j$:
$$
\align
&\aligned
\tilde\nabla_mF_j=&-2\,\mu_m\,v_j-\sum^n_{s=1}2\,\mu_s\,v^s\,g_{mj}-\\
&-2\,z_j\,v_m+\tilde\nabla_mH\,N_j+\sum^n_{s=1}\frac{H}{|\bold v|}
\,P^s_m\,g_{sj},
\endaligned\\
&\nabla_iF_j=-\sum^n_{s=1}2\,\nabla_i\mu_s\,v^s\,v_j-\nabla_iz_j
\,|\bold v|^2+\nabla_iH\,N_j.
\endalign
$$
Then accomplish the following calculations:
$$
\align
&\sum^n_{i=1}\sum^n_{j=1}\nabla_iF_j\,P^i_\varepsilon\,P^j_\sigma=
-|\bold v|^2\sum^n_{i=1}\sum^n_{j=1}\nabla_iz_j\,P^i_\varepsilon
\,P^j_\sigma,\\
\vspace{1ex}
&\sum^n_{i=1}\sum^n_{j=1}\sum^n_{m=1}N^m\,\frac{F_i\,\tilde\nabla_mF_j}
{|\bold v|}\,P^i_\varepsilon\,P^j_\sigma=-|\bold v|^2\sum^n_{i=1}
\sum^n_{j=1}z_i\,z_j\,P^i_\varepsilon\,P^j_\sigma.
\endalign
$$
If we take into account the above calculations, then, substituting
components of force field \thetag{4.46} into \thetag{1.3}, we get
$$
\sum^n_{i=1}\sum^n_{j=1}(\nabla_iz_j-\nabla_jz_i)\,P^i_\varepsilon
\,P^j_\sigma=0.
\hskip -3em
\tag4.47
$$
Components of projector field in \thetag{4.47} depend on $\bold v$,
while difference $\nabla_iz_j-\nabla_jz_i$ doesn't depend. Let's
recall again that here we consider multidimensional case
$n\geqslant 3$. Two-dimensional case is considered in {\bf other
thesis} \cite{36}. Suppose that $\bold a$ and $\bold b$ are two
arbitrary vectors from tangent space to $M$ at some point $p$.
For $n\geqslant 3$ we can direct $\bold v$ so that it will be
perpendicular to $\bold a$ and $\bold b$. Then 
$$
\xalignat 2
&\sum^n_{\varepsilon=1}P^i_\varepsilon\,a^\varepsilon=a^i,
&&\sum^n_{\sigma=1}P^j_\sigma\,b^\sigma=b^j.
\endxalignat
$$
Contracting \thetag{4.47} with the product $a^\varepsilon\,b^\sigma$,
now we get
$$
\varphi(\bold a,\bold b)=\sum^n_{i=1}\sum^n_{j=1}(\nabla_iz_j-\nabla_jz_i)
\,a^i\,b^j=0.
$$
Since $\bold a$ and $\bold b$ are arbitrary vectors, the above equation
means that skew symmetric bilinear form $\varphi$ with components
$\varphi_{ij}=\nabla_iz_j-\nabla_jz_i$ is identically zero. Hence
$$
\nabla_iz_j-\nabla_jz_i=0.
\hskip -3em
\tag4.48
$$
Note that quantities $z_1,\,\ldots,\,z_n$ in \thetag{4.48}, which are
covariant components of vector field $\bold z$, do not depend on
$\bold v$. Let's calculate $\nabla_iz_j$ and $\nabla_jz_i$ according
top the formula \thetag{7.3} from Chapter
\uppercase\expandafter{\romannumeral 3}, and let's take into account
the symmetry of connection components $\Gamma^k_{ij}$ for metric
connection. As a result from \thetag{4.48} we derive
$$
\frac{\partial z_i}{\partial x^j}=\frac{\partial z_j}{\partial x^i}.
\hskip -3em
\tag4.49
$$
The relationship \thetag{4.49} means that differential $1$-form
$\bold z$ with components $z_1,\,\ldots,\,z_n$ is closed. Its
known that any closed form is locally exact (see proof in \cite{79}).
When applied to differential $1$-form $\bold z$, this fact yields
$$
z_i=\frac{\partial f}{\partial x^i}=\nabla_if,
\hskip -3em
\tag4.50
$$
where $f$ is a scalar field, which is defined locally in the neighborhood
of any point.\par
    Let's substitute the expression \thetag{4.50} for $z_i$ into the
formula \thetag{4.46}. The result of this substitution is written as
follows:
$$
F_i=-\nabla_if\,|\bold v|^2+2\sum^n_{s=1}\nabla_sf\,v^s\,v_i+
\left(H-\shave{\sum^n_{s=1}}2\,|\bold v|\,(\mu_s+\nabla_sf)\,v^s\right)
\,N_i.
$$
Now it's convenient to make the following redesignation, since the
equations \thetag{1.3} and \thetag{1.4} do not specify the
form of scalar field $\bold H$:
$$
H-\sum^n_{s=1}2|\bold v|\,(\mu_s+\nabla_s)\,v^s\longrightarrow H.
$$
As a result of this redesignation we get simpler formula for the
force field $\bold F$:
$$
F_i=-\nabla_if\,|\bold v|^2+2\sum^n_{s=1}\nabla_sf\,v^s\,v_i+H\,N_i.
\hskip -3em
\tag4.51
$$
In essential, formula \thetag{4.51} coincides with formula \thetag{2.11},
which we studied earlier. The only difference is that field $F$ here is
defined locally. Substituting formula \thetag{4.51} into weak normality 
equations
\thetag{1.1} and \thetag{1.2}, we can specify $H$ in it:
$$
F_i=-|\bold v|^2\,\nabla_if+2\sum^n_{s=1}
\nabla_sf\,v^s\,v_i+H(v\,e^{-f})\,e^f\,N_i.
\hskip -3em
\tag4.52
$$
This is the result of \S\,3 (see formula \thetag{3.26} and theorem~3.1).
Main result of this section is formulated as follows.
\proclaim{Theorem 4.1} Newtonian dynamical system on Riemannian manifold
$M$ of the dimension $n\geqslant 3$ admitting the normal shift is
metrized by geodesic flow of affine connection $\Gamma$ if and only if
locally this connection is a metric connection for metric $\tilde\bold g
=e^{-2f}\,\bold g$ conformally equivalent to basic metric of manifold
$M$.
\endproclaim
\pagebreak
    This result justifies our first approach, when we restricted ourselves
to considering only conformally equivalent metrics. It was obtained in
paper \cite{85}, which is not yet published.
\head
\S\,5. Examples of non-metrizable dynamical systems.
\endhead
     Some examples of dynamical systems admitting the normal shift in
Euclidean space $\Bbb R^2$ were constructed in paper \cite{35} (see also
preprint \cite{34}), where the theory of such systems originates.
Later it was found that some of them are non-metrizable. This indicates
non-triviality of theory in the dimension $n=2$.\par
     Problem of finding multidimensional non-metrizable systems was
open for a while. The importance of solving this problem was pointed
out by academician A\.~T\. Fomenko, when I reported at the seminar in
Moscow State University in the beginning of 1994. Some example of
multidimensional non-metrizable dynamical systems were found in
\cite{42}. Here we shall describe these examples.\par
     Suppose that $A=A(v)$ is some smooth scalar function of one
variable, and suppose that $\boldsymbol\mu$ is some vector field
on Riemannian manifold $M$ of the dimension $n\geqslant 3$. Let's
consider the force field $\bold F$ on $M$ with the following
components:
$$
F^k=A(|\bold v|)\left(\,\shave{\sum^n_{s=1}}2\,\mu^s\,N_s\,N^k-\mu^k
\right).\hskip -2em
\tag5.1
$$
\proclaim{Theorem 5.1} Newtonian dynamical system with force field
\thetag{5.1} on Riemannian manifold $M$ of the dimension $n\geqslant 3$
is a system admitting the normal shift if and only if $\boldsymbol\mu$
is (at least locally) a gradient of some scalar field, i\.~e\.
$\mu_i=\nabla_if$.
\endproclaim
\demo{Proof} Theorem~12.1 from Chapter
\uppercase\expandafter{\romannumeral 5} reduces proof to substituting
\thetag{5.1} into normality equations \thetag{1.1}, \thetag{1.2},
\thetag{1.3}, and \thetag{1.4}. Let's begin with the first of these
equations. In \thetag{1.1} we need to evaluate the following
expressions:
$$
\aligned
&\sum^n_{j=1}N^j\,F_j=A\sum^n_{s=1}\mu_s\,N^s,\\
\vspace{1ex}
&\sum^n_{i=1}\frac{F_i\,P^i_q}{|\bold v|}=-\sum^n_{i=1}
\frac{A\,\mu_i\,P^i_q}{|\bold v|}.
\endaligned\hskip -2em
\tag5.2
$$
Let's differentiate first of these two equalities \thetag{5.2}.
Applying $\tilde\nabla_i$, we get
$$
\sum^n_{j=1}\tilde\nabla_i(N^j\,F_j)=A'\sum^n_{s=1}\mu_s\,N^s
\,N_i+A\sum^n_{s=1}\frac{\mu_s\,P^s_i}{|\bold v|}.\hskip -2em
\tag5.3
$$
Here $A'$ is a derivative of the function $A=A(v)$ with respect
to its unique argument $v$. By contracting \thetag{5.3} with
$P^i_q$ we obtain
$$
\sum^n_{i=1}\sum^n_{j=1}\tilde\nabla_i(N^j\,F_j)\,P^i_q=
\sum^n_{i=1}\frac{A\,\mu_i\,P^i_q}{|\bold v|}.\hskip -2em
\tag5.4
$$
Comparing \thetag{5.4} with second equality \thetag{5.2}, we find
that \thetag{5.1} satisfies the equation \thetag{1.1} identically.
\par
     Now let's substitute \thetag{5.1} into \thetag{1.2}. This
require larger amount of calculations. For the first, let's
evaluate covariant derivatives $\nabla_iF_j$ and $\tilde\nabla_rF_j$:
$$
\align
&\nabla_iF_j=A\left(\,\shave{\sum^n_{s=1}}2\,\nabla_i\mu_s\,N^s
\,N_j-\nabla_i\mu_j\right),\hskip -2em
\tag5.5\\
\vspace{2ex}
&\aligned
\tilde\nabla_j&F_i=A'\left(\,\shave{\sum^n_{s=1}}2\,\mu_s\,N^s
\,N_i-\mu_i\right)N_j\,+\\
&+\,\sum^n_{s=1}A\left(\frac{2\,\mu_s\,P^s_j\,N_i}{|\bold v|}
+\shave{\sum^n_{k=1}}\frac{2\,\mu_s\,N^s\,P^k_i\,g_{kj}}
{|\bold v|}\right).
\endaligned\hskip -2em
\tag5.6
\endalign
$$
Then, using formulas \thetag{5.5} and \thetag{5.6}, we evaluate the
following expressions:
$$
\align
&\aligned
&\sum^n_{i=1}\sum^n_{j=1}\nabla_iF_j\,N^j\,P^i_k=
A\sum^n_{i=1}\sum^n_{j=1}\nabla_i\mu_j\,N^j\,P^i_k,\\
\vspace{1ex}
&\sum^n_{i=1}\sum^n_{j=1}\nabla_jF_i\,N^j\,P^i_k=
-A\sum^n_{i=1}\sum^n_{j=1}\nabla_j\mu_i\,N^j\,P^i_k,
\endaligned\hskip -2em
\tag5.7\\
\vspace{2ex}
&\sum^n_{j=1}\sum^n_{r=1}\tilde\nabla_jF_r\,N^j\,N^r=
A'\sum^n_{s=1}\mu_s\,N^s.\hskip -2em
\tag5.8
\endalign
$$
Apart from \thetag{5.7} and \thetag{5.8}, we need to evaluate
few more expressions:
$$
\align
&\sum^n_{i=1}\tilde\nabla_jF_i\,P^i_k=-A'\sum^n_{s=1}\mu_s\,P^s_k
\,N_j+\sum^n_{s=1}\sum^n_{r=1}\frac{2\,A\,\mu_s\,N^s\,P^r_j\,g_{rk}}
{|\bold v|},\\
\vspace{2ex}
&\sum^n_{i=1}\sum^n_{j=1}\tilde\nabla_jF_i\,P^i_k=-\left(A\,A'+
\frac{2\,A^2}{|\bold v|}\right)\sum^n_{r=1}\sum^n_{s=1}\mu_s\,N^s
\,\mu_r\,P^r_k,\\
\vspace{2ex}
&-\sum^n_{i=1}\sum^n_{j=1}\frac{2\,F_i\,F_j\,N^j\,P^i_k}{|\bold v|^2}=
\frac{2\,A^2}{|\bold v|}\sum^n_{r=1}\sum^n_{s=1}\mu_s\,N^s\,\mu_r
\,P^r_k,\\
\vspace{2ex}
&-\sum^n_{i=1}\sum^n_{j=1}\sum^n_{r=1}\frac{N^r\,N^j\,\tilde\nabla_jF_r}
{|\bold v|}\,F_i\,P^i_k=A\,A'\sum^n_{r=1}\sum^n_{s=1}\mu_s\,N^s
\,\mu_r\,P^r_k.
\endalign
$$
Taking into account all above calculations, we reduce \thetag{1.2} to
the following equation
$$
\sum^n_{i=1}\sum^n_{j=1}(\nabla_i\mu_j-\nabla_j\mu_i)
\,N^j\,P^i_k=0.\hskip -2em
\tag5.9
$$
Note that \thetag{5.9} is the equation with respect to $\mu_i$. It
doesn't contain $A$. The quantities $\nabla_i\mu_j$ and $\nabla_j\mu_i$
in \thetag{5.9} do not depend on $\bold v$. All dependence on
velocity vector is concentrated in $N^j$ and $P^i_k$. Suppose that
$\bold a$ and $\bold b$ are two arbitrary vectors from tangent
space $T_p(M)$ at some point $p\in M$. We can direct $\bold v$
along vector $\bold a$ in \thetag{5.9}. Then we have the following
relationships:
$$
\xalignat 2
&\qquad a^j=\alpha\,N^j,
&&b^i=\beta\,N^i+\sum^n_{k=1}P^i_k\,b^k.
\hskip -3em
\tag5.10
\endxalignat
$$
First relationship \thetag{5.10} simply means that $\bold a$ and
$\bold N$ are collinear vectors. Second relationship means that
$\bold b$ is represented as a sum of two components, one directed
along $\bold N$, and other being perpendicular to $\bold N$,
i\.~e\. $\bold b=\beta\,\bold N+\bold P(\bold b)$. Now denote
$\varphi_{ij}=\nabla_i\mu_j-\nabla_j\mu_i$. Then let's do the
following calculations:
$$
\gather
\sum^n_{i=1}\sum^n_{j=1}\varphi_{ij}\,b^i\,a^j
=\alpha\,\beta\sum^n_{i=1}\sum^n_{j=1}\varphi_{ij}\,N^i\,N^j\,+\\
+\,\alpha\sum^n_{i=1}\sum^n_{j=1}\sum^n_{k=1}(\nabla_i\mu_j
-\nabla_j\mu_i)\,N^j\,P^i_k\,b^k=0.
\endgather
$$
Here took into account \thetag{5.9} and used skew symmetry: $\varphi_{ij}=
-\varphi_{ji}$. As a result of above calculations we obtained the
relationship 
$$
\varphi(\bold b,\bold a)=\sum^n_{i=1}\sum^n_{j=1}\varphi_{ij}\,b^i
\,a^j=0,
$$
which means that bilinear form $\varphi$ with components $\varphi_{ij}$
is identically zero. Therefore
$$
\nabla_i\mu_j=\nabla_j\mu_i.\hskip -3em
\tag5.11
$$
The equation \thetag{5.11} is exact analog of the equation \thetag{4.48}.
By the same arguments as in \S\,4 from \thetag{5.11} we derive the
following formula for the components of $\boldsymbol\mu$:
$$
\mu_i=\frac{\partial f}{\partial x^i}=\nabla_if.\hskip -3em
\tag5.12
$$
Here $f$ is a scalar field, which is defined locally in the neighborhood
of each point $p$ on the manifold $M$.\par
     Next step consists in substituting \thetag{5.1} into the equation
\thetag{1.3}. With this purpose, let's do appropriate calculations:
$$
\sum^n_{j=1}\sum^n_{m=1}P^j_\sigma\,N^m\tilde\nabla_mF_j=
-A'\sum^n_{j=1}\mu_j\,P^j_\sigma.\hskip -3em
\pagebreak
\tag5.13
$$
Let's combine the equality \thetag{5.13} and second relationship in
\thetag{5.2}. This yields
$$
\sum^n_{i=1}\sum^n_{j=1}\sum^n_{m=1}P^i_\varepsilon\,
P^j_\sigma\,N^m\frac{F_i\,\tilde\nabla_mF_j}{|\bold v|}=
A\,A'\sum^n_{i=1}\sum^n_{j=1}\frac{\mu_i\,P^i_\varepsilon\,
\mu_j\,P^j_\sigma}{|\bold v|}.
$$
Now we evaluate the expression obtained by contracting
$\nabla_iF_j$ with $P^i_\varepsilon$ and $P^j_\sigma$:
$$
-\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\,
\nabla_iF_j=\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon
\,P^j_\sigma\,\nabla_i\mu_j.
$$
Taking into account above calculations, we can write \thetag{1.3}
in form of the following equations for the components of
$\boldsymbol\mu$:
$$
\sum^n_{i=1}\sum^n_{j=1}(\nabla_i\mu_j-\nabla_j\mu_i)\,
P^i_\varepsilon\,P^j_\sigma=0.\hskip -3em
\tag5.14
$$
These equations are fulfilled unconditionally due to the relationships
\thetag{5.11} and \thetag{5.12}. Thus, the equations \thetag{1.3}
do not impose additional limitations upon the choice of parameters
$A$ and $\boldsymbol\mu$ in \thetag{5.1}.\par
    Last step in proving theorem~5.1 consists in substituting
\thetag{5.1} into the equations \thetag{1.4}. For to do it let's
evaluate the expression from left hand side of this equation:
$$
\sum^n_{i=1}\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF^i\,
P^\varepsilon_i=\sum^n_{s=1}\frac{2\,A\,\mu_s\,N^s}{|\bold v|}\,
P^\varepsilon_\sigma.\hskip -3em
\tag5.15
$$
Right hand side of \thetag{5.15} differs from $P^\varepsilon_\sigma$
by numeric multiple 
$$
\lambda=\sum^n_{s=1}\frac{2\,A\,\mu_s\,N^s}{|\bold v|}.
$$
Right hand side of \thetag{1.4} is equal to $\lambda\,P^\varepsilon_\sigma$
as well. Now one can see that the equations \thetag{1.4 } are fulfilled
by \thetag{5.1} unconditionally. They do not impose additional
limitations on the choice of $A$ and $\boldsymbol\mu$. Thus, the
only restriction due to the normality equations consists in
\thetag{5.11}. This restriction is resolved explicitly in
form of \thetag{5.12}. Theorem~5.1 is proved.
\qed\enddemo
    In paper \cite{42} we chose flat space $M=\Bbb R^n$ with
standard Euclidean metric. Therefore, choosing constant field
$\boldsymbol\mu$ with $\mu_i=\const$, in \cite{42} we had the
situation, when relationships \thetag{5.11} are unconditionally
fulfilled.\par
    Now suppose that vector field $\boldsymbol\mu$ is chosen
according to the theorem~5.1 just proved. Then dynamical system with
force field \thetag{5.1} is a system admitting the normal shift.
Is it metrizable or not\,? Force field of metrizable dynamical system
that admits normal shift is given by formula \thetag{3.26}.
\pagebreak If formulas \thetag{5.1} and \thetag{3.26} define the same
force field, then we should have the following equality:
$$
\gathered
|\bold v|^2\left(\,\shave{\sum^n_{s=1}}2\,\nabla_sf\,N^s\,N_k
-\nabla_kf\right)+H(|\bold v|\,e^{-f})\,e^f\,N_k=\\
\vspace{1ex}
=A(|\bold v|)\left(\,\shave{\sum^n_{s=1}}2\,\mu_s\,N^s\,N_k
-\mu_k\right).
\endgathered\hskip -3em
\tag5.16
$$
Let's multiply \thetag{5.16} by $N^k$, and then contract it with
respect to index $k$. As a result we get the equality binding
two special scalar fields from extended algebra:
$$
H(|\bold v|\,e^{-f})\,e^f=\sum^n_{s=1}(\mu_s\,A(|\bold v|)-|\bold v|^2
\,\nabla_sf)\,N^s.\hskip -3em
\tag5.17
$$
Left hand side of \thetag{5.17} doesn't depend on $\bold v$. Difference
$\mu_s\,A(|\bold v|)-|\bold v|^2\,\nabla_sf$ in right hand side of
\thetag{5.17} doesn't depend on $\bold v$ as well. Therefore we can
keep $|\bold v|$ unchanged and rotate vector $\bold v$ so that right
hand side of \thetag{5.17} turns to be zero. Hence $H$ is identically
zero function. This reduces \thetag{5.17} to the form
$$
\sum^n_{s=1}(\mu_s\,A(|\bold v|)-|\bold v|^2
\,\nabla_sf)\,N^s=0.\hskip -3em
\tag5.18
$$
Due to the arbitrariness in the choice of $\bold N$ from \thetag{5.18}
we derive
$$
\mu_s\,A(|\bold v|)-|\bold v|^2\,\nabla_sf=0.
$$
It's obvious that this covectorial equality breaks into two parts:
$$
\xalignat 2
&\qquad\mu_s=\const\cdot\nabla_sf,&&A(v)=\frac{v^2}{\const}.\hskip -3em
\tag5.19
\endxalignat
$$
As a result of \thetag{5.19} we conclude: {\it Newtonian dynamical
system with force field \thetag{5.1} is metrizable if and only if
parameter $A$ in \thetag{5.1} coincides with $|\bold v|^2$ up to a
constant multiple}. Any other choice of this parameter, say $A=
|\bold v|^3$, and the choice $\mu_s=\nabla_sf$ give an example of
non-metrizable dynamical system.
\enddocument
\end

ediss-07.tex
\input amstex
\documentstyle{amsppt}
%
% Author:   Ruslan Abdulovich Sharipov
% Title:    Dynamical systems admitting the normal shift
% Comment:  Thesis for the degree of Doctor of Sciences in Russia
% Address:  Rabochaya street 5, Ufa, 450003, Russia
% E-mail:   R_Sharipov@ic.bashedu.ru
%           ruslan-sharipov@usa.net
% Web-page: http://www.geocities.com/CapeCanaveral/Lab/5341
%
\nopagenumbers
\catcode`@=11
\redefine\logo@{\relax}
\catcode`\@=\active
\def\startpage#1{\pageno=#1}
\pagewidth{12.8cm}
\pageheight{20cm}
\def\const{\operatorname{const}}
\def\compos{\,\raise 1pt\hbox{$\sssize\circ$} \,}
\accentedsymbol\tx{\tilde x}
\leftheadtext{\uppercase\expandafter{\romannumeral 7}.
REDUCTION OF NORMALITY EQUATIONS \dots}
\Monograph
\topmatter
\title\chapter{7}
Reduction of normality equations\\for the dimension $n\geqslant 3$.
\endtitle
\endtopmatter
\loadbold
\startpage{155}
\document
\head
\S\,1. Scalar ansatz\footnotemark.
\endhead
\rightheadtext{\S\,1. Scalar ansatz.}
\footnotetext{{\bf Ansatz}, word of german origin, in Russian mathematical
texts means {\bf substitution}, i\.~e\. something to be substituted into
the equation with the hope that it's a solution.}
     Further analysis of Newtonian dynamical systems admitting the normal
shift on Riemannian manifold $M$ requires the analysis of corresponding
differential equations for the force field $\bold F$, which are called
{\bf normality equations}. They are subdivided into two groups of
equations. First group is formed by {\bf weak normality} equations (see
\thetag{6.16} and \thetag{6.17} in Chapter
\uppercase\expandafter{\romannumeral 5}). They are written as
$$
\align
&\sum^n_{i=1}\left(v^{-1}\,F_i+\shave{\sum^n_{j=1}}
\tilde\nabla_i\left(N^j\,F_j\right)\right)P^i_k=0,\hskip -2em
\tag1.1\\
\vspace{2ex}
&\aligned
&\sum^n_{i=1}\sum^n_{j=1}\left(\nabla_iF_j+\nabla_jF_i-2\,v^{-2}
\,F_i\,F_j\right)N^j\,P^i_k\,+\\
\vspace{0.4ex}
&+\sum^n_{i=1}\sum^n_{j=1}\left(\frac{F^j\,\tilde\nabla_jF_i}{v}
-\sum^n_{r=1}\frac{N^r\,N^j\,\tilde\nabla_jF_r}{v}\,F_i\right)P^i_k=0.
\endaligned\hskip -3em
\tag1.2
\endalign
$$
Second group is formed by {\bf additional normality conditions}
(see \thetag{6.16} and \thetag{6.17} in Chapter
\uppercase\expandafter{\romannumeral 5}). They are written only in
multidimensional case $n\geqslant 3$:
$$
\align
&\aligned
&\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_i\,\tilde\nabla_mF_j}{v}
-\nabla_iF_j\right)=\\
\vspace{0.4ex}
&\quad\,=\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_j\,\tilde\nabla_mF_i}{v}
-\nabla_jF_i\right),
\endaligned\hskip -3em
\tag1.3\\
\vspace{1ex}
&\sum^n_{i=1}\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF^i\,
P^\varepsilon_i=\sum^n_{i=1}\sum^n_{j=1}\sum^n_{m=1}
\frac{P^j_m\,\tilde\nabla_jF^i\,P^m_i}{n-1}\,P^\varepsilon_\sigma.
\hskip -3em\tag1.4
\endalign
$$
Normality equations \thetag{1.1}, \thetag{1.2}, \thetag{1.3},
\thetag{1.4} assume that we choose some local map $(U,x^1,\ldots,
x^n)$ on $M$ and associated map $(\pi^{-1}(U),x^1,\ldots,x^n,v^1,
\ldots,v^n)$ on tangent bundle $TM$ (see \thetag{4.2} in Chapter
\uppercase\expandafter{\romannumeral 2}). This choice determines
auxiliary parameters in these equations. These are: components of
unit vector $\bold N$ directed along the vector of velocity;
components of orthogonal projector $\bold P$ to the hyperplane
perpendicular to the velocity vector; scalar field $v=|\bold v|$
of modulus of velocity vector.\par
    We begin the study of normality equations \thetag{1.1}, \thetag{1.2},
\thetag{1.3}, \thetag{1.4} with the first of them, i\.~e\. first we
consider the equations \thetag{1.1}. Denote by $A$ the following sum:
$$
A=\sum^n_{i=1}N^i\,F_i.\hskip -2em
\tag1.5
$$
The quantity $A$ is an extended scalar field on $M$. Scalar field $A$
is interpreted as a projection of $\bold F$ to the direction of velocity
vector $\bold v$. Taking into account formula \thetag{1.5}, let's rewrite
the equation \thetag{1.1} as follows:
$$
\sum^n_{i=1}\left(v^{-1}\,F_i+\tilde\nabla_iA\right)P^i_k=0.
\hskip -2em
\tag1.6
$$

Now let's open brackets in the equations \thetag{1.6} and recall
explicit expression for the components of projector $\bold P$ (see
formula \thetag{5.4} in Chapter
\uppercase\expandafter{\romannumeral 5}). This yields
$$
\gathered
\sum^n_{i=1}\frac{F_i\,P^i_k}{|\bold v|}+\sum^n_{i=1}
\tilde\nabla_iA\,P^i_k=\sum^n_{i=1}\frac{F_i\,\delta^i_k}
{|\bold v|}-\sum^n_{i=1}\frac{F_i\,N^i\,N_k}{|\bold v|}\,+\\
\vspace{1ex}
+\,\sum^n_{i=1}\tilde\nabla_iA\,P^i_k=\frac{F_k}{|\bold v|}
-\sum^n_{i=1}\frac{A\,N_k}{|\bold v|}+\sum^n_{i=1}
\tilde\nabla_iA\,P^i_k=0.
\endgathered\hskip -2em
\tag1.7
$$
The last equality in \thetag{1.7} allows us to express components of
force vector $\bold F$ through scalar field $A$. Indeed, we easily
find that
$$
F_k=A\,N_k-|\bold v|\sum^n_{i=1}\tilde\nabla_iA\,P^i_k.\hskip -2em
\tag1.8
$$
Due to what was said above this formula is called {\bf scalar ansatz}.
\par
    Scalar ansatz turns to identity first normality equation \thetag{1.1}.
Let's apply it to second equation, i\.~e\. let's substitute \thetag{1.8}
into \thetag{1.2}. For $\nabla_iF_j$ we get
$$
\nabla_iF_j=\nabla_iA\,N_j-|\bold v|\sum^n_{s=1}
\nabla_i\tilde\nabla_sA\,P^s_j.\hskip -2em
\tag1.9
$$
In deriving \thetag{1.9} we used formulas \thetag{5.5}, \thetag{5.6},
\thetag{5.7}, \thetag{5.8}, and \thetag{5.9} from Chapter
\uppercase\expandafter{\romannumeral 5}. The same formulas are used in
evaluating $\tilde\nabla_iF_j$:
$$
\gathered
\tilde\nabla_jF_i=\tilde\nabla_jA\,N_i+\frac{A\,P_{ij}}{|\bold v|}
+\sum^n_{s=1}\nabla_sA\,N^s\,P_{ij}\,+\\
\vspace{1ex}
+\,\sum^n_{s=1}\tilde\nabla_sA\,P^s_j\,N_i-\sum^n_{s=1}
\tilde\nabla_sA\,P^s_i\,N_j-|\bold v|\sum^n_{s=1}\tilde\nabla_j
\tilde\nabla_sA\,P^s_i.
\endgathered\hskip -2em
\tag1.10
$$
For the sake of brevity we denoted by $P_{ij}$ covariant components
of projector $\bold P$:
$$
P_{ij}=\sum^n_{s=1}P^s_i\,g_{sj}=g_{ij}-N_i\,N_j.\hskip -3em
\tag1.11
$$
Now let's substitute \thetag{1.8} into the following expressions
that have entries of $\bold F$:
$$
\gather
\sum^n_{i=1}F_i\,P^i_k=-|\bold v|\sum^n_{s=1}\tilde\nabla_sA\,P^s_k,
\hskip -3em
\tag1.12\\
\vspace{1ex}
-\sum^n_{i=1}\sum^n_{i=1}\frac{2\,F_i\,F_j}{|\bold v|^2}\,N^j
\,P^i_k=\sum^n_{s=1}\frac{2\,A\,\tilde\nabla_sA\,P^s_k}{|\bold v|}.
\hskip -3em
\tag1.13
\endgather
$$
Further we use formula \thetag{1.9} in order to evaluate the
following expressions:
$$
\gather
\sum^n_{i=1}\sum^n_{j=1}\nabla_iF_j\,N^j\,P^i_k=\sum^n_{s=1}
\nabla_sA\,P^s_k,\hskip -3em
\tag1.14\\
\vspace{1ex}
\sum^n_{i=1}\sum^n_{j=1}\nabla_jF_i\,N^j\,P^i_k=-|\bold v|
\sum^n_{r=1}\sum^n_{s=1}\nabla_r\tilde\nabla_sA\,N^r\,P^s_k.
\hskip -3em
\tag1.15
\endgather
$$
And finally, we use formula \thetag{1.10} in order to accomplish the
following calculations:
$$
\gather
\aligned
\sum^n_{i=1}\frac{\tilde\nabla_jF_i}{|\bold v|}\,&P^i_k=
\frac{A\,P_{kj}}{|\bold v|^2}+\sum^n_{s=1}\frac{\tilde
\nabla_sA\,N^s\,P_{kj}}{|\bold v|}\,-\\
\vspace{1ex}
&-\,\sum^n_{s=1}\frac{\tilde\nabla_sA\,P^s_k\,N_j}{|\bold v|}
-\sum^n_{s=1}\tilde\nabla_j\tilde\nabla_sA\,P^s_k,
\endaligned\hskip -3em
\tag1.16\\
\vspace{2ex}
-\sum^n_{j=1}\sum^n_{r=1}\frac{N^r\,N^j\,\tilde\nabla_jF_r}
{|\bold v|}=-\sum^n_{s=1}\frac{\tilde\nabla_sA\,N^s}{|\bold v|}.
\hskip -3em
\tag1.17
\endgather
$$
Before substituting \thetag{1.16} into the equation \thetag{1.2}
we should contract it with $F^j$, while the equality \thetag{1.17}
should be multiplied by \thetag{1.12}. Upon completing these
evaluations we get
$$
\gather
\quad\aligned
&\sum^n_{i=1}\sum^n_{j=1}\frac{F^j\,\tilde\nabla_jF_i}{|\bold v|}
\,P^i_k=|\bold v|\sum^n_{q=1}\sum^n_{s=1}\sum^n_{r=1}P^{qr}\,
\tilde\nabla_qA\,\tilde\nabla_r\tilde\nabla_sA\,P^s_k\,-\\
\vspace{1ex}
&-\sum^n_{s=1}\left(\,\shave{\sum^n_{r=1}}N^r\left(\tilde\nabla_rA\,
\tilde\nabla_sA+A\,\tilde\nabla_r\tilde\nabla_sA\right)
+\frac{2\,A\,\tilde\nabla_sA}{|\bold v|}\right)P^s_k.
\endaligned\hskip -3em
\tag1.18\\
\vspace{2ex}
-\sum^n_{j=1}\sum^n_{r=1}\frac{N^r\,N^j\,\tilde\nabla_jF_r}
{|\bold v|}=\sum^n_{s=1}\sum^n_{r=1}N^r\,\nabla_rA\,\nabla_sA\,P^s_k.
\hskip -3em
\tag1.19
\endgather
$$
For the sake of brevity in \thetag{1.18} by $P^{qr}$ we denoted purely
contravariant components of the field of orthogonal projectors $\bold P$:
$$
P^{qr}=\sum^n_{s=1}g^{qs}\,P^r_s=g^{qr}-N^q\,N^r.\hskip -3em
\tag1.20
$$
Now let's add the expressions \thetag{1.13}, \thetag{1.14}, \thetag{1.15},
\thetag{1.18}, \thetag{1.19} and let's take into account the equation
\thetag{1.2}. Then \thetag{1.2} appears to be written as the following
equation for the scalar field $A$ introduced by formula \thetag{1.5}:
$$
\gathered
\sum^n_{s=1}\left(\nabla_sA+|\bold v|\sum^n_{q=1}\sum^n_{r=1}
P^{qr}\,\tilde\nabla_qA\,\tilde\nabla_r\tilde\nabla_sA\right.-\\
\vspace{1ex}
-\left.\shave{\sum^n_{r=1}}N^r\,A\,\tilde\nabla_r\tilde\nabla_sA
-|\bold v|\shave{\sum^n_{r=1}}N^r\,\nabla_r\tilde\nabla_sA\right)
P^s_k=0.
\endgathered\hskip -3em
\tag1.21
$$
This equation \thetag{1.21} is a result of applying scalar ansatz
\thetag{1.8} to the second normality equation \thetag{1.2}.\par
    Next step consists in applying scalar ansatz to the third normality
equation \thetag{1.3}. First of all, let's evaluate the following
expressions by virtue of \thetag{1.8}:
$$
\gather
-\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma
\nabla_jF_i=
|\bold v|\sum^n_{s=1}P^r_\sigma\,\nabla_r\tilde\nabla_sA
\,P^s_\varepsilon,\\
\vphantom{\vrule height 25pt}
\sum^n_{i=1}\sum^n_{m=1}P^i_\varepsilon\,N^m\,
\frac{\tilde\nabla_mF_i}{|\bold v|}=
-\sum^n_{s=1}\frac{\tilde\nabla_sA\,P^s_\varepsilon}{|\bold v|}
-\sum^n_{s=1}\sum^n_{q=1}N^q\,\tilde\nabla_q\tilde\nabla_sA
\,P^s_\varepsilon.
\endgather
$$
Taking into account \thetag{1.12}, from these two equalities we derive
$$
\gather
\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_i\,\tilde\nabla_mF_j}{v}
-\nabla_iF_j\right)=\\
\vspace{1ex}
=\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\left(\tilde
\nabla_rA\,\tilde\nabla_sA+|\bold v|\,\nabla_r\tilde\nabla_sA
\vphantom{\shave{\sum^n_{q=1}}}+|\bold v|\shave{\sum^n_{q=1}}
\tilde\nabla_rA\,N^q\,\tilde\nabla_q\tilde\nabla_sA\right).
\endgather
$$
Now normality equations \thetag{1.3} are written in form of
the following equations for scalar field $A$ that determines
force field according to \thetag{1.8}:
$$
\aligned
\sum^n_{s=1}&\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\left(
\nabla_r\tilde\nabla_sA+\shave{\sum^n_{q=1}}\tilde\nabla_rA\,N^q\,
\tilde\nabla_q\tilde\nabla_sA\right)=\\
\vspace{1ex}
&=\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\left(
\nabla_s\tilde\nabla_rA+\shave{\sum^n_{q=1}}\tilde\nabla_sA\,N^q\,
\tilde\nabla_q\tilde\nabla_rA\right).
\endaligned\hskip -3em
\tag1.22
$$\par
    The last step consists in applying scalar ansatz to the fourth normality
equation  \thetag{1.4}.  With  this  goal,  let's  evaluate   the 
following
expression by virtue of \thetag{1.8}:
$$
\aligned
\sum^n_{i=1}&\sum^n_{j=1}P^j_\sigma\,\tilde\nabla_jF^i\,P^\varepsilon_i=
\sum^n_{q=1}N^q\,\tilde\nabla_qA\,P^\varepsilon_\sigma\,+\\
\vspace{1ex}
&+\,\frac{A}{|\bold v|}\,P^\varepsilon_\sigma-|\bold v|\sum^n_{r=1}
\sum^n_{s=1}P^r_\sigma\,\tilde\nabla_r\tilde\nabla_sA\,P^{s\varepsilon}.
\endaligned\hskip -3em
\tag1.23
$$
Here $P^{s\varepsilon}$ are defined by \thetag{1.20}. First two
terms in right hand side of \thetag{1.23} differ from 
$P^\varepsilon_\sigma$ only by scalar multiple. Therefore
they are unessential for resulting equation:
$$
\sum^n_{r=1}\sum^n_{s=1}P^r_\sigma\,\tilde\nabla_r\tilde\nabla_sA\,
P^{s\varepsilon}=\lambda\,P^\varepsilon_\sigma.\hskip -3em
\tag1.24
$$
Here $\lambda$ is a scalar parameter that could be defined by the following 
formula:
$$
\lambda=\sum^n_{r=1}\sum^n_{s=1}\frac{P^{rs}\,\tilde\nabla_r
\tilde\nabla_sA}{n-1}.\hskip -3em
\tag1.25
$$
The equations \thetag{1.24} with scalar parameter $\lambda$ from 
\thetag{1.25} are obtained from \thetag{1.4} by applying scalar 
ansatz \thetag{1.8}. They play key role in further analysis of 
whole set of normality equations.
\head
\S\,2. Fiber spherical coordinates.
\endhead
     The equations \thetag{1.21}, \thetag{1.22}, \thetag{1.24} form
complete set of normality equations for scalar field $A$. They are
equivalent to \thetag{1.1}, \thetag{1.2}, \thetag{1.3}, \thetag{1.4} 
due to scalar ansatz \thetag{1.8}. Let's study last one of them.
In order to do this we write explicit expressions for covariant
derivatives $\tilde\nabla_r$ and $\tilde\nabla_s$, applying formula
\thetag{7.4} from Chapter \uppercase\expandafter{\romannumeral 3}:
$$
\sum^n_{r=1}\sum^n_{s=1}P^r_\sigma\,P^s_\varepsilon
\frac{\partial^2 A}{\partial v^r\,\partial v^s}=\lambda
\,P_{\sigma\varepsilon}.\hskip -2em
\tag2.1
$$
The equations \thetag{2.1} contain partial derivatives with respect
to variables $v^1,\,\ldots,\,v^n$ in the fiber of tangent bundle
over each point $p\in M$, but they do not contain partial derivatives
respective to variables $x^1,\,\ldots,\,x^n$ in base manifold $M$.
Fiber $\pi^{-1}(p)$ is $n$-dimensional linear space with metric
$\bold g$. Components of this metric $g_{ij}$ in coordinates $v^1,\,
\ldots,\,v^n$ are constants within fixed fiber. Therefore equations
\thetag{2.1} are the equations in linear space with Euclidean metric.
\par
    Let's fix some point $p\in M$. Condition $|\bold v|=\const$ defines
a sphere of the radius $v=|\bold v|$ in tangent space $T_p(M)=\pi^{-1}
(p)$ being the fiber of tangent bundle. Tangent space $T_p(M)$ is 
disjoint union of spheres of various radii and the point $\bold v=0$,
which can be treated as a sphere of zero radius. Let $u^1,\,\ldots,\,
u^{n-1}$ be local coordinates on the sphere of unit radius $S^{n-1}$,
and suppose that vector-function $\bold v=\bold N(u^1,\ldots,u^{n-1})$
realize its embedding into the tangent space $T_p(M)$. Let's add one
more coordinate $u^n=v=|\bold v|$ to $u^1,\,\ldots,\,u^{n-1}$. Then
the relationships
$$
\align
&v^1=u^n\,N^1(u^1,\ldots,u^{n-1}),\hskip -2em\\
&.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
.\ .\ .\ .\ .\ .\ .\ \hskip -2em\tag2.2\\
&v^n=u^n\,N^n(u^1,\ldots,u^{n-1}).\hskip -2em
\endalign
$$
define transition function binding Cartesian coordinates $v^1,\,\ldots,
\,v^n$ and spherical coordinates $u^1,\,\ldots,\,u^n$ in $T_p(M)$. Such
spherical coordinates $u^1,\,\ldots,\,u^n$ are called {\bf fiber
spherical} coordinates, since they are defined separately in the fiber
of tangent bundle over fixed point $p\in M$. In fact, components $N^1,\,
\ldots,\,N^n$ of the vector $\bold N$ in \thetag{2.2} depend on
coordinates $x^1,\,\ldots,\,x^n$ of the point $p$ as well. However, in
the analysis of the equations \thetag{2.1} this circumstance plays no
role, since $p$ is fixed.\par
     Let's make the change of variables \thetag{2.2} in the equations
\thetag{2.1}. As it's used to do in such situation, we denote by $S$
and $T$ transition matrices corresponding to the change of variables
\thetag{2.2}. For their components we have
$$
\xalignat 2
&\quad S^k_i=\frac{\partial v^k}{\partial u^i},
&&T^k_i=\frac{\partial u^k}{\partial v^i}.
\hskip -2em
\tag2.3
\endxalignat
$$
Partial derivatives $\partial/\partial v^i$ in \thetag{2.1} coincide
with covariant derivatives with respect to metric connection for
metric $\bold g$, which is flat Euclidean metric in $T_p(M)$ with
constant components $g_{ij}$ in Cartesian coordinates $v^1,\,\ldots,
\,v^n$. In curvilinear coordinates components of metric $\bold g$ are
not constant, therefore corresponding connection components
$\vartheta^k_{ij}$ are nonzero, and the equations \thetag{2.1}
take the form
$$
\sum^n_{r=1}\sum^n_{s=1}\tilde P^r_\sigma\,\tilde P^s_\varepsilon
\left(\frac{\partial^2 A}{\partial u^r\,\partial u^s}
-\shave{\sum^n_{k=1}}\vartheta^k_{rs}\,\frac{\partial A}{\partial u^k}
\right)=\lambda\,\tilde P_{\sigma\varepsilon}.\hskip -2em
\tag2.4
$$
Components of projection operator $\bold P$ are transformed to curvilinear
coordinates $u^1,\,\ldots,\,u^n$ by standard formula
$$
\tilde P^k_q=\sum^n_{i=1}\sum^n_{j=1}T^k_i\,S^j_q\,P^i_j,
$$
where transition matrix components are defined by \thetag{2.3}.\par
    Spherical coordinates \thetag{2.2} have some features. Vector
$\bold N=\bold N(u^1,\ldots,u^{n-1})$, that defines transition to
spherical coordinates $u^1,\,\ldots,\,u^n$ in \thetag{2.2}, coincides
with normal vector of spheres $u^n=\const$. Due to
$$
S^i_n=\frac{\partial v^i}{\partial u^n}=N^i
$$
vector $\bold N$ is $n$-th base vector in moving frame of curvilinear
coordinates $u^1,\,\ldots,\,u^n$.
Other vectors of this frame are perpendicular to $\bold N$ and tangent
to spheres the $u^n=\const$. This determines the structure of components
of metric tensor in spherical coordinates $u^1,\,\ldots,\,u^n$ defined
by \thetag{2.2}:
$$
\tilde g_{ij}=\Vmatrix
\tilde g_{11}     & \hdots & \tilde g_{1\,n-1}   & 0      \\
\vspace{2ex}
\vdots     & \ddots & \vdots       & \vdots \\
\vspace{2ex}
\tilde g_{n-1\,1} & \hdots & \tilde g_{n-1\,n-1} & 0      \\
\vspace{2ex}
0      & \hdots & 0      & 1
\endVmatrix.\hskip -2em
\tag2.5
$$
Quantities $u^1,\,\ldots,\,u^{n-1}$ can be considered as local
coordinates on sphere $u^n=\const$. In this approach
we find that upper diagonal block in matrix \thetag{2.5} is
made of components of metric tensor for induced metric on
sphere $|\bold v|=u^n=\const$.\par
    Bloc structure of matrix \thetag{2.5} determines similar
structure of components of metric connection $\vartheta^k_{rs}$.
For $i$, $j$, $k$ ranging $1\leqslant i,j,k\leqslant n-1$ the
quantities $\vartheta^k_{rs}$ coincide with components of metric
connection for induced metric on spheres $u^n=\const$.
Other components $\vartheta^k_{rs}$ are calculated explicitly.
Part of them are zero:
$$
\vartheta^i_{nn}=\vartheta^n_{in}=\vartheta^n_{ni}=0
\text{ \ for all \ }i=1,\ldots,n.\hskip -2em
\tag2.6
$$
Moreover, for $1\leqslant i,j\leqslant n-1$ and for $k=n$ we have
$$
\xalignat 2
&\vartheta^n_{ij}=-\frac{\tilde g_{ij}}{|\bold v|},
&&\vartheta^i_{nj}=\vartheta^i_{jn}=\frac{\delta^i_j}{|\bold v|}.
\hskip -2em
\tag2.7
\endxalignat
$$
Operator $\bold P$ is a projector to hyperplane perpendicular to
the vector $\bold N$ that coincides with $n$-th coordinate vector
in moving frame of curvilinear coordinates $u^1,\,\ldots,\,u^n$.
Therefore, taking into account the structure of metric tensor
\thetag{2.5}, we get
$$
\tilde N^i=\tilde N_i=
\cases 0&\text{for \ }i<n,\\
1&\text{for \ }i=n.\endcases
$$
In order to evaluate components of projector field $\bold P$ we
use formula \thetag{5.4} from Chapter
\uppercase\expandafter{\romannumeral 7}: $\tilde P^i_j=\delta^i_j
-\tilde N^i\,\tilde N_j$. This yields
$$
\tilde P^i_j=\Vmatrix
1 & \hdots & 0 & 0 \\
\vspace{0.5ex}
\vdots & \ddots & \vdots & \vdots \\
\vspace{0.5ex}
0 & \hdots & 1 & 0 \\
\vspace{0.5ex}
0 & \hdots & 0 & 0
\endVmatrix.\hskip -2em
\tag2.8
$$
The relationships \thetag{2.6}, \thetag{2.7}, and \thetag{2.8}
allow to make the equations \thetag{2.4} more specific. For
$1\leqslant\varepsilon,\sigma\leqslant n-1$ we have
$$
\frac{\partial^2 A}{\partial u^\varepsilon\,\partial u^\sigma}
-\sum^{n-1}_{k=1}\vartheta^k_{\varepsilon\sigma}\,\frac{\partial A}
{\partial u^k}+\frac{\tilde g_{\varepsilon\sigma}}{|\bold v|}
\,\frac{\partial A}{\partial u^n}=\lambda\,\tilde g_{\varepsilon
\sigma}.\hskip -2em
\tag2.9
$$
In case of $\varepsilon=n$ and in case of $\sigma=n$ the equations
\thetag{2.4} turn to identity. They add nothing to \thetag{2.9}.
As for $\lambda$, it is an arbitrary scalar parameter. Upon
substituting
$$
\lambda=\mu+\frac{1}{|\bold v|}\,\frac{\partial A}{\partial u^n}=
\mu+\frac{1}{|\bold v|}\sum^n_{i=1}N^i\,\tilde\nabla_iA\hskip -3em
\tag2.10
$$
into \thetag{2.9} we replace $\lambda$ by other scalar parameter
$\mu$. This simplifies our equations
$$
\frac{\partial^2 A}{\partial u^\varepsilon\,\partial u^\sigma}
-\sum^{n-1}_{k=1}\vartheta^k_{\varepsilon\sigma}\,\frac{\partial A}
{\partial u^k}=\mu\,\tilde g_{\varepsilon\sigma}.\hskip -3em
\tag2.11
$$\par
    Due to $1\leqslant\varepsilon,\sigma\leqslant n-1$ the equations
\thetag{2.11} describe only the dependence of $A$ upon coordinate
$u^1,\,\ldots,\,u^{n-1}$ on spheres $|\bold v|=\const$. The quantities
$\vartheta^k_{\varepsilon\sigma}$ in \thetag{2.11} are components of
metric connection on such spheres. Therefore we can write \thetag{2.11}
in terms of covariant derivatives on spheres 
$$
\bar\nabla_\varepsilon\bar\nabla_\sigma A=\mu
\,\tilde g_{\varepsilon\sigma}.\hskip -3em
\tag2.12
$$
In multidimensional case $n\geqslant 3$, which we are studying now,
system of equations \thetag{2.12} remains overdetermined yet. This
allows us to consider it's differential consequences. To do
it we use well-known identity (see \cite{77}, or identity \thetag{7.12}
in Chapter \uppercase\expandafter{\romannumeral 3}):
$$
[\bar\nabla_i,\,\bar\nabla_j]\bar\nabla_qA=-\sum^{n-1}_{k=1}
\tilde R^k_{qij}\,\bar\nabla_kA.\hskip -3em
\tag2.13
$$
Here $\tilde R^k_{qij}$ are the components of curvature tensor for
induced metric on sphere $|\bold v|=v=\const$. This sphere is a space
of constant sectional curvature (see \cite{13}). Curvature tensor of
the space of constant sectional curvature is determined by one scalar
quantity. In case of sphere this quantity is its radius:
$$
\tilde R^k_{qij}=\frac{\delta^k_i\,\tilde g_{qj}-\delta^k_j\,\tilde g_{qi}}
{|\bold v|^2}.
$$
This specifies right hand side in the identity \thetag{2.13}. Now this
identity is written as
$$
[\bar\nabla_i,\,\bar\nabla_j]\bar\nabla_qA=\frac{\tilde g_{qi}\,
\bar\nabla_jA-\tilde g_{qj}\,\bar\nabla_iA}{|\bold v|^2}.\hskip -3em
\tag2.14
$$
Left hand side of \thetag{2.14} is calculated by virtue of \thetag{2.12}:
$$
[\bar\nabla_i,\,\bar\nabla_j]\bar\nabla_qA=\bar\nabla_i
\bar\nabla_j\bar\nabla_qA-\bar\nabla_j\bar\nabla_i\bar\nabla_qA
=\tilde g_{jq}\,\bar\nabla_i\mu-\tilde g_{iq}\,\bar\nabla_j\mu.
$$
Upon substituting this expression into \thetag{2.14} we obtain the
required differential consequence of the equations \thetag{2.12}:
$$
\tilde g_{qj}\,\bar\nabla_i\mu-\tilde g_{qi}\,\bar\nabla_j\mu=
-\frac{\tilde g_{qj}\,\bar\nabla_iA-\tilde g_{qi}\,\bar\nabla_jA}
{|\bold v|^2}.\hskip -3em
\tag2.15
$$
These equations \thetag{2.15} can be made more simple, if we multiply
them by $g^{qj}$ and contract with respect to indices $q$ and $j$.
This yields
$$
(n-2)\,\bar\nabla_i\mu=-\frac{(n-2)}{|\bold v|^2}\bar\nabla_iA.
$$
In multidimensional case $n\geqslant 3$, which we are studying now,
the multiple $(n-2)$ in the above equation is nonzero. Hence
$$
\bar\nabla_iA=-|\bold v|^2\,\bar\nabla_i\mu.\hskip -3em
\tag2.16
$$
We are to note that differential consequences \thetag{2.16}, obtained
just above, are of lower order than initial equations \thetag{2.12}
by themselves.\par
     Next step consists in substituting \thetag{2.16} back to the
equations \thetag{2.12}. As a result of this substitution we obtain
the following equations:
$$
\bar\nabla_i\bar\nabla_j\mu=-\frac{\tilde g_{ij}}
{|\bold v|^2}\,\mu.\hskip -3em
\tag2.17
$$
Similar to \thetag{2.12}, the equations \thetag{2.17} can be treated
as the equations on spheres $|\bold v|=\const$ in tangents space
$T_p(M)$ at the fixed point $p$. They describe the dependence ob
$\mu$ upon spherical coordinates $u^1,\,\ldots,\,u^{n-1}$, but do not
determine the dependence of $\mu$ on radial variable $u^n=|\bold v|$.
\proclaim{Theorem 2.1} Each smooth function $\mu$ on sphere of the
radius $u^n=|\bold v|=\const$ satisfying differential equations 
\thetag{2.17} has the form 
$$
\mu=\sum^n_{i=1}m_i\,N^i,\hskip -3em
\tag2.18
$$
where $m_1,\,\ldots,\,m_n$ are some constants, and $N^1,\,\ldots,\,N^n$
are components of vector $\bold N$ that determines  embedding  of 
sphere
into Euclidean space according to \thetag{2.2}.
\endproclaim
\demo{Proof} The equations \thetag{2.17} have tensorially covariant
form, i\.~e\. they are written in covariant derivatives. They do not
change their appearance under the changes of local coordinates on
spheres $|\bold v|=\const$. Right hand side of \thetag{2.18} also
is not sensible to the changes of local coordinates on spheres.
Therefore we can choose some special (most convenient) coordinates,
assuming that metric $\bold g$ in the fiber $T_p(M)$ over fixed point $p$
in Cartesian coordinates $v^1,\,\ldots,\,v^n$ is brought
to canonical form, so that its matrix is determined by Kronecker delta
symbol: 
$$
g_{ij}=\delta _{ij}=\cases
1&\text{for \ }i=j,\\
0&\text{for \ }i\neq j.
\endcases\hskip -3em
\tag2.19
$$
This can be done at the expense of proper choice of local
coordinates $x^1,\,\ldots,\,x^n$ on the base manifold $M$.
Now spherical coordinates $u^1,\,\ldots,\,u^{n-1}$ can be chosen
so that in metric \thetag{2.19} change of variables \thetag{2.2}
specifies by the following relationships:
$$
\matrix
\format\l&\,\l&\l\\
v^1&=v\,\sin(u_{n-1})\,\sin(u_{n-2})\cdot\ldots\cdot\sin(u_3)
\,&\sin(u_2)\,\sin(u_1),\\
\vspace{1ex}
v^2&=v\,\sin(u_{n-1})\,\sin(u_{n-2})\cdot\ldots\cdot\sin(u_3)
\,&\sin(u_2)\,\cos(u_1),\\
\vspace{1ex}
v^3&=v\,\sin(u_{n-1})\,\sin(u_{n-2})\cdot\ldots\cdot\sin(u_3)
\,&\cos(u_2),\\
\vspace{1ex}
\hdotsfor 2\\
\vspace{2ex}
v^{n-1}&=v\,\sin(u_{n-1})\,\cos(u_{n-2}),\\
\vspace{1ex}
v^n&=v\,\cos(u_{n-1}).
\endmatrix\hskip -2em
\tag2.20
$$
Here $v=|\bold v|=u^n$ is the radius of sphere. The following
inequalities determine ranges for spherical coordinates
$u^1,\,\ldots,\,u^n$:
$$
\align
0\leqslant\,\, &u^1<2\pi,\\
0\leqslant\,\, &u^2<\pi,\\
.\ .\ .\ &.\ .\ .\ .\ .\ .\ \\
0\leqslant\,\, &u^{n-1}<\pi,\\
0\leqslant\,\, &u^n<2<+\infty.
\endalign
$$
In spherical coordinates $u^1,\,\ldots,\,u^n$ given by \thetag{2.20}
the matrix of metric tensor is diagonal. Its diagonal elements are
positive: $g_{ii}>0$. They define the quantities $H_i=\sqrt{g_{ii}}$,
which are called {\bf scaling factors} or {\bf Lame coefficients}:
$$
\matrix
\format\l&\,\l&\l\\
H_1&=v\,\sin(u_{n-1})\,\sin(u_{n-2})\cdot\ldots&\cdot\sin(u_3)
\sin(u_2),\\
\vspace{1ex}
H_2&=v\,\sin(u_{n-1})\,\sin(u_{n-2})\cdot\ldots&\cdot\sin(u_3),\\
\vspace{1ex}
\hdotsfor 2\\
\vspace{2ex}
H_{n-2}&=v\,\sin(u_{n-1}),\\
\vspace{1ex}
H_{n-1}&=v,\\
\vspace{1ex}
H_n&=1.
\endmatrix\hskip -3em
\tag2.21
$$
Components of metric connection $\vartheta^k_{ij}$ are defined
according to well-known formula
$$
\vartheta^k_{ij}=\frac{1}{H_k}\,\frac{\partial H_k}{\partial u^j}
\,\delta_{ik}+\frac{1}{H_k}\,\frac{\partial H_k}{\partial u^i}\,
\delta_{jk}-\frac{H_i}{(H_k)^2}\,\frac{\partial H_i}{\partial u^k}\,
\delta_{ij},
$$
which is derived from formula \thetag{1.6} in Chapter 
\uppercase\expandafter{\romannumeral 6}. Most connection components
$\vartheta^k_{ij}$ appear to be zero. This follows from explicit
formula for them:
$$
\vartheta^k_{ij}=\cases
\cot(u_j)\,\delta^k_i&\text{for \ }i<j\leqslant n-1,\\
\cot(u_i)\,\delta^k_j&\text{for \ }j<i\leqslant n-1.
\endcases
$$
The case of coinciding indices $i=j$ subdivides into two cases.
For the first of them, when $k\leqslant i$, we have $\vartheta^k_{ii}=0$.
Tn second subcase, when $i<k\leqslant n-1$, we get
$$
\vartheta^k_{ii}=-\frac{\cos(u^k)}{\sin(u^k)}\,\prod^{k}_{s=i+1}
\sin^2(u^s).
$$
Using these formulas, we can write the equations \thetag{2.17} explicitly.
We obtain the equations arranged in three groups. First group corresponds
to the case $i<j\leqslant n-1$:
$$
\mu_{ij}=\frac{\cos(u^j)}{\sin(u^j)}\,\mu_i.
\tag2.22
$$
The equations of second group correspond to the case $i=j\leqslant n-2$:
$$
\mu_{ii}+\sum^{n-1}_{k=i+1}\frac{\cos(u^k)}{\sin(u^k)}\,
\left(\,\shave{\prod^{k}_{s=i+1}}\sin^2(u^s)\right)\,\mu_k=
-\left(\,\shave{\prod^{n-1}_{s=i+1}}\sin^2(u^s)\right)\,\mu.
\hskip -3em
\tag2.23
$$
Here by $\mu_i$ and $\mu_{ij}$ de denoted partial derivatives of $\mu$
of the first and second order with respect to variables $u^i$ and $u^j$:
$$
\xalignat 2
&\mu_i=\frac{\partial\mu}{\partial u^i},
&&\mu_{ij}=\frac{\partial\mu}{\partial u^i\,\partial u^j}.
\endxalignat
$$
And finally, third group of equations corresponds to $i=j=n-1$. It
consists of exactly one equation,which is very simple:
$$
\mu_{ii}=-\mu\text{, \ where \ }i=n-1.\hskip -3em
\tag2.24
$$\par
    In further proof of theorem we use the method of reducing dimension
$n\to n-1$. Let's examine the equation \thetag{2.24}. It describes the
dependence of $\mu$ on the last coordinate on sphere $u^{n-1}$. One can
easily write the general solution of this equation:
$$
\mu=\tilde\mu\,\sin(u^{n-1})+m_n\,\cos(u^{n-1}).\hskip -3em
\tag2.25
$$
Parameters $\tilde\mu$ and $m_n$ can depend on $u^1,\,\ldots,\,u^{n-2}$,
but they do not depend on $u^{n-1}$. Let's substitute \thetag{2.25} into
the equations \thetag{2.22}, taking $j=n-1$ and $i<n-1$:
$$
\gather
\frac{\partial\tilde\mu}{\partial u^i}\,\cos(u^{n-1})-
\frac{\partial\tilde m_n}{\partial u^i}\,\sin(u^{n-1})=\\
\vspace{1ex}
=\frac{\cos(u^{n-1})}{\sin(u^{n-1})}\left(
\frac{\partial\tilde\mu}{\partial u^i}\,\sin(u^{n-1})+
\frac{\partial\tilde m_n}{\partial u^i}\,\cos(u^{n-1})\right).
\endgather
$$
Opening brackets and collecting similar terms in both sides of
this equality, we get
$$
\frac{\partial\tilde m_n}{\partial u^i}=0\text{ \ for all \ }
i<n-1.
$$
This means that parameter $m_n$ in \thetag{2.25} does not depend on
$u^1,\,\ldots,\,u^{n-1}$. It is a constant on sphere
$|\bold v|=v=\const$.\par
    Keeping in mind that $m_n$ is constant, let's substitute the expression
\thetag{2.25} for $\mu$ into the equations \thetag{2.23}. For $i=n-2$ we
obtain
$$
\gather
\tilde\mu_{ii}\,\sin(u^{n-1})+\frac{\cos(u^{n-1})}{\sin(u^{n-1})}\,
\sin^2(u^{n-1})\cdot\left(\tilde\mu\,\cos(u^{n-1})\,\right.-\\
\vspace{1ex}
-\,\left.m_n\,\sin(u^{n-1})\right)=-\sin^2(u^{n-1})\cdot\left(
\tilde\mu\,\sin(u^{n-1})+m_n\,\cos(u^{n-1})\right).
\endgather
$$
Opening brackets and collecting similar terms, we get the equation
$$
\tilde\mu_{ii}=-\tilde\mu\text{, \ where \ }i=n-2.\hskip -3em
\tag2.26
$$
If the dimension $n$ is high enough ($n>3$), we have more equations,
these are the equations \thetag{2.22} with $i<j\leqslant n-2$ and
the equations \thetag{2.23} with $i\leqslant n-3$. Substituting
\thetag{2.25} into the equations \thetag{2.22} for $i<j\leqslant n-2$,
we get
$$
\tilde\mu_{ij}=\frac{\cos(u^j)}{\sin(u^j)}\,\tilde\mu_i.
\tag2.27
$$
Substituting \thetag{2.25} into \thetag{2.23}, we get more bulk
expressions:
$$
\gather
\left(\tilde\mu_{ii}+\shave{\sum^{n-2}_{k=i+1}}\frac{\cos(u^k)}{\sin(u^k)}
\left(\,\shave{\prod^{k}_{s=i+1}}\sin^2(u^s)\right)\,\tilde\mu_k
\right)\cdot\sin(u^{n-1})\,+\\
\vspace{1ex}
+\,\frac{\cos(u^{n-1})}{\sin(u^{n-1})}\left(\,\shave{\prod^{n-1}_{s=i+1}}
\sin^2(u^s)\right)\cdot\left(\tilde\mu\,\cos(u^{n-1})-m_n\,\sin(u^{n-1})
\right)=\\
\vspace{1ex}
=-\left(\,\shave{\prod^{n-1}_{s=i+1}}\sin^2(u^s)\right)
\cdot\left(\tilde\mu\,\sin(u^{n-1})-m_n\,\cos(u^{n-1})\right).
\endgather
$$
Let's open brackets and collect similar terms in the above expression.
All entries of parameter $m_n$ then cancel out. Further let's take into
account the identity $\cos^2(u^{n-1})+\sin^2(u^{n-1})=1$ and divide
both sides of resulting equality by $\sin(u^{n-1})\neq 0$. As a result
we get the equation for $\tilde\mu$, where $i\leqslant n-3$:
$$
\gathered
\tilde\mu_{ii}+\shave{\sum^{n-2}_{k=i+1}}\frac{\cos(u^k)}{\sin(u^k)}
\left(\,\shave{\prod^{k}_{s=i+1}}\sin^2(u^s)\right)\,\tilde\mu_k=\\
\vspace{1ex}
=-\left(\,\shave{\prod^{n-2}_{s=i+1}}\sin^2(u^s)\right)\tilde\mu.
\endgathered\hskip -3em
\tag2.28
$$\par
     The equations \thetag{2.26}, \thetag{2.27}, and \thetag{2.28}
are analogous to \thetag{2.24}, \thetag{2.22}, and \thetag{2.23} with
the only difference that $n$ is replaced by $n-1$ and $\mu$ is replaced
by $\tilde\mu$. This means that the equations \thetag{2.24}, \thetag{2.22},
and \thetag{2.23} admit the reduction $n\to n-1$ by substituting
\thetag{2.25}. Let's consider the series of reductions:
$$
n\to n-1\to n-2\to\ \ldots\ \to 3\to 2.\hskip -3em
\tag2.29
$$
All reductions in the series \thetag{2.29} are similar to each other,
except for the last one. In making last reduction we break the condition
$n>3$. Therefore the equations \thetag{2.27} and \thetag{2.28} do not
arise, and we get the only equation \thetag{2.26} with $i=1$:
$$
\tilde\mu_{11}=-\tilde\mu.\hskip -3em
\tag2.30
$$
General solution of the equation \thetag{2.30} is determined by two
constants $m_1$ and $m_2$: 
$$
\tilde\mu=m_1\,\sin(u^1)+m_2\,\cos(u^1).\hskip -3em
\tag2.31
$$
Now let's step back along the series of reductions \thetag{2.29},
using \thetag{2.25}. As a result we get initial function $\mu$
$$
\mu=m_1\prod^{n-1}_{s=1}\sin(u^s)+
\sum^n_{k=2}m_k\left(\,\shave{\prod^{n-1}_{s=k}}\sin(u^s)\right)
\cos(u^{k-1}).\hskip -3em
\tag2.32
$$
Here $m_1,\,\ldots,\,m_n$ are constants. Comparing \thetag{2.2} and
\thetag{2.20}, we get
$$
\align
N^1&=\prod^{n-1}_{s=1}\sin(u^s),\\
\vspace{1ex}
N^k&=\left(\,\shave{\prod^{n-1}_{s=k}}\sin(u^s)\right)
\cos(u^{k-1})\text{ \ for \ } 1<k<n,\\
\vspace{1ex}
N^n&=\cos(u^{n-1}).
\endalign
$$
And finally, comparing the above expression for components of vector
$\bold N$ with \thetag{2.32} we conclude, that formula \thetag{2.14}
is exactly the same as formula \thetag{2.18} in the statement of
theorem~2.1. Proof is over.\qed\enddemo
     Having proved the theorem~2.1 and thereby having studied the
equation \thetag{2.17}, let's return to the equations \thetag{2.16}.
Let's write them as
$$
\bar\nabla_i(A+|\bold v|^2\,\mu)=0.\hskip -3em
\tag2.33
$$
Covariant derivative $\bar\nabla_i$, when applied to scalar field
$A+|\bold v|^2\,\mu$, coincides with partial derivative
$\partial/\partial u^i$ with respect to local coordinate $u^i$
on sphere $|\bold v|=\const$. Therefore from the equation
\thetag{2.33} we derive
$$
\pagebreak
A+|\bold v|^2\,\mu=\const.\hskip -3em
\tag2.34
$$
Note that right hand side of \thetag{2.34} and parameters $m_1,\,\ldots,
\,m_n$ in \thetag{2.18} are constant only within each particular sphere
$|\bold v|=\const$ in $T_p(M)$. Therefore, denoting $b_i=-|\bold v|\,m_i$,
we can formulate the following theorem. 
\proclaim{Theorem 2.2} Scalar field $A$ from extended algebra of tensor
fields  on  Riemannian  manifold  $\bold  M$,  which  is   smooth 
everywhere on
$TM$, except for the points of zero section $\bold v=0$,
satisfies the equations \thetag{1.24} if and only if it is
given by formula
$$
A=a+\sum^n_{i=1}b_i\,v^i,\hskip -3em
\tag2.35
$$
where extended scalar field $a$ and the components $b_i$ of extended
covectorial field $\bold b$ depend only on modulus of velocity vector
$v=|\bold v|$ in fibers of tangent bundle $TM$.
\endproclaim
    Necessity of the conclusion of theorem~2.2 is already proved by
discussion preceding the statement of the theorem. In order to prove
sufficiency one should only substitute \thetag{2.35} into the equations
\thetag{1.24}. Let's use that $a=a(x^1,\ldots,x^n,v)$ and $b_i=b_i(x^1,
\ldots,x^n,v)$, where $v=|\bold v|$, and denote by $a'$ and $b'_i$
the derivatives
$$
\xalignat 2
&a'=\frac{\partial a}{\partial v},
&&b'_i=\frac{\partial b_i}{\partial v}.
\endxalignat
$$
Then for the derivatives  $\tilde\nabla_sA$ and
$\tilde\nabla_r\tilde\nabla_sA$ in the equations
\thetag{1.24} we get
$$
\gather
\tilde\nabla_sA=\left(a'+\shave{\sum^n_{i=1}}b'_i\,v^i\right)
N_s+b_s,\hskip -3em
\tag2.36\\
\vspace{2ex}
\aligned
\tilde\nabla_r&\tilde\nabla_sA=\left(a''+\shave{\sum^n_{i=1}}b''_i
\,v^i\right)N_r\,N_s\,+\\
\vspace{1ex}
&+\,b'_s\,N_r+b'_r\,N_s+\left(\frac{a'}{v}+\shave{\sum^n_{i=1}}b'_i
\,N^i\right)P_{rs}.
\endaligned\hskip -3em
\tag2.37
\endgather
$$
Further let's contract \thetag{2.37} with $P^r_\sigma$, and
with $P^{s\varepsilon}$ respective to indices $r$ and $s$:
$$
\sum^n_{r=1}\sum^n_{s=1}P^r_\sigma\,\tilde\nabla_r\tilde\nabla_sA\,
P^{s\varepsilon}=
\left(\frac{a'}{v}+\shave{\sum^n_{i=1}}b'_i\,N^i\right)
P^\varepsilon_\sigma.
$$
The relationship just obtained shows that the quantity $A$ from
\thetag{2.35} satisfies the equations \thetag{1.24}, where scalar
parameter $\lambda$ is given by 
$$
\pagebreak
\lambda=\frac{a'}{v}+\shave{\sum^n_{i=1}}b'_i\,N^i.
$$
We conclude the study of the equations \thetag{1.24} by calculating
$\mu$ on a base of formula \thetag{2.10} binding $\lambda$ and $\mu$:
$$
\mu=\lambda-\frac{1}{|\bold v|}\sum^n_{i=1}N^i\,\tilde\nabla_iA=
-\sum^n_{i=1}\frac{b_i}{|\bold v|}\,N^i.
$$
This expression is in agreement with formula \thetag{2.18}, if we
remember that $b_i=|\bold v|\,m_i$.
\head
\S\,3. Specification of scalar ansatz.
\endhead
     Normality equations \thetag{1.24}, as it follows from theorem~2.2,
determine the dependence of $A$ upon velocity vector $\bold v$. Dependence
of $A$ upon spatial coordinates $x^1,\,\ldots,\,x^n$ should be specified
by substituting \thetag{2.35} into other two normality equations
\thetag{1.21} and \thetag{1.22}. Let's make calculations needed for it:
$$
\gather
\nabla_sA=\nabla_sa+\sum^n_{i=1}\nabla_sb_i\,v^i,
\hskip -2em
\tag3.1\\
\vspace{1ex}
\nabla_r\tilde\nabla_sA=\left(\nabla_ra'+\shave{\sum^n_{i=1}}
\nabla_rb'_i\,v^i\right)N_s+\nabla_rb_s.\hskip -2em
\tag3.2
\endgather
$$
From \thetag{2.36} and \thetag{2.37} we derive the following relationships:
$$
\gather
\sum^n_{r=1}P^r_\sigma\,\tilde\nabla_rA=\sum^n_{r=1}P^r_\sigma\,b_r,
\hskip -2em
\tag3.3\\
\vspace{1ex}
\sum^n_{s=1}\sum^n_{q=1}P^s_\varepsilon\,N^q\,\tilde\nabla_q
\,\tilde\nabla_sA=\sum^n_{s=1}P^s_\varepsilon\,b'_s.\hskip -2em
\tag3.4
\endgather
$$
Let's combine \thetag{3.3} and \thetag{3.4}. As a result we get the
relationship
$$
\sum^n_{s=1}\sum^n_{r=1}\sum^n_{q=1}P^r_\sigma\,P^s_\varepsilon
\,\tilde\nabla_rA\,N^q\,\tilde\nabla_q\tilde\nabla_sA=\sum^n_{s=1}
\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\,b_r\,b'_s.
$$
Then let's multiply both sides of \thetag{3.2} by  $P^r_\sigma\,
P^s_\varepsilon$ and let's contract it with respect to $r$ and
$s$. This yields one more relationship:
$$
\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\,
\nabla_r\tilde\nabla_sA=\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,
P^s_\varepsilon\,\nabla_rb_s.
$$
Now let's add two above relationships. \pagebreak This determines
the result of substituting \thetag{2.35} into the left hand side of
the normality equation \thetag{1.22}:
$$
\aligned
\sum^n_{s=1}&\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\left(
\nabla_r\tilde\nabla_sA+\shave{\sum^n_{q=1}}\tilde\nabla_rA\,
N^q\,\tilde\nabla_q\tilde\nabla_sA\right)=\\
\vspace{1ex}
&=\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\,\nabla_rb_s
+\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\,b_r\,b'_s.
\endaligned\hskip -2em
\tag3.5
$$
In a similar way we can calculate right hand side of normality equation
\thetag{1.22}:
$$
\aligned
\sum^n_{s=1}&\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\left(
\nabla_s\tilde\nabla_rA+\shave{\sum^n_{q=1}}\tilde\nabla_sA\,
N^q\,\tilde\nabla_q\tilde\nabla_rA\right)=\\
\vspace{1ex}
&=\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\,\nabla_sb_r
+\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\,b_s\,b'_r.
\endaligned\hskip -2em
\tag3.6
$$
From \thetag{3.5} and \thetag{3.6} we conclude that the equation
\thetag{1.22} reduces to 
$$
\sum^n_{s=1}\sum^n_{r=1}P^r_\sigma\,P^s_\varepsilon\,\bigl(\nabla_rb_s
+b_r\,b'_s-\nabla_sb_r-b_s\,b'_r\bigr)=0.\hskip -2em
\tag3.7
$$
For the further analysis of the obtained equations \thetag{3.7} we
should use the peculiarity of extended covector field $\bold b$.
Components $b_1,\,\ldots,\,b_n$ of this field depend only on modulus
of velocity vector, they do not depend on its direction. In
calculating derivatives $\nabla_rb_s$ and $\nabla_sb_r$ in \thetag{3.7}
we shall use the following theorem.
\proclaim{Theorem 3.1} Let $(U,x^1,\ldots,x^n)$ be some local map on
Riemannian manifold $M$, and let $(\pi^{-1}(U),x^1,\ldots,x^n,v^1,
\ldots,v^n)$ be associated local map on tangent bundle $TM$. If
components of extended tensor field $\bold X$ in $U$ depend only on
modulus of velocity vector $\bold v$, then components of spatial
gradient in $U$ are calculated by formula 
$$
\aligned
\nabla_mX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=&\frac{\partial
X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial x^m}+
\sum^r_{k=1}\sum^n_{a_k=1}\!\Gamma^{i_k}_{m\,a_k}\,X^{i_1\ldots\,
a_k\ldots\,i_r}_{j_1\ldots\,\ldots\,\ldots\,j_s}\,-\\
\vspace{1ex}
&-\sum^s_{k=1}\sum^n_{b_k=1}\!\Gamma^{b_k}_{m\,j_k}
X^{i_1\ldots\,\ldots\,\ldots\,i_r}_{j_1\ldots\,b_k\ldots\,j_s}.
\endaligned\hskip -2em
\tag3.8
$$
\endproclaim
\demo{Proof} Formula \thetag{3.8} is obtained as a reduction of the
formula \thetag{7.3} from Chapter \uppercase\expandafter{\romannumeral 3}.
For the components of tensor field $\bold X$, which is mentioned in the
theorem, their natural arguments are $x^1,\,\ldots,\,x^n$ and $v$, where 
$$
v=|\bold v|=\sqrt{\shave{\sum^n_{i=1}\sum^n_{j=1}}g_{ij}(x^1,\ldots,
x^n)\,v^i\,v^j\,}.\hskip -2em
\tag3.9
$$
However, in formula \thetag{7.3} from Chapter
\uppercase\expandafter{\romannumeral 3} all derivatives are assumed to
be calculated with respect to $x^1,\,\ldots,\,x^n,\,v^1,\,\ldots,\,v^n$.
Therefore we should substitute 
$$
\gather
\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial v^b}
\text{ \ for \ }\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}
{\partial v}\cdot\frac{\partial v}{\partial v^b},\hskip -3em
\tag3.10\\
\vspace{1ex}
\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial x^m}
\text{ \ for \ }\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}
{\partial x^m}+\frac{\partial X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}
{\partial v}\cdot\frac{\partial\,v}{\partial x^m}.\hskip -3em
\tag3.11
\endgather
$$
Partial derivatives $\partial\,v/\partial v^b$ and $\partial\,v/\partial
x^m$ are calculated by virtue of \thetag{3.9}. Upon calculating these
derivatives and upon making substitutions \thetag{3.10} and \thetag{3.11}
into the formula \thetag{7.3} from Chapter
\uppercase\expandafter{\romannumeral 3} we find two terms there:
$$
\sum^n_{a=1}\sum^n_{b=1}\frac{X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}
{\partial v}\,\frac{1}{2}\,\frac{\partial g_{ab}}{\partial x^m}\,
\frac{v^a\,v^b}{v}\text{ \ \ and \ \ }
-\sum^n_{a=1}\sum^n_{b=1}v^a\,\Gamma^b_{ma}\,\frac{\partial
X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{\partial v}\,N_b.
$$
If we take into account explicit formula for components of metric
connection $\Gamma^b_{ma}$ (see \thetag{1.2} in Chapter
\uppercase\expandafter{\romannumeral 5}), then we easily see
that these two terms cancel out each other. As a result formula
\thetag{7.3} from Chapter \uppercase\expandafter{\romannumeral 3}
reduces to \thetag{3.8}. \qed\enddemo
    {\bf Remark}. Calculations similar to that, which we used in proving
theorem~3.1, were applied to scalar field $H$ (see formula \thetag{3.16}
in Chapter  \uppercase\expandafter{\romannumeral 6}).
\proclaim{Corollary} Components $\omega_{rs}=\nabla_rb_s+b_r\,b'_s
-\nabla_sb_r-b_s\,b'_r$ of skew-symmetric extended tensor field
$\boldsymbol\omega$ in \thetag{3.7} depend only on modulus of
velocity vector $\bold v$ within fibers of tangent bundle $TM$.
\endproclaim
     This fact is an immediate consequence of formula \thetag{3.8}.
It allows us to make further simplifications in the equations \thetag{3.7}.
Let $\bold c$ and $\bold d$ be two arbitrary vectors from tangent space 
$T_p(M)$. In multidimensional case $n\geqslant 3$ we can keep $|\bold v|$
unchanged and rotate velocity vector $\bold v$ so that will be perpendicular
to $\bold c$ and $\bold d$ simultaneously. Then the following equalities
will be fulfilled
$$
\xalignat 2
&\sum^n_{\sigma=1}P^r_\sigma\,c^{\sigma}=c^r,
&&\sum^n_{\varepsilon=1}P^s_\varepsilon\,d^{\varepsilon}=d^s.
\endxalignat
$$
Therefore we can bring the equations \thetag{3.7} to the form 
$$
\sum^n_{s=1}\sum^n_{r=1}c^r\,d^s\,\bigl(\nabla_rb_s
+b_r\,b'_s-\nabla_sb_r-b_s\,b'_r\bigr)=0.\hskip -3em
\tag3.12
$$
Since $\bold c$ and $\bold d$ are two arbitrary vectors, the
equations \thetag{3.12} are equivalent to
$$
\pagebreak
\nabla_rb_s+b_r\,b'_s=\nabla_sb_r+b_s\,b'_r.\hskip -3em
\tag3.13
$$\par
    Next step consists in reducing the equations \thetag{1.21}.
For this purpose let's substitute \thetag{2.35} into \thetag{1.21}.
Due to \thetag{3.1} then we have
$$
\sum^n_{s=1}\nabla_sA\,P^s_k=\sum^n_{s=1}\nabla_sa\,P^s_k+
\sum^n_{s=1}\sum^n_{r=1}\nabla_sb_r\,v^r\,P^s_k.\hskip -3em
\tag3.14
$$
Further from \thetag{2.36} and \thetag{2.37} we derive the following
two relationships:
$$
\gather
\sum^n_{r =1}P^{qr}\,\tilde\nabla_qA=\sum^n_{s=1}b_s\,P^{sr},
\hskip -3em
\tag3.15\\
\vspace{1ex}
|\bold v|\sum^n_{r=1}\sum^n_{s=1}P^{qr}\,\tilde
\nabla_r\tilde\nabla_sA\,P^s_k=\left(a'+\shave{\sum^n_{r=1}}b'_r\,v^r
\right)P^q_k.\hskip -3em
\tag3.16
\endgather
$$
In \thetag{3.15} we have free index $r$, and in \thetag{3.16} we have
free index is $q$. Let's multiply the equalities \thetag{3.15} and
\thetag{3.16}, then contract resulting product with respect to indices
$r$ and $q$ upon multiplying it by $g_{rq}$. This yields
$$
|\bold v|\sum^n_{q=1}\sum^n_{r=1}\sum^n_{s=1}P^{qr}\,\tilde\nabla_qA
\,\tilde\nabla_r\tilde\nabla_sA\,P^s_k=
\left(a'+\shave{\sum^n_{r=1}}b'_r\,v^r\right)\sum^n_{s=1}b_s\,P^s_k.
$$
Let's multiply \thetag{2.37} by $N^r\,A\,P^s_q$ and contract it
with respect to indices $r$ and $s$:
$$
-\sum^n_{r=1}\sum^n_{s=1}N^r\,A\,\tilde\nabla_r\tilde\nabla_sA\,P^s_k=
-\left(a+\shave{\sum^n_{r=1}}b_r\,v^r\right)\sum^n_{s=1}b'_s\,P^s_k.
$$
And finally, let's contract \thetag{3.2} with $N^r$ and with $P^s_k$
by indices $r$ and $s$. This yields
$$
-|\bold v|\sum^n_{r=1}\sum^n_{s=1}N^r\,\nabla_r\tilde\nabla_sA
\,P^s_k=-\sum^n_{r=1}\sum^n_{s=1}v^r\,\nabla_rb_s\,P^s_k.
$$
Now, in order to get the result of substituting \thetag{2.35} into the
equations \thetag{1.21} we should add \thetag{3.14} and three above
equalities:
$$
\sum^n_{s=1}\left(\nabla_sa+b_s\,a'-a\,b'_s+\shave{\sum^n_{r=1}}
v^r\bigl(\nabla_sb_r+b_s\,b'_r-\nabla_rb_s-b_r\,b'_s\bigr)\right)
P^s_k=0.
$$
Taking into account \thetag{3.13} we find that the expression enclosed into
smaller round brackets is zero. Therefore \thetag{1.21} is written as
$$
\pagebreak
\sum^n_{s=1}\left(\nabla_sa+b_s\,a'-a\,b'_s\right)P^s_k=0.\hskip -3em
\tag3.17
$$
The equations \thetag{3.17} are similar to \thetag{3.7}, contraction
with projector components $P^s_k$ in them can be omitted. Then the
equations \thetag{3.17} are written as
$$
\nabla_sa+b_s\,a'=a\,b'_s.\hskip -3em
\tag3.18
$$
Reasons used to get \thetag{3.18} are similar to those used to bring
\thetag{3.13} to the form \thetag{3.7}.
\proclaim{Theorem 3.2} Force field $\bold F$ defined by scalar ansatz
\thetag{1.8} corresponds to Newtonian dynamical system admitting the
normal shift on Riemannian manifold $M$ if and only if scalar field $A$
in \thetag{1.8} is defined by formula \thetag{2.35}, where extended
fields $a$ and $\bold b$ depending only on modulus of velocity vector
within fibers of tangent bundle $TM$ are such that they satisfy the
equation \thetag{3.13} and \thetag{3.18}.
\endproclaim
    Thus, the scalar ansatz \thetag{1.8} and its specification
\thetag{2.35} reduce the system of normality equations \thetag{1.1},
\thetag{1.2}, \thetag{1.3}, and \thetag{1.4} to the equations
\thetag{3.13} and \thetag{3.18}. These two equations are called
{\bf reduced normality equations}.
\head
\S\,4. Analysis of the reduced equations.
\endhead
    For the purpose of further study of the equations \thetag{3.13} and
\thetag{3.18} let's express covariant derivatives in them through
corresponding partial derivatives. In order to do it we use formula
\thetag{3.8} and take into account symmetry of connection components:
$$
\gather
\left(\frac{\partial}{\partial x^s}+b_s\,\frac{\partial}{\partial v}
\right)a=\left(a\,\frac{\partial}{\partial v}\right)b_s,\hskip -2em
\tag4.1\\
\vspace{1ex}
\left(\frac{\partial}{\partial x^s}+b_s\,\frac{\partial}{\partial v}
\right)b_r=\left(\frac{\partial}{\partial x^r}+b_r\,\frac{\partial}
{\partial v}\right)b_s.\hskip -2em
\tag4.2
\endgather
$$
The equations \thetag{4.2} form closed system of equations with respect
to $b_1,\,\ldots,\,b_n$. Let's study them separately. Consider the
differential operators used in them:
$$
\bold L_i=\frac{\partial}{\partial x^i}+b_i\,\frac{\partial}{\partial v}
\text{, \ where \ }i=1,\,\ldots,\,n.\hskip -2em
\tag4.3
$$
Now by means of direct calculations we verify that the system of
equations \thetag{4.2} is exactly the commutativity condition for
the operators \thetag{4.3}:
$$
[\bold L_s,\,\bold L_r]=0\text{ \ for \ }r,s=1,\,\ldots,\,n.
\hskip -2em
\tag4.4
$$
Let $\Bbb R^+=(0,\,+\infty)$ be positive half-axis on the axis of real
numbers $\Bbb R$. Linear operators \thetag{4.3} have natural interpretation
as vector field on Cartesian product $M\times\Bbb R^+$. Let's complete
vector fields $\bold L_1,\,\ldots,\,\bold L_n$ with one more vector
field $\bold L_{n+1}$, which possibly doesn't commutate with $\bold L_1,
\,\ldots,\,\bold L_n$, but complete them up to a moving frame on the
manifold $M\times\Bbb R^+$. Each field $\bold L_i$ generates its own
one-parametric local group of local diffeomorphisms (see \cite{12}).
Denote by $y^i$ parameter of such group:
$$
\varphi_i(y^i):\,M\times\Bbb R^+\to M\times\Bbb R^+.\hskip -2em
\tag4.5
$$
Let's fix some point $p_0\in M\times\Bbb R^+$  and  consider  the 
composition
of such diffeomorphisms applied to the fixed point $p_0$:
$$
p(y^1,\ldots,y^n,w)=\varphi_1(y^1)\compos\,\ldots\,\compos\varphi_n(y^n)
\compos\varphi_{n+1}(w)(p_0).\hskip -2em
\tag4.6
$$
In  left  hand  side  of  \thetag{4.6}  we  have  the  point  $p$ 
parameterized
by the set of real numbers $y^1,\,\ldots,\,y^n,\,w=y^{n+1}$. This is
equivalent to defining local coordinates on the manifold $M\times\Bbb
R^+$ in some neighborhood of the point $p_0$. Commutativity condition
\thetag{4.4} implies commutativity of first $n$ maps \thetag{4.5} in
composition \thetag{4.6}. When applied to vector fields $\bold L_1,\,
\ldots,\,\bold L_n$, this yields
$$
\align
&\bold L_1=\frac{\partial}{\partial y^1}=\sum^n_{i=1}\frac{\partial x^i}
{\partial y^1}\,\frac{\partial}{\partial x^i}+\frac{\partial v}
{\partial y^1}\,\frac{\partial}{\partial v},\hskip -2em\\
&.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\
.\ .\ .\ .\ \hskip -2em
\tag4.7\\
&\bold L_n=\frac{\partial}{\partial y^n}=\sum^n_{i=1}\frac{\partial x^i}
{\partial y^n}\,\frac{\partial}{\partial x^i}+\frac{\partial v}
{\partial y^n}\,\frac{\partial}{\partial v}.\hskip -2em
\endalign
$$
Let's compare formulas \thetag{4.7} and \thetag{4.3} for the fields
$\bold L_1,\,\ldots,\,\bold L_n$. Then 
$$
\frac{\partial x^i}{\partial y^k}=\delta^i_k=
\cases 1&\text{for \ }i=k,\\
0&\text{for \ }i\neq k.
\endcases\hskip -2em
\tag4.8
$$
From the same comparison of \thetag{4.7} and \thetag{4.3} for the
functions $b_k=b_k(x^1,\ldots,x^n,v)$ expressed in variables $y^1,
\,\ldots,\,y^n,\,w$ we get
$$
b_k=\frac{\partial v}{\partial y^k}.\hskip -2em
\tag4.9
$$
The relationships \thetag{4.8} shows that transition from new
coordinates $y^1,\,\ldots,\,y^n,\,w$ back to coordinates
$x^1,\,\ldots,\,x^n,\,v$ is defied by the only one function
$V(y^1,\ldots,y^n,w)$:
$$
\cases
x^1=y^1,\,\ldots,\,x^n=y^n,\\
\vspace{1ex}
v=V(y^1,\ldots,y^n,w).
\endcases\hskip -3em
\tag4.10
$$
Direct transition from $x^1,\,\ldots,\,x^n,\,v$ to new variables
$y^1,\,\ldots,\,y^n,\,w$ is also defined by the only one function
$W(x^1,\ldots,x^n,v)$:
$$
\cases
y^1=x^1,\,\ldots,\,y^n=x^n,\\
\vspace{1ex}
w=W(x^1,\ldots,x^n,v).
\endcases\hskip -3em
\tag4.11
$$
Functions $V(y^1,\ldots,y^n,w)$ and $W(x^1,\ldots,x^n,v)$ are bound by
obvious relationships expressing the fact that transitions \thetag{4.10}
and \thetag{4.11} are inverse to each other:
$$
\aligned
&V(x^1,\ldots,x^n,W(x^1,\ldots,x^n,v))=v,
\vspace{1ex}
&W(x^1,\ldots,x^n,V(x^1,\ldots,x^n,w))=w.
\endaligned\hskip -3em
\tag4.12
$$
For the sake of brevity let's introduce the following quite natural
notations for partial derivatives of the function $V(x^1,\ldots,x^n,w)$:
$$
\aligned
&V_i(x^1,\ldots,x^n,w)=\frac{\partial V(x^1,\ldots,x^n,w)}
{\partial x^i},\\
\vspace{1ex}
&V_w(x^1,\ldots,x^n,w)=\frac{\partial V(x^1,\ldots,x^n,w)}
{\partial w}.
\endaligned\hskip -3em
\tag4.13
$$
Similar notations are used for second order partial derivatives
of $V(x^1,\ldots,x^n,w)$:
$$
\aligned
&V_{ij}(x^1,\ldots,x^n,w)=\frac{\partial^2 V(x^1,\ldots,x^n,w)}
{\partial x^i\,\partial x^j},\\
\vspace{1.5ex}
&V_{iw}(x^1,\ldots,x^n,w)=\frac{\partial^2 V(x^1,\ldots,x^n,w)}
{\partial x^i\,\partial w}.
\endaligned\hskip -3em
\tag4.14
$$
Now we can rewrite formula \thetag{4.9} for $b_k$ in initial variables
$x^1,\,\ldots,\,x^n,\,v$:
$$
b_k=V_k(x^1,\ldots,x^n,W(x^1,\ldots,x^n,v)).\hskip -3em
\tag4.15
$$
\proclaim{Theorem 4.1}\!Functions $b_1,\,\ldots,\,b_n$ satisfy
nonlinear system of differential equations \thetag{4.2} if and
only if they are defined according to the formula \thetag{4.15}
by some function $V(x^1,\ldots,x^n,w)$ such that $\partial V/
\partial w\neq 0$.
\endproclaim
\demo{Proof} Each solution of the system of equations \thetag{4.2}
is given by \thetag{4.15}. This was proved above. Conversely,
suppose that we have a function $V(x^1,\ldots,x^n,w)$ with nonzero
derivative $\partial V/\partial w$. From $\partial V/\partial w
\neq 0$ we conclude that there exists (at least locally) some function
$W(x^1,\ldots,x^n,v)$ related to $V(x^1,\ldots,x^n,w)$ according to
the formulas \thetag{4.12} (see theorems on implicit functions in \cite{86}
or in \cite{87}). From formulas \thetag{4.12} we derive the relationships 
$$
\align
&\frac{\partial W}{\partial v}=\frac{1}{V_w(x^1,\ldots,x^n,W(x^1,\ldots,
x^n,v))},\hskip -3em
\tag4.16\\
\vspace{1.5ex}
&\frac{\partial W}{\partial x^s}=-\frac{V_s(x^1,\ldots,x^n,W(x^1,\ldots,
x^n,v))}{V_w(x^1,\ldots,x^n,W(x^1,\ldots,x^n,v))},\hskip -3em
\tag4.17
\endalign
$$
written in terms of above notations \thetag{4.13}. Let's substitute 
functions $V$ and $W$ into \thetag{4.15} and get functions $b_1,\,\ldots,
\,b_n$. Then by direct substitution we verify that functions \thetag{4.15}
satisfy the equations \thetag{4.2}. Indeed, we have
$$
\gather
\aligned
\frac{\partial b_r}{\partial x^s}=V_{rs}&(x^1,\ldots,x^n,W(x^1,
\ldots,x^n,v))\,+\\
&+\,V_{rw}(x^1,\ldots,x^n,W(x^1,\ldots,x^n,v))\,\frac{\partial W}
{\partial x^s},
\endaligned\hskip -3em
\tag4.18\\
\displaybreak
\aligned
b_s\,\frac{\partial b_r}{\partial v}=V_s&(x^1,\ldots,x^n,W(x^1,\ldots,
x^n,v))\,\times\\
&\times\,V_{rw}(x^1,\ldots,x^n,W(x^1,\ldots,x^n,v))\,\frac{\partial W}
{\partial v}.
\endaligned\hskip -3em
\tag4.19
\endgather
$$
Let's add the equalities \thetag{4.18} and \thetag{4.19}. Then take
into account formulas \thetag{4.16} and \thetag{4.17} for
partial derivatives of $W$. As a result we get
$$
\left(\frac{\partial}{\partial x^s}+b_s\,\frac{\partial}{\partial v}
\right)b_r=V_{rs}(x^1,\ldots,x^n,W(x^1,\ldots,x^n,v)).\hskip -3em
\tag4.20
$$
Due to \thetag{4.14} indices $r$ and $s$ in right hand side of
\thetag{4.20} can be transposed, i\.~e\. $V_{rs}=V_{sr}$. Hence
the equations \thetag{4.2} for the functions \thetag{4.15} are
fulfilled.\qed\enddemo
    Having constructed general solution for the system of equations
\thetag{4.2}, we now turn to the equations \thetag{4.1}. Let's make
change of variables \thetag{4.10} in them. In left hand side of these
equations we have the same differential operator $\bold L_s$ as in
\thetag{4.2}. In variables $y^1,\,\ldots,\,y^n,\,w$ it has the form
$\bold L_s=\partial/\partial y^s$ (see relationships \thetag{4.7}).
Then let's transform to variables $y^1,\,\ldots,\,y^n,\,w$ the operator
from right hand side of \thetag{4.1}:
$$
\frac{\partial}{\partial v}=\sum^n_{i=1}\frac{\partial y^i}
{\partial v}\,\frac{\partial}{\partial y^i}+
\frac{\partial w}{\partial v}\,\frac{\partial}{\partial w}=
\frac{\partial W}{\partial v}\,\frac{\partial}{\partial w}.
$$
For further transformation of this expression we apply formula
\thetag{4.16}:
$$
\frac{\partial}{\partial v}=\frac{1}{V_w(y^1,\ldots,y^n,w)}
\,\frac{\partial}{\partial w}.
$$
Now, in terms of notations introduced in \thetag{4.13} and \thetag{4.14},
the equations \thetag{4.1} transformed to new variables $y^1,\,\ldots,\,
y^n,\,w$ are written as follows:
$$
\frac{\partial a}{\partial y^s}=\frac{V_{sw}}{V_w}\,a.\hskip -3em
\tag4.21
$$
These equations \thetag{4.21} are easily solved, if we write them in
the form
$$
\frac{\partial}{\partial y^s}\left(\frac{a}{V_w}\right)=0.\hskip -3em
\tag4.22
$$
General solution of \thetag{4.22} contain an arbitrary function of one
variable $h(w)$:
$$
a=h(w)\ V_w(y^1,\ldots,y^n,w).\hskip -3em
\tag4.23
$$
Upon returning to the variables $x^1,\,\ldots,\,x^n,\,v$ in formula
\thetag{4.23} we have
$$
\aligned
a=h(&W(x^1,\ldots,x^n,v))\,\times\\
\vspace{1ex}
&\times\,V_w(x^1,\ldots,x^n,W(x^1,\ldots,x^n,v)).
\endaligned\hskip -3em
\pagebreak
\tag4.24
$$\par
    For the purpose of additional verification let's substitute
the expression \thetag{4.24} for $a$ and the expressions \thetag{4.15}
for $b_s$ into the equation \thetag{4.1}:
$$
\align
&\frac{\partial a}{\partial x^s}=\bigl(h'(W)\,V_w+h(W)\,V_{ww}\bigr)
\,\frac{\partial W}{\partial x^s}+h(W)\,V_{sw},\\
\vspace{1.5ex}
&b_s\,\frac{\partial a}{\partial v}=V_s\,\bigl(h'(W)\,V_w+h(W)\,V_{ww}
\bigr)\,\frac{\partial W}{\partial v}.
\endalign
$$
Then we add these equalities and take into account formulas \thetag{4.16}
and \thetag{4.17} for the derivatives $\partial W/\partial x^s$ and
$\partial W/\partial v$:
$$
\left(\frac{\partial}{\partial x^s}+b_s\,\frac{\partial}{\partial v}
\right)a=h(W)\,V_{sw}.\hskip -3em
\tag4.25
$$
Similar operations in right hand side of the equations \thetag{4.1}
yield
$$
\left(a\,\frac{\partial}{\partial v}\right)b_s=h(W)\,V_w\,
V_{sw}\frac{\partial W}{\partial v}.\hskip -3em
\tag4.26
$$
Comparing \thetag{4.25} with \thetag{4.26} and taking into account
formula \thetag{4.16} for $\partial W/\partial v$, we complete the
proof of the following theorem. 
\proclaim{Theorem 4.2} Functions $b_1,\,\ldots,\,b_n$ and $a$
satisfy the system of differential equations \thetag{4.1} and \thetag{4.2}
if and only if they are given by formulas \thetag{4.15} and
\thetag{4.24}.
\endproclaim
\head
\S\,5. The general formula for the force field.
\endhead
    Analysis of reduced normality equations, accomplished in \S\,4,
allows us to construct general solution for these equations. Now
we are able to write {\bf formula} defining force field for
{\bf arbitrary} Newtonian dynamical system {\bf admitting the normal
shift} on Riemannian manifold of the dimension $n\geqslant 3$. Let's
substitute \thetag{2.35} into the formula \thetag{1.8} for components
of force field $\bold F$. Thereby let's take into account that fields
$a$ and $\bold b$ determining parameter $A$ in scalar ansatz depend
only on modulus of velocity vector $\bold v$ within fibers of tangent
bundle $TM$:
$$
F_k=a\,N_k+|\bold v|\sum^n_{i=1}b_i\,\bigl(2\,N^i\,N_k-\delta^i_k\bigr).
\hskip -2em
\tag5.1
$$
Formula \thetag{5.1} completely determines the dependence of $\bold F$
on velocity vector $\bold v=v\,\bold N$. It is similar to formula
\thetag{4.51} from Chapter
\uppercase\expandafter{\romannumeral 6}. However the dependence on
modulus of velocity vector in \thetag{5.1} is much more complicated.
It is described by formulas \thetag{4.15} and \thetag{4.24}. Finally,
for $\bold F$ we have
$$
\aligned
F_k&=h(W)\,\,V_w(x^1,\ldots,x^n,W)\,\,N_k\,+\\
\vspace{1ex}
&+\,|\bold v|\sum^n_{i=1}V_i(x^1,\ldots,x^n,W)\,\,\bigl(2\,N^i\,N_k
-\delta^i_k\bigr).
\endaligned\hskip -2em
\tag5.2
$$
Here $h(W)$ is an arbitrary function of one variable, through
$V_i$ and $V_w$ we denoted partial derivatives \thetag{4.13},
while functions $V(x^1,\ldots,x^n,W)$ and $W(x^1,\ldots,x^n,v)$
are related by \thetag{4.12}.\par
    Formula \thetag{5.2} for the force field of Newtonian dynamical
system admitting the normal shift contains arbitrariness given by
one function of $(n+1)$ variables, namely by function
$V(x^1,\ldots,x^n,w)$. Arbitrariness given by $h(w)$ can be excluded
due to gauge transformation that preserves $a$ and $\bold b$:
$$
\align
&V(x^1,\ldots,x^n,w)\to V(x^1,\ldots,x^n,\rho^{-1}(w)),\hskip -2em\\
\vspace{1ex}
&W(x^1,\ldots,x^n,v)\to \rho(W(x^1,\ldots,x^n,v)),\hskip -2em
\tag5.3\\
\vspace{1ex}
&h(w)\to h(\rho^{-1}(w))\,\,\rho'(\rho^{-1}(w)).
\endalign
$$
Hence this gauge transformation \thetag{5.3} changes $V$, $W$, and $h$
in \thetag{5.2}, but preserves force field $\bold F$. For $h(w)\neq 0$
we can choose function $\rho(w)$ such that $h(w)\,\rho'(w)=1$. Then,
upon making gauge transformation \thetag{5.3}, in this case we get
formula
$$
\aligned
F_k&=V_w(x^1,\ldots,x^n,W)\,\,N_k\,+\\
\vspace{1ex}
&+\,|\bold v|\sum^n_{i=1}V_i(x^1,\ldots,x^n,W)\,\,\bigl(2\,N^i\,N_k
-\delta^i_k\bigr).
\endaligned\hskip -2em
\tag5.4
$$
Formula \thetag{5.4} is almost as general as formula \thetag{5.2}.
The only exception is the case, when $h=0$. This case cannot be
described by \thetag{5.4}.
\head
\S\,6. Effectivization of the general formula.
\endhead
    Formulas \thetag{5.2} and \thetag{5.4} yield explicit expressions
for the force field ob Newtonian dynamical system admitting the normal
shift on Riemannian manifold $M$. However, they have one common fault:
non-efficiency due to the fact that one should calculate implicit
function $W(x^1,\ldots,x^n,v)$ from the equations \thetag{4.12}. In order
to get more effective formula let's use relationships \thetag{4.16} and
\thetag{4.17}. We rewrite them as follows:
$$
\align
&V_w(x^1,\ldots,x^n,W(x^1,\ldots,x^n,v))=\frac{1}{\partial W
/\partial v},\hskip -2em
\tag6.1\\
\vspace{1.5ex}
&V_k(x^1,\ldots,x^n,W(x^1,\ldots,x^n,v))=-\frac{\partial W
/\partial x^k}{\partial W/\partial v}.\hskip -2em
\tag6.2
\endalign
$$
Substituting \thetag{6.2} into formula \thetag{4.15} for components
of the field $\bold b$, we get 
$$
b_k=-\frac{\partial W(x^1,\ldots,x^n,v)/\partial x^k}
{\partial W(x^1,\ldots,x^n,v)/\partial v}.\hskip -2em
\tag6.3
$$
Here $W(x^1,\ldots,x^n,v)$ is an arbitrary function satisfying the only
condition that the derivative in denominator of fraction \thetag{6.3}
is nonzero. \pagebreak This function, upon substituting \thetag{6.1} into
the formula \thetag{4.24}, determines scalar field $a$ as well:
$$
a=\frac{h(W(x^1,\ldots,x^n,v))}{\partial W(x^1,\ldots,x^n,v)
/\partial v}.\hskip -2em
\tag6.4
$$
Now let's substitute \thetag{6.3} and \thetag{6.4} into the formula
\thetag{5.1} for the field $\bold F$:
$$
\aligned
F_k&=\frac{h(W(x^1,\ldots,x^n,v))}{\partial W(x^1,\ldots,x^n,v)
/\partial v}\,\,N_k\,+\\
\vspace{1ex}
&-\,|\bold v|\sum^n_{i=1}\frac{\partial W(x^1,\ldots,x^n,v)/\partial x^i}
{\partial W(x^1,\ldots,x^n,v)/\partial v}\,\,\bigl(2\,N^i\,N_k
-\delta^i_k\bigr).
\endaligned\hskip -2em
\tag6.5
$$
Formula \thetag{6.5} can be rewritten in terms of covariant derivatives.
We formulate this result as a theorem.
\proclaim{Theorem 6.1} Newtonian dynamical system on Riemannian manifold
$M$ of the dimension $n\geqslant 3$ admits the normal shift if and only
if its force field $\bold F$ is given by
$$
F_k=\frac{h(W)\,N_k}{W_v}-|\bold v|\sum^n_{i=1}\frac{\nabla_iW}{W_v}
\,\bigl(2\,N^i\,N_k-\delta^i_k\bigr),\hskip -2em
\tag6.6
$$
where $W$ is an arbitrary extended scalar field on $M$ with nonzero
derivative 
$$
W_v=\frac{\partial W}{\partial v}=\sum^n_{i=1}N^i\,\,\tilde\nabla_i W
\neq 0
$$
depending only on modulus of velocity vector $v=|\bold v|$ within
fibers of tangent bundle $TM$, while $h=h(W)$ is an arbitrary scalar
function of one variable.
\endproclaim
    In  various  particular  cases  formula  \thetag{6.6}  yields 
various
special examples of dynamical systems admitting the normal shift,
which were considered above. For $W=v$ and $h=0$ we get $\bold F=0$.
This case corresponds to {\bf geodesic flow} on $M$.\par
    Let $f=f(x^1,\ldots,x^n)$ be some scalar field on $M$. Let's
take $W=v\,e^{-f}$. The condition $W_v=\partial W/\partial v\neq 0$
then is fulfilled, since $W_v=e^{-f}$. Now assume that $h=H(w)$, and
substitute $h$ and $W$ into \thetag{6.6}. As a result we obtain
force field
$$
F_k=H(v\,e^{-f})\,e^f\,N_k-|\bold v|^2\,\nabla_kf+2\sum^n_{s=1}
\nabla_sf\,v^s\,v_k,
$$
corresponding to {\bf metrizable dynamical system} admitting the
normal shift (see formula \thetag{3.26} in Chapter
\uppercase\expandafter{\romannumeral 6}).\par
    Let $A=A(v)$ be some nonzero scalar function of one variable,
and suppose that $f=f(x^1,\ldots,x^n)$ is some scalar field on $M$.
We take 
$$
\pagebreak
W=\exp\left(\shave{\int^v_0}\frac{v\,dv}{A(v)}-f\right)
$$
and choose $h(W)=0$. Then formula \thetag{6.6} yields the field
$\bold F$ of the form 
$$
F_k=A(|\bold v|)\sum^n_{i=1}\nabla_if\,\bigl(2\,N^i\,N_k
-\delta^i_k\bigr).
$$
This field correspond s to {\bf non-metrizable dynamical system}
considered in \S\,5 of Chapter \uppercase\expandafter{\romannumeral 6}.
Note that dynamical systems with force field \thetag{6.6} in generic
case are also {\bf non-metrizable}.
\head
\S\,7. Kinematics of normal shift.
\endhead
    Having explicit formula \thetag{6.6} for the force field of
dynamical system admitting the normal shift, we can describe in more
details the process of normal shift for given hypersurface $S$.
According to the definition~11.2 from Chapter
\uppercase\expandafter{\romannumeral 5} one choose some point $p_0$,
then on some smaller part of $S'$ of $S$ one should find the function
$\nu$ that determines the modulus of initial velocity on trajectories
of shift. This function is normalized at the point $p_0$ by the
condition
$$
\nu(p_0)=\nu_0,\hskip -2em
\tag7.1
$$
where $\nu_0$ is nonzero real number (see condition \thetag{11.4} in
Chapter \uppercase\expandafter{\romannumeral 5}). Let's choose local
coordinates $x^1,\,\ldots,\,x^n$ on $M$ in some neighborhood of $p_0$
and local coordinates $u^1,\,\ldots,\,u^{n-1}$ on $S$ in the
neighborhood of the same point. This choice determines coordinate
tangent vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$
to $S$. Denote by $\tau^i_k$ components of $k$-th tangent vector
$\boldsymbol\tau_k$. If $S$ is defined by parametric equations
$$
\align
x^1&=x^1(u^1,\ldots,u^{n-1}),\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .
\hskip -2em\tag7.2\\
x^n&=x^n(u^1,\ldots,u^{n-1}),\hskip -2em
\endalign
$$
then components of tangent vectors $\boldsymbol\tau_1,\,\ldots,
\,\boldsymbol\tau_{n-1}$ are defined by derivatives 
$$
\tau^i_k=\frac{\partial x^i}{\partial u^k}.\hskip -2em
\tag7.3
$$
Function $\nu=\nu(p)=\nu(u^1,\ldots,u^{n-1})$ determines initial
velocity on trajectories of shift (see \thetag{2.2} in Chapter
\uppercase\expandafter{\romannumeral 5}):
$$
\bold v(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=
\nu(p)\cdot\bold n(p).\hskip -2em
\tag7.4
$$
Here $\bold n(p)$ is unitary normal vector to $S$ at the point $p$.
Condition \thetag{7.4} provides normality of shift at the initial
instant of time $t=0$. For to keep normality of shift at all other
instants of time we should choose function $\nu(u^1,\ldots,u^{n-1})$
in some special way. It should satisfy the system of differential
equations \thetag{7.16} that was derived in Chapter 
\uppercase\expandafter{\romannumeral 5}. These equations have the
following form:
$$
\frac{\partial\nu}{\partial u^k}=-\nu^{-1}\,
(\bold F\,|\,\boldsymbol\tau_k).
\hskip -2em
\tag7.5
$$
Let's substitute force field \thetag{6.6} into the equations
\thetag{7.5} and let's take into account that due to \thetag{7.4},
for $t=0$, vector $\bold N$ coincides with unitary normal vector
$\bold n(p)$ on $S$, and $|\bold v|$ coincides with $\nu$. As a
result we get:
$$
\frac{\partial\nu}{\partial u^k}=-\sum^n_{i=1}\frac{\nabla_iW}{W_v}
\,\tau^i_k.
\hskip -2em
\tag7.6
$$
Let's multiply \thetag{7.6} by $W_v$ and let's collect all terms in
left hand side of this equation. Moreover, let's write all derivatives
in explicit form. Then we obtain
$$
\frac{\partial W}{\partial v}\,\frac{\partial\nu}{\partial u^k}
+\sum^n_{i=1}\frac{\partial W}{\partial x^i}\,\frac{\partial x^i}
{\partial u^k}=0.\hskip -2em
\tag7.7
$$
Now it is easy to detect that left hand side of \thetag{7.7} is a
derivative of function $W(x^1,\ldots,x^n,v)$ with respect to the
variable $u^k$, when functions \thetag{7.2} are substituted for
arguments $x^1,\,\ldots,\,x^n$ and function $\nu(u^1,\ldots,u^{n-1})$
is substituted for argument $v$ in this function. Therefore the
equations \thetag{7.7} are easily integrated. They yield the
following implicit equation for the function $\nu=\nu(p)$ on $S$:
$$
W(x^1(p),\ldots,x^n(p),\nu(p))=W_0=\const.\hskip -2em
\tag7.8
$$
The value of constant $W_0$ in \thetag{7.8} is determined by normalization
condition \thetag{7.1}:
$$
W_0=W(x^1(p_0),\ldots,x^n(p_0),\nu_0).
$$
\proclaim{Theorem 7.1} Let $S$ be arbitrary hypersurface $S$ in Riemannian
manifold $M$ given by functions $x^1(p),\,\ldots,\,x^n(p)$ in parametric
form \thetag{7.2}. In order to construct the normal shift of $S$
along trajectories of Newtonian dynamical system with force field
\thetag{6.6} we should determine the function $\nu=\nu(p)$ in \thetag{7.4}
by functional equation \thetag{7.8} in implicit form.
\endproclaim
    Having constructed the normal shift $f_t\!:\,S\to S_t$ of $S$ along
trajectories of dynamical system with force field \thetag{6.6}, we get
the one parametric family of hypersurfaces $S_t$. By changing reference
instant of time $t\to t+t_0$ we can consider each of them as initial
hypersurface. Therefore on each hypersurface $S_t$ we have the equality 
$$
W(p,|\bold v|)=W(t)=\const\hskip -2em
\tag7.9
$$
similar to \thetag{7.8}. However, the values of constants $W(t)$
in \thetag{7.9} can differ for different hypersurfaces $S_t$. Let's
find the dynamics of $W(t)$ in time. In order to do this we use the
formula, which is the consequence of formula \thetag{4.2} from Chapter
\uppercase\expandafter{\romannumeral 4} applied to extended scalar
field $v=|\bold v|$:
$$
\pagebreak 
\frac{d|\bold v|}{dt}=\nabla_tv=\sum^n_{i=1}\tilde\nabla_iv\,
\nabla_tv^i=\sum^n_{i=1}N_i\,F^i,\hskip -3em
\tag7.10
$$
Let's substitute \thetag{6.6} into \thetag{7.10}. As a result of this
substitution we get
$$
\frac{dv}{dt}=\frac{h(W)}{W_v}-\sum^n_{i=1}\frac{\nabla_iW}{W_v}
\,\frac{dx^i}{dt}.
$$
Let's multiply \thetag{7.6} by $W_v$ and collect all terms in left
hand side of this equation. Moreover, let's write all derivatives
in explicit form. Then we obtain
$$
\sum^n_{i=1}\frac{\partial W}{\partial x^i}\,\frac{dx^i}{dt}+
\frac{\partial W}{\partial v}\,\frac{dv}{dt}=h(W).\hskip -3em
\tag7.11
$$
It is easy to see that left hand side of \thetag{7.11} is a time
derivative of scalar field $W$ along the trajectories of normal
shift. Thus the equality \thetag{7.11} is the very equality
determining the dynamics of parameters $W(t)$ in \thetag{7.9}.
Let's write it as
$$
\frac{dW}{dt}=h(W).\hskip -3em
\tag7.12
$$
If function $h(W)$ in \thetag{6.6} is zero, then due to \thetag{7.12}
scalar function $W$ not only is constant on each separate hypersurface
$S_t$, but have the same values on all such hypersurfaces. For
$h(W)\neq 0$ differential equation \thetag{7.12} is easily integrated.
Therefore, if we know the value of $W$ on initial hypersurface $S$,
then we can calculate its value $W(t)$ on any shifted hypersurface
$S_t$.
\enddocument
\end

ediss-08.tex
\input amstex
\documentstyle{amsppt}
%
% Author:   Ruslan Abdulovich Sharipov
% Title:    Dynamical systems admitting the normal shift
% Comment:  Thesis for the degree of Doctor of Sciences in Russia
% Address:  Rabochaya street 5, Ufa, 450003, Russia
% E-mail:   R_Sharipov@ic.bashedu.ru
%           ruslan-sharipov@usa.net
% Web-page: http://www.geocities.com/CapeCanaveral/Lab/5341
%
\nopagenumbers
\catcode`@=11
\redefine\logo@{\relax}
\catcode`\@=\active
\def\startpage#1{\pageno=#1}
\pagewidth{12.8cm}
\pageheight{20cm}
\def\const{\operatorname{const}}
\def\compos{\,\raise 1pt\hbox{$\sssize\circ$} \,}
\accentedsymbol\tx{\tilde x}
%\def\tr{\operatorname{tr}}
%\accentedsymbol\DotTildePhi{\Dot{\Tilde\varphi}}
%\accentedsymbol\ty{\tilde y}
%\def\msum{\operatornamewithlimits{\sum\!{\ssize\ldots}\!\sum}}
%\leftheadtext{\uppercase\expandafter{\romannumeral 7}.
%REDUCTION OF NORMALITY EQUATIONS \dots}
\Monograph
\topmatter
\title\chapter{8}
Normal shift in Finslerian Geometry.
\endtitle
\endtopmatter
\loadbold
\startpage{183}
\document
\head
\S\,1. Basic concepts of Finslerian geometry.
\endhead
    Finslerian geometry is more complicated geometric media, where
we can develop the theory of Newtonian dynamical systems admitting
the normal shift. Finslerian geometry stems from \cite{88}. It arises
in considering extremum problem for functional
$$
I=\int\limits^{\,\,t_2}_{t_1}\Cal F(x^1,\ldots,x^n,v^1,\ldots,v^n)\,dt,
\hskip -2em
\tag1.1
$$
where $x^i=x^i(t)$ and $v^i=\dot x^i=dx^i/dt$. In geometric situation
variables $x^1,\,\ldots,\,x^n$ are interpreted as local coordinates
on some manifold $M$, while integral \thetag{1.1} is a path integral
along some curve connecting two points of this manifold. In order to
keep previous terminology we interpret parameter $t$ as time variable.
Time derivatives $v^i=\dot x^i$ for $x^1,\,\ldots,\,x^n$ are components
of vector $\bold v$ tangent to the path of integration. It is called
velocity vector. Function $\Cal F$ in \thetag{1.1} is assumed to be
satisfying the following three conditions:\par
\parshape 2 0cm 12.8cm 1.1cm 11.7cm\noindent
\,\,$\Phi$1).\ \ function $\Cal F$ is positively homogeneous with the
   degree of homogeneity $1$ respective to components of velocity vector,
   i\.~e\.
   $$
   \Cal F(x^1,\ldots,x^n,k\cdot\bold v)=|k|\cdot\Cal F(x^1,\ldots,x^n,
   \bold v);
   $$\par
\parshape 2 0cm 12.8cm 1.1cm 11.7cm\noindent
\,\,$\Phi$2).\ \ function $\Cal F$ is non-negative, i\.~e\. $\Cal F(x^1,
    \ldots,x^n,\bold v)\geqslant 0$, and such that
   $$
   \Cal F(x^1,\ldots,x^n,\bold v)=0\text{\ \ implies \ }\bold v=0;
   $$\par
\parshape 2 0cm 12.8cm 1.1cm 11.7cm\noindent
\,\,$\Phi$3).\ \ for the function $\Cal H(x^1,\ldots,x^n,\bold v)=
    \Cal F(x^1,\ldots,x^n,\bold v)^2$ corresponding matrix
   $$
   g_{ij}(x^1,\ldots,x^n,\bold v)=\frac{1}{2}\,
   \frac{\partial^2\Cal H}{\partial v^i\,\partial v^j},\hskip -2em
   \tag1.2
   $$
   given by partial derivatives of the function $\Cal H$, is positively
   definite.\par
\definition{Definition 1.1} Smooth manifold $M$ equipped with
smooth\footnotemark\ extended scalar field $\Cal F$ satisfying three
above conditions $\Phi$1, $\Phi$2, $\Phi$3 is called {\bf Finslerian
manifold}.
\enddefinition
\footnotetext{Scalar field $\Cal F$ in definition~1.1 is assumed smooth
on tangent bundle $TM$ everywhere, except for the points of  zero 
section
$\bold v=0$.}
      Definition~1.1 shows that concept of extended tensor fields
described in Chapters \uppercase\expandafter{\romannumeral 2},
\uppercase\expandafter{\romannumeral 3}, and
\uppercase\expandafter{\romannumeral 4} is quite natural and lies
in the basis of whole Finslerian geometry. Path integral \thetag{1.1}
defines length of curves on Finslerian manifold, while matrix
$g_{ij}$ in \thetag{1.2} define components of metric tensor of
Finslerian metric $\bold g$. Finslerian metric is extended tensor
tensor field of type $(0,2)$. Its components 
$$
g_{jk}=\frac{1}{2}\,\tilde\nabla_j\tilde\nabla_k\Cal H\hskip -2em
\tag1.3
$$
depend on velocity vector, they are homogeneous functions of degree
$0$ respective to $\bold v$. Applying covariant derivative $\tilde
\nabla_i$ to \thetag{1.3}, we get tensor field of type $(0,3)$:
$$
C_{ijk}=\frac{1}{2}\,\tilde\nabla_ig_{jk}=\frac{1}{4}\,\tilde\nabla_i
\tilde\nabla_j\tilde\nabla_k\Cal H.\hskip -2em
\tag1.4
$$
Tensor field $\bold C$ with components \thetag{1.4} is called {\bf Cartan
tensor}, its symmetric in all three indices. In Finslerian geometry Cartan
tensor is non-zero, therefore $\tilde\nabla\bold g\neq 0$. This means
that covariant differentiation is not in concordance with metric. We should
take into account this fact in raising and lowering indices of tensor
fields.\par
     Let's use Euler's theorem on homogeneous functions (see \cite{87})
to components of metric tensor $g_{ij}$, which are homogeneous functions
of degree $0$. As a result we get
$$
\sum^n_{k=1}\frac{\partial g_{ij}}{\partial v^k}\,v^k=0.
$$
This relationship is used to write as a property of Cartan tensor,
which partially compensate non-concordance of Finslerian metric
and covariant differentiation $\tilde\nabla$:
$$
\sum^n_{k=1}C_{ijk}\,v^k=0.\hskip -2em
\tag1.5
$$
Applying Euler's theorem on homogeneous functions to the function
$\Cal H=\Cal F^2$, which is homogeneous of degree $2$ respective
to $\bold v$, we get
$$
\sum^n_{i=1}\frac{\partial\Cal H}{\partial v^i}\,v^i=2
\,\Cal H.\hskip -2em
\tag1.6
$$
Left and right hand sides of \thetag{1.6} are homogeneous functions
of degree $2$ in $v$. Applying Euler's theorem once more, we get
$$
\sum^n_{i=1}\sum^n_{j=1}\frac{\partial^2\Cal H}{\partial v^i\,
\partial v^j}\,v^i\,v^j+\sum^n_{i=1}\frac{\partial\Cal H}{\partial v^i}
\,v^i=4\,\Cal H.
$$
Taking into account \thetag{1.6} and \thetag{1.2}, we can write this
relationship as
$$
\sum^n_{i=1}\sum^n_{j=1}g_{ij}\,v^i\,v^j=\Cal H.\hskip -2em
\tag1.7
$$
In other words, the value of quadratic form given by metric tensor
$\bold g$, when applied to vector $\bold v$, is equal to square of
the function $\Cal F$. Therefore we can write path integral \thetag{1.1}
for Finslerian length of curve in the form, which is exactly the same
as in case of such integral in Riemannian geometry:
$$
I=\int\limits^{\,\,t_2}_{t_1}\sqrt{\sum^n_{i=1}\sum^n_{j=1}g_{ij}
\,v^i\,v^j}\,dt.\hskip -2em
\tag1.8
$$
\head
\S\,2. Dynamic connection of Cartan.
\endhead
    Connection of Levi-Civita (or metric connection), which plays
central role in Riemannian geometry, is defined as symmetric
connection such that $\nabla\bold g=0$. In Finslerian geometry
similar condition $\nabla\bold g=0$ defines {\bf dynamic connection
of Cartan}, which was introduced in \cite{89} (see also \cite{90},
\cite{91}, and \cite{61}). Let's write this condition $\nabla\bold g=0$
in local coordinates $M$:
$$
\nabla_ig_{jk}=\frac{\partial g_{jk}}{\partial x^i}
-\sum^n_{s=1}\Gamma^s_{ij}\,g_{sk}-\sum^n_{s=1}\Gamma^s_{ik}\,g_{js}
-\sum^n_{s=1}\sum^n_{r=1}\Gamma^s_{ir}\,\frac{\partial g_{jk}}
{\partial v^s}\,v^r=0.
$$
Here components of connection $\Gamma^k_{ij}$ depend not only on
coordinates $x^1,\,\ldots,\,x^n$, but on components of velocity
vector $\bold v$. This is complete agreement with what was said
in \S\,4 of Chapter \uppercase\expandafter{\romannumeral 3}. Let's
introduce the following notation:
$$
\Gamma_{kij}=\sum^n_{s=1}g_{ks}\,\Gamma^s_{ij}.\hskip -2em
\tag2.1
$$
This simplifies the relationship written above:
$$
\frac{\partial g_{jk}}{\partial x^i}
-\Gamma_{kij}-\Gamma_{jik}-\sum^n_{s=1}\sum^n_{r=1}\Gamma^s_{ir}
\,\frac{\partial g_{jk}}{\partial v^s}\,v^r=0.\hskip -2em
\tag2.2
$$
Let's make cyclic rearrangement of indices $i\to j\to k\to 
i$ in formula
\thetag{2.2}:
$$
\align
&\frac{\partial g_{ki}}{\partial x^j}-\Gamma_{ijk}-\Gamma_{kji}
-\sum^n_{s=1}\sum^n_{r=1}\Gamma^s_{jr}
\,\frac{\partial g_{ki}}{\partial v^s}\,v^r=0.\hskip -2em
\tag2.3\\
\vspace{1ex}
&\frac{\partial g_{ij}}{\partial x^k}-\Gamma_{jki}-\Gamma_{ikj}
-\sum^n_{s=1}\sum^n_{r=1}\Gamma^s_{kr}\,
\frac{\partial g_{ij}}{\partial v^s}\,v^r=0.\hskip -2em
\tag2.4
\endalign
$$
Let's add \thetag{2.2} and \thetag{2.3}, then subtract \thetag{2.4}
from the sum obtained. Thereby we take into account symmetry of
Cartan connection, i\.~e\. $\Gamma_{kij}=\Gamma_{kji}$. Then we get
$$
\gathered
\frac{\partial g_{jk}}{\partial x^i}+\frac{\partial g_{ki}}{\partial x^j}
-\frac{\partial g_{ij}}{\partial x^k}-2\,\Gamma_{kij}=\\
\vspace{1.5ex}
=\sum^n_{s=1}\sum^n_{r=1}\left(\Gamma^s_{ir}\,\frac{\partial g_{jk}}
{\partial v^s}+\Gamma^s_{jr}\,\frac{\partial g_{ki}}{\partial v^s}
-\Gamma^s_{kr}\,\frac{\partial g_{ij}}{\partial v^s}\right)v^r.
\endgathered\hskip -2em
\tag2.5
$$
For the sake of convenience in calculations let's introduce the
following notations:
$$
\gather
\gamma_{kij}=\frac{1}{2}\left(\frac{\partial g_{jk}}{\partial x^i}
+\frac{\partial g_{ki}}{\partial x^j}-\frac{\partial g_{ij}}
{\partial x^k}\right),\hskip -2em
\tag2.6\\
\vspace{1.5ex}
\gamma^k_{ij}=\frac{1}{2}\sum^n_{s=1}g^{ks}\,\left(\frac{\partial
g_{js}}{\partial x^i}+\frac{\partial g_{si}}{\partial x^j}
-\frac{\partial g_{ij}}{\partial x^s}\right),\hskip -2em
\tag2.7
\endgather
$$
Formulas for $\gamma_{kij}$ and $\gamma^k_{ij}$ coincide with well-known
formulas for components metric connection in Riemannian geometry
(see \thetag{1.2} in Chapter \uppercase\expandafter{\romannumeral 5}).
However, in Finslerian geometry they do not define connection
components.\par
    Taking into account notations \thetag{2.6} and \thetag{2.7}, we
transform the relationship \thetag{2.5} for components of Cartan
connection as follows:
$$
\Gamma_{kij}=\gamma_{kij}-\sum^n_{s=1}\sum^n_{r=1}
\left(\Gamma^s_{ir}\,C_{sjk}+\Gamma^s_{jr}\,C_{ski}-\Gamma^s_{kr}
\,C_{sij}\right)v^r.\hskip -3em
\tag2.8
$$
Here we also took into account the relationship \thetag{1.4}, that
determines components of Cartan tensor. The equalities \thetag{2.8}
are algebraic equations for components of Cartan connection
$\Gamma_{kij}$. They should be resolved. Let's multiply \thetag{2.8}
by $v^i$ and $v^j$, then contract it with respect to indices $i$ and
$j$. If we take into account \thetag{1.5}, we get
$$
\sum^n_{i=1}\sum^n_{j=1}\Gamma_{kij}\,v^i\,v^j=\sum^n_{i=1}\sum^n_{j=1}
\gamma_{kij}\,v^i\,v^j.\hskip -2em
\tag2.9
$$
The relationship \thetag{2.9} prompts the following notations:
$$
\aligned
&G_k=\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1}\gamma_{kij}\,v^i\,v^j,\\
\vspace{1ex}
&G^k=\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1}\gamma^k_{ij}\,v^i\,v^j.
\endaligned\hskip -3em
\tag2.10
$$
Now we multiply \thetag{2.8} and contract with respect to index $j$.
Thereby we take into account \thetag{1.5} and notations \thetag{2.10}:
$$
\sum^n_{j=1}\Gamma_{kij}\,v^j=\sum^n_{j=1}\gamma_{kij}\,v^j
-\sum^n_{s=1}2\,G^s\,C_{ski}.\hskip -3em
\tag2.11
$$
Upon raising index $k$ in formula \thetag{2.11} we obtain
$$
\sum^n_{r=1}\Gamma^s_{ir}\,v^r=\sum^n_{r=1}\gamma^s_{ir}\,v^r
-\sum^n_{q=1}2\,G^q\,C^s_{qi}.\hskip -3em
\tag2.12
$$
Further let's note that in right hand side of \thetag{2.8} we have
the same sums as in left hand side of formula \thetag{2.12}.
Therefore the relationship \thetag{2.8} can be rewritten as:
$$
\aligned
&\Gamma_{kij}=\gamma_{kij}-\sum^n_{s=1}\sum^n_{r=1}
\left(\gamma^s_{ir}\,C_{sjk}+\gamma^s_{jr}\,C_{ski}-\gamma^s_{kr}
\,C_{sij}\right)v^r\,+\\
\vspace{1ex}
&\,\,+\,\sum^n_{s=1}\sum^n_{q=1}
2\left(G^q\,C^s_{qi}\,\,C_{sjk}+G^q\,C^s_{qj}\,C_{ski}-
G^q\,C^s_{qk}\,C_{sij}\right).
\endaligned\hskip -2em
\tag2.13
$$
This is required explicit formula for components of Cartan connection
in Finslerian geometry. With the aim of simplifying formula \thetag{2.13}
let's make some further transformations in it. Let's calculate derivatives
of $G^s$ in \thetag{2.10}:
$$
\frac{\partial G^s}{\partial v^i}=\frac{1}{2}\sum^n_{r=1}
\sum^n_{j=1}\frac{\partial \gamma^s_{rj}}{\partial v^i}
\,v^r\,v^j+\sum^n_{r=1}\gamma^s_{ir}\,v^r.\hskip -3em
\tag2.14
$$
Partial derivatives $\partial\gamma^s_{rj}/\partial v^i$ in \thetag{2.14}
are calculated by direct differentiation of formula \thetag{2.7}
determining quantities $\gamma^s_{rj}$:
$$
\aligned
\frac{\partial \gamma^s_{rj}}{\partial v^i}&=
\frac{1}{2}\sum^n_{a=1}\frac{\partial g^{sa}}{\partial v^i}
\left(\frac{\partial g_{ja}}{\partial x^r}+\frac{\partial g_{ar}}
{\partial x^j}-\frac{\partial g_{rj}}{\partial x^a}\right)\,+\\
\vspace{1ex}
&+\,\frac{1}{2}\sum^n_{a=1}g^{sa}\left(\frac{\partial^2 g_{ja}}
{\partial x^r\,\partial v^i}+\frac{\partial^2 g_{ar}}
{\partial x^j\,\partial v^i}-\frac{\partial^2 g_{rj}}
{\partial x^a\,\partial v^i}\right).
\endaligned\hskip -3em
\tag2.15
$$
Matrices with components $g^{ij}$ and $g_{ij}$ are inverse to each
other. Therefore we can calculate partial derivatives $\partial g^{sa}/
\partial v^i$:
$$
\frac{\partial g^{sa}}{\partial v^i}=-\sum^n_{k=1}\sum^n_{m=1}
g^{sk}\,\frac{\partial g_{km}}{\partial v^i}\,g^{ma}=
-\sum^n_{k=1}\sum^n_{m=1}2\,g^{sk}\,C_{ikm}\,g^{ma}.
$$
Let's substitute this expression into \thetag{2.15} and let's take
into account \thetag{1.4}:
$$
\aligned
\frac{\partial \gamma^s_{rj}}{\partial v^i}&=
-\sum^n_{k=1}\sum^n_{m=1}2\,g^{sk}\,C_{ikm}\gamma^m_{rj}\,+\\
\vspace{1ex}
&+\,\sum^n_{a=1}g^{sa}\left(\frac{\partial C_{ija}}{\partial x^r}
+\frac{\partial C_{iar}}{\partial x^j}
-\frac{\partial C_{irj}}{\partial x^a}\right).
\endaligned\hskip -3em
\tag2.16
$$
By substituting \thetag{2.16} into the formula \thetag{2.14} we
contract derivatives $\partial C_{ija}/\partial x^r$, $\partial
C_{iar}/\partial x^j$, and $\partial C_{irj}/\partial x^a$
with $v^j$ and $v^r$ respective to $r$ and $j$. All such contractions
yield zero due to \thetag{1.5}. Therefore \thetag{2.14} can be written
as follows:
$$
\frac{\partial G^s}{\partial v^i}=-\sum^n_{m=1}2\,C^s_{im}\,G^m
+\sum^n_{r=1}\gamma^s_{ir}\,v^r.\hskip -3em
\tag2.17
$$
Let's compare formulas \thetag{2.17} and \thetag{2.12}. As a result of
comparison we derive
$$
\sum^n_{r=1}\Gamma^s_{ir}\,v^r=\frac{\partial G^s}{\partial v^i}.
\hskip -3em
\tag2.18
$$
Substituting \thetag{2.18} into \thetag{2.8}, we get another formula
for components of Cartan connection. It is simpler than formula
\thetag{2.13} above:
$$
\Gamma_{kij}=\gamma_{kij}-\sum^n_{s=1}\left(\frac{\partial G^s}
{\partial v^i}\,C_{sjk}+\frac{\partial G^s}{\partial v^j}\,C_{ski}
-\frac{\partial G^s}{\partial v^k}\,C_{sij}\right).\hskip -3em
\tag2.19
$$
From \thetag{2.13} or \thetag{2.19} one can derive that components
of Cartan connection are homogeneous functions of degree $0$ with respect
to components of velocity vector $\bold v$ in fibers of tangent bundle
$TM$. These formulas in Finslerian geometry are well-known. The above
derivation of them can be found in \cite{61}.
\head
\S\,3. Projectors defined by velocity vector and some identities
for curvature tensors.
\endhead
\rightheadtext{\S\,3. Projectors defined by velocity vector \dots}
     Let's consider the velocity vector $\bold v$. It is interpreted as
extended vector field on the manifold $M$. Let's consider its covariant
derivatives in Finslerian geometry. By direct calculations according to
formulas \thetag{7.3} and \thetag{7.4} from Chapter
\uppercase\expandafter{\romannumeral 3} we get:
$$
\xalignat 2
&\nabla_kv^i=0,&&\tilde\nabla_kv^i=\delta^i_k.
\hskip -2em
\tag3.1
\endxalignat
$$
These relationships coincide with relationships \thetag{5.5} in Chapter
\uppercase\expandafter{\romannumeral 5}. They are fulfilled for all
extended connections on $M$, including Cartan connection.\par
    Let's apply lowering index procedure to the vector field of velocity
$\bold v$, and obtain covector field of velocity. In order to do it we use
Finslerian metric:
$$
v_i=\sum^n_{s=1}g_{is}(x^1,\ldots,x^n,\bold v)\,v^s.\hskip -2em
\tag3.2
$$
Now let's apply covariant derivatives to this covector field:
$$
\xalignat 2
&\nabla_kv_i=0,&&\tilde\nabla_kv_i=g_{ik}.
\hskip -2em
\tag3.3
\endxalignat
$$
First relationship \thetag{3.3} is a consequence of the relationship
$\nabla_kg_{is}=0$ that express concordance of Finslerian metric and
Cartan connection. Second is less trivial:
$$
\tilde\nabla_kv_i=\sum^n_{s=1}g_{is}\tilde\nabla_kv^s+\sum^n_{s=1}
\tilde\nabla_kg_{is}\,v^s=g_{ik}+\sum^n_{s=1}2\,C_{iks}\,v^s.
$$
Indeed, last term in the above expression vanishes due to the
identity \thetag{1.5} for Cartan tensor. As a result we prove second
relationship in \thetag{3.3}.\par
    Denote by $v$ modulus of velocity vector: $v=|\bold v|$. This is
extended scalar field on $M$, square of this field is determined by
formula 
$$
|\bold v|^2=(v\,|\,v)=\sum^n_{i=1}\sum^n_{j=1}g_{ij}(x^1,\ldots,x^n,
\bold v)\,v^i\,v^j=\sum^n_{i=1}v_i\,v^i.
$$
Formulas \thetag{3.1} and  \thetag{3.3}  allow  us  to  calculate 
covariant
derivatives of $v=|\bold v|$:
$$
\xalignat 2
&\nabla_kv=0,&&\tilde\nabla_kv=N_k.
\hskip -2em
\tag3.4
\endxalignat
$$
Here $N_k$ are covariant components of unitary vectorial field $\bold N$
directed along $\bold v$:
$$
\xalignat 2
&N^i=\frac{v^i}{|\bold v|},&&N_i=\frac{v_i}{|\bold v|}.\hskip -2em
\tag3.5
\endxalignat
$$
Let's calculate covariant derivatives of the fields \thetag{3.5}. In
order to do it we use above formulas  \thetag{3.1}, \thetag{3.3}, and
\thetag{3.4}:
$$
\xalignat 2
&\nabla_kN^i=0,&&\nabla_kN_i=0.\hskip -2em
\tag3.6
\endxalignat
$$
In calculating covariant derivatives $\tilde\nabla_kN^i$ tensor field
$\bold P$ arises. Its values are interpreted as projectors to hyperplane
perpendicular to velocity vector $\bold v$:
$$
\align
\tilde\nabla_kN^i&=\frac{1}{|\bold v|}\left(\delta^i_k-N_k\,N^i\right)
=\frac{1}{|\bold v|}\,P^i_k,\hskip -2em
\tag3.7\\
\vspace{1.5ex}
\tilde\nabla_kN_i&=\frac{P_{ik}}{|\bold v|}=\sum^n_{s=1}\frac{g_{is}\,
P^s_k}{|\bold v|}.\hskip -2em
\tag3.8
\endalign
$$
Indeed, if $\bold X$ is some arbitrary vector field, and $\bold P$ is
operator field with components
$$
P^i_k=\delta^i_k-N_k\,N^i,\hskip -2em
\tag3.9
$$
then  for  the  field  $\bold  Y=\bold  P\bold  Y$  we  have  the 
orthogonality
$$
(\bold Y\,|\,\bold v)=\sum^n_{i=1}\sum^n_{j=1}Y^i\,g_{ij}(x^1,\ldots,x^n,
\bold v)\,v^j=0.\hskip -3em
\tag3.10
$$
Scalar product $(\bold Y\,|\,\bold v)$ in \thetag{3.10} is linear with
respect to $\bold Y$, however, it is not linear with respect to $\bold v$.
This is due to special role of velocity vector: it defines the point of
tangent bundle $TM$ and it forms extended vector field simultaneously.\par
     Further let's calculate covariant derivatives for projector field
$\bold P$. From the relationships \thetag{3.6} and \thetag{3.9} we
derive that 
$$
\pagebreak
\nabla_kP^j_i=0.\hskip -3em
\tag3.11
$$
Similarly from the relationships \thetag{3.6} and \thetag{3.9} we get
$$
\tilde\nabla_kP^j_i=-\frac{N_i\,P_k^j+P_{ik}\,N^j}{|\bold v|}.\hskip -3em
\tag3.12
$$
Formulas \thetag{3.11} and \thetag{3.12} are identical to corresponding
formulas in \S\,5 of Chapter \uppercase\expandafter{\romannumeral 5}.
The same is true for formulas \thetag{3.3}, \thetag{3.4}, \thetag{3.6},
\thetag{3.7}, and \thetag{3.8}. This coincidence  is  because  we 
choose
Cartan connection, which is concordant with Finslerian metric, and since
we have identity \thetag{1.5} for Cartan tensor.
\proclaim{Theorem 3.1} Tensor of dynamic curvature $\bold D$, which is
defined by Cartan connection \thetag{2.19} according to formula \thetag{6.5}
from Chapter \uppercase\expandafter{\romannumeral 3}, satisfies the
relationship 
$$
(\bold v\,|\,\bold D(\bold X,\bold Y)\bold v)=
\sum^n_{m=1}\sum^n_{i=1}\sum^n_{j=1}\sum^n_{k=1}
D^m_{kij}(x^1,\ldots,x^n,\bold v)\,X^i\,Y^j\,v^k\,v_m=0,
$$
where $\bold X$ and $\bold Y$ are two arbitrary vector fields on Finslerian
manifold $M$.
\endproclaim
\demo{Proof} Let's use one of the commutational relationships \thetag{7.8}
from Chapter \uppercase\expandafter{\romannumeral 3}:
$$
[\nabla_i,\,\tilde\nabla_j]\varphi=-\sum^n_{m=1}\sum^n_{k=1}
v^k\,D^m_{kij}\,\tilde\nabla_m\varphi.\hskip -3em
\tag3.13
$$
Here $\varphi$ is arbitrary scalar field from extended algebra of
tensor fields. Let's substitute $\varphi=|\bold v|^2$ into \thetag{3.13}.
In left hand side we get
$$
[\nabla_i,\,\tilde\nabla_j]\varphi=\nabla_i\tilde\nabla_j\varphi
-\tilde\nabla_j\nabla_i\varphi=\nabla_i(2\,v\,N_j)=0.
$$
Similar calculations with $\varphi=|\bold v|^2$ for right hand side of
formula \thetag{3.13} yield
$$
\sum^n_{m=1}\sum^n_{k=1}v^k\,D^m_{kij}\,\tilde\nabla_m\varphi=
\sum^n_{m=1}\sum^n_{k=1}2\,D^m_{kij}\,v^k\,v_m.
$$
Equating these two expressions for left and right hand sides of
\thetag{3.13}, we get
$$
\sum^n_{m=1}\sum^n_{k=1}D^m_{kij}\,v^k\,v_m=0.
$$
Now, for to complete proof of theorem one should multiply this equality
by $X^i\,Y^j$, and should accomplish summation in indices $i$ and $j$.
\qed\enddemo
\proclaim{Theorem 3.2} Tensor of Riemannian curvature $\bold R$,
defined by Cartan connection \thetag{2.19} according to formula
\thetag{6.8} from Chapter \uppercase\expandafter{\romannumeral 3},
satisfies the relationship 
$$
(\bold v\,|\,\bold R(\bold X,\bold Y)\bold v)=
\sum^n_{m=1}\sum^n_{i=1}\sum^n_{j=1}\sum^n_{k=1}
R^m_{kij}(x^1,\ldots,x^n,\bold v)\,X^i\,Y^j\,v^k\,v_m=0,
$$
where $\bold X$ and $\bold Y$ are two arbitrary vector fields on Finslerian
manifold $M$.
\endproclaim
\demo{Proof} Proof of this theorem is similar to that of previous one.
However, instead of \thetag{7.8} here we use formula \thetag{7.10}
from Chapter \uppercase\expandafter{\romannumeral 3}. Here it is
written as 
$$
[\nabla_i,\,\nabla_j]\varphi=-\sum^n_{m=1}\sum^n_{k=1}v^m\,R^k_{mij}
\,\tilde\nabla_k\varphi,\hskip -3em
\tag3.14
$$
since torsion field for Cartan connection is equal to zero. Let's
substitute $\varphi=|\bold v|^2$ into \thetag{3.14}. In left hand
side we have
$$
[\nabla_i,\,\nabla_j]\varphi=\nabla_i\nabla_j\varphi-
\nabla_j\nabla_i\varphi=0,
$$
this follows from \thetag{3.4}. In right hand side of \thetag{3.14}
we get
$$
\sum^n_{m=1}\sum^n_{k=1}v^m\,R^k_{mij}\,\tilde\nabla_k\varphi=
\sum^n_{m=1}\sum^n_{k=1}2\,R^k_{mij}\,v^m\,v_k.
$$
Equating these two expressions for left and right hand sides of
\thetag{3.14}, we get
$$
\sum^n_{m=1}\sum^n_{k=1}R^m_{kij}\,v^k\,v_m=0.
$$
Now, for to complete proof of theorem one should multiply this equality
by $X^i\,Y^j$, and should accomplish summation in indices $i$ and $j$.
\qed\enddemo
     Cartan connection \thetag{2.19} is symmetric: $\Gamma^m_{ij}
=\Gamma^m_{ji}$. Therefore tensor of dynamic curvature $\bold D$
is also symmetric. This is expressed by the equality
$$
D^m_{kij}=D^m_{ikj}.\hskip -3em
\tag3.15
$$
Due to \thetag{3.15} we can transpose vectors $\bold X$ and $\bold v$
in formula from theorem~3.1:
$$
(\bold v\,|\,\bold D(\bold v,\bold Y)\bold X)=0.
\tag3.16
$$
\head
\S\,4. Hypersurfaces in Finslerian space and their normal vector.
\endhead
\rightheadtext{\S\,4. Hypersurfaces in Finslerian space \dots}
     Suppose that $S$ is a hypersurface in Finslerian manifold $M$,
and let $u^1,\,\ldots,\,u^{n-1}$ be local coordinates on $S$. In
local coordinates $x^1,\,\ldots,\,x^n$ on $M$ this hypersurface
can be represented by parametric equations:
$$
\align
x^1&=x^1(u^1,\ldots,u^{n-1}),\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .
\hskip -2em\tag4.1\\
x^n&=x^n(u^1,\ldots,u^{n-1}).\hskip -2em
\endalign
\pagebreak
$$
Tangent hyperplane to $S$ is defined by corresponding coordinate
tangent vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$.
In local coordinates $x^1,\,\ldots,\,x^n$ they are expanded as
follows:
$$
\boldsymbol\tau_k=\sum^n_{i=1}\tau^i_k\,\frac{\partial}{\partial x^i}.
\hskip -2em
\tag4.2
$$
\definition{Definition 4.1} Let $S$ be hypersurface in Finslerian
manifold $M$, and suppose that $p\in S$. Vector $\bold n$ from tangent
space $T_p(M)$ is called {\bf normal vector} to $S$, if it is perpendicular
to vectors \thetag{4.2} with respect to Finslerian metric \thetag{1.2}:
$$
\sum^n_{i=1}\sum^n_{j=1}\tau^i_k\,g_{ij}(x^1,\ldots,x^n,\bold n)\,n^j=0.
\hskip -2em
\tag4.3
$$
\enddefinition
    Problem of existence of normal vector in the sense of definition~4.1
in Finslerian manifold is nontrivial, since the equations \thetag{4.3}
are non-linear with respect to components of vector $\bold n$. In order
to solve this problem we consider the map $\bold n\to\boldsymbol
\xi(\bold n)$, that maps vector $\bold n$ to the covector $\boldsymbol
\xi$ by means lowering index procedure:
$$
\xi_i=\sum^n_{j=1}g_{ij}(x^1,\ldots,x^n,\bold n)\,n^j.
\hskip -2em
\tag4.4
$$
According to the equations \thetag{4.3} the contraction of $\boldsymbol\xi$
with coordinate tangent vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol
\tau_{n-1}$ from \thetag{4.2} should yield zero result:
$$
\sum^n_{i=1}\tau^i_k\,\,\xi_i=0.
\hskip -2em
\tag4.5
$$
We see that \thetag{4.5} is a system of homogenized linear equations
with matrix of rank $n-1$ respective to $\xi_1,\,\ldots,\,\xi_n$. They
define $\boldsymbol\xi$ uniquely up to a numeric multiple. Therefore
problem of existence of normal vector reduces to inverting map $\bold n
\to\boldsymbol\xi(\bold n)$ defined by \thetag{4.4}. By differentiating
\thetag{4.4} we calculate Jacoby matrix of this map:
$$
\aligned
J_{ij}&=\frac{\partial\xi_i}{\partial n^j}=g_{ij}+\sum^n_{k=1}
\frac{\partial g_{ik}}{\partial n^j}\,n^k=g_{ij}(x^1,\ldots,x^n,
\bold n)\,+\\
\vspace{1ex}
&+\,\sum^n_{k=1}2\,C_{jik}(x^1,\ldots,x^n,\bold n)\,n^k=
g_{ij}(x^1,\ldots,x^n,\bold n).
\endaligned
$$
The result of contracting $\bold n$ with Cartan tensor $\bold C$ here
is zero due to \thetag{1.5}. The equality $J_{ij}=g_{ij}$ means that
$\det J\neq 0$ since matrix of metric tensor is positively definite
(see axiom $\Phi$3 in definition of Finslerian manifold).\par
     Thus, the map $\bold n\to\boldsymbol\xi(\bold n)$ in \thetag{4.4}
is locally bijective. It is homogeneous of degree $1$:
$\boldsymbol\xi(k\cdot\bold n)=k\cdot\boldsymbol\xi(\bold n)$. If we
fix some Euclidean metric in vector space $T_p(M)$ and some other metric
in covector space $T^*_p(M)$, then we can treat $\bold n\to\boldsymbol
\xi(\bold n)$ as a map from one unit sphere $S^{n-1}$ to other unit
sphere $S^{n-1}$. Due to local bijectivity this map is open, hence
the image of $S^{n-1}$ is an open set in other sphere. But, at the same
time, this image is closed as an image of bicompact space $S^{n-1}$
under the map into Hausdorf space $S^{n-1}$ (see \cite{92} or \cite{93})
Hence, since sphere $S^{n-1}$ is connected topological space, map
$\bold n\to\boldsymbol\xi(\bold n)$ is surjective. It is non-ramified
covering due to local bijectiveness. Then remember, that non-ramified
covering of multidimensional sphere $S^{n-1}$ is one-sheeted, because
sphere $S^{n-1}$ is simply connected space for $n\geqslant 3$ (see
\cite{94}). This means that $\bold n\to\boldsymbol\xi(\bold n)$ is a
bijection of spheres (diffeomorphism). This result form a theorem.
\proclaim{Theorem 4.1} Each smooth hypersurface $S$ in Finslerian manifold
$M$ of the dimension $n\geqslant 3$ possess unitary normal vector
$\bold n$ in the sense of definition~4.1, this vector is fixed uniquely
up to a choice of orientation on $S$.
\endproclaim
     This result is true in two-dimensional case $n=2$ as well, i\.~e\.
we have the following theorem.
\proclaim{Theorem 4.2} Each smooth hypersurface $S$ in Finslerian manifold
$M$ possess unitary normal vector $\bold n$ in the sense of definition~4.1,
this vector is fixed uniquely up to a choice of orientation on $S$.
\endproclaim
     In two dimensional case $n=2$ let's fix a point $p\in M$ and consider
the map $\bold n\to\boldsymbol\xi(\bold n)$ from $T_p(M)$ to $T^*_p(M)$
given by formula \thetag{4.4}. Apart from Finslerian metric in tangent
space $T_p(P)$ we fix Euclidean metric $\tilde g$ with unit matrix in
some local coordinates $x^1,\,\ldots,\,x^n$:
$$
\tilde g_{ij}=\tilde g^{ij}=\Vmatrix 1&0\\ 0&1\endVmatrix.
\hskip -2em
\tag4.6
$$
Metric \thetag{4.6} defines raising index procedure. Let's apply it to
covector field $\boldsymbol\xi$. Then formula \thetag{4.4} can be replaced
by the following one:
$$
\xi^i=\sum^2_{k=1}\sum^2_{j=1}\tilde g^{ik}\,g_{kj}(x^1,x^2,\bold n)\,n^j.
\hskip -2em
\tag4.7
$$
Written as \thetag{4.7}, $\bold n\to\boldsymbol\xi(\bold n)$ is a map
from $T_p(M)$ to $T_p(M)$. Let's take $\bold n$ to be unitary vector in
metric \thetag{4.6}. Then $\bold n$ and $\boldsymbol\xi(\bold n)$ are
written as:
$$
\xalignat 2
&\bold n=\Vmatrix \cos(\varphi)\\ \vspace{2ex} \sin(\varphi\endVmatrix,
&&\boldsymbol\xi(\bold n)=|\boldsymbol\xi|\cdot \Vmatrix \cos(\tilde
\varphi)\\ \vspace{2ex}\sin(\tilde\varphi)\endVmatrix.\hskip -2em
\tag4.8
\endxalignat
$$
Map $\bold n\to\boldsymbol\xi(\bold n)$ defines non-ramified covering
of unit circle by some other unit circle. In terms of angles $\varphi$
and $\tilde\varphi$ it is given by strictly monotonous smooth function
$\tilde\varphi(\varphi)$. The difference $\tilde\varphi(\varphi+2\pi)
-\tilde\varphi(\varphi)$ is an integer multiple of $2\pi$:
$$
\tilde\varphi(\varphi+2\pi)-\tilde\varphi(\varphi)=2\pi\,m.\hskip -2em
\tag4.9
$$
Integer number $m\neq 0$ in \thetag{4.9} is positive, if function
$\tilde\varphi(\varphi)$ is increasing function; it is negative, if
$\tilde\varphi(\varphi)$ is decreasing function. Modulus of this number
is equal to the number of sheets in non-ramified covering $S^1\to S^1$.
\par
    Suppose that $m\neq 1$. Let $\psi(\varphi)=\tilde\varphi(\varphi)
-\varphi$. Then for the function $\psi(\varphi)$  from the relationship
\thetag{4.9} we derive that 
$$
\psi(\varphi+2\pi)-\psi(\varphi)=2\pi(m-1)\neq 0.
$$
This means that function has indefinitely large values of one sign as
$\varphi\to+\infty$, and it has indefinitely large values of opposite
sign as $\varphi\to-\infty$. Hence, since it is continuous, there exist
an angle $\varphi_0$ such that $\psi(\varphi_0)=\pi/2$. For such angle
$\varphi=\varphi_0$ we have $\tilde\varphi=\varphi+\pi/2$, i\.~e\.
vectors $\bold n$ and $\boldsymbol\xi(\bold n)$ from \thetag{4.8} are
perpendicular to each other in Euclidean metric \thetag{4.6}:
$$
\tilde\bold g(\bold n,\boldsymbol\xi)=\sum^2_{i=1}\sum^2_{j=1}
\tilde g_{ij}\,n^i\,\xi^j=\sum^2_{i=1}\sum^2_{j=1}
g_{kj}(x^1,x^2,\bold n)\,n^k\,n^j=0.
$$
This relationship contradicts to the fact that matrix $g_{kj}(x^1,x^2,
\bold n)$ in \thetag{1.2} is positively definite (see axiom $\Phi$3 from
the definition of Finslerian manifold). This contradiction proves that
$m=1$, i\.~e\. $\bold n\to\boldsymbol\xi(\bold n)$ is one-sheeted covering
$S^1\to S^1$. Thus, theorem~4.2 is proved.\par
     Existence of normal vectors allows us to generalize the theory of
dynamical systems admitting the normal shift for the case of Finslerian
manifolds.
\head
\S\,5. Geodesic normal shift.
\endhead
    Let's consider some Newtonian dynamical system on Finslerian manifold
$M$. In local coordinates $x^1,\,\ldots,\,x^n$ on $M$ it is given by
ordinary differential equations
$$
\xalignat 2
&\quad\dot x^i=v^i,
&&\nabla_tv^i=F^i(x^1,\ldots,x^n,v^1,\ldots,v^n).\hskip -2em
\tag5.1
\endxalignat
$$
Here $F^1,\,\ldots,\,F^n$ are components of force vector. These
equations \thetag{5.1} differ from corresponding equations in
Riemannian geometry only in covariant derivatives $\nabla_tv^i$,
here they are calculated respective to extended Cartan connection:
$$
\nabla_tv^i=\dot v^i+\sum^n_{j=1}\sum^n_{k=1}\dot x^j\,
\Gamma^i_{jk}(x^1,\ldots,x^n,v^1,\ldots,v^n)\,v^k.\hskip -3em
\tag5.2
$$
Derivative $\nabla_t\bold v$ with components given by \thetag{5.2}
is a covariant derivative of velocity vector $\bold v$ with respect
to parameter $t$ along the curve (compare formula \thetag{5.2} with
formulas \thetag{3.10} and \thetag{3.13} in Chapter
\uppercase\expandafter{\romannumeral 4}).\par
     The simplest Newtonian dynamical system of the form \thetag{5.1}
is a geodesic flow of Finslerian metric \thetag{1.2}. It corresponds
to purely zero force field in $\bold F=0$:
$$
\xalignat 2
&\quad\dot x^i=v^i,
&&\nabla_tv^i=0.\hskip -2em
\tag5.3
\endxalignat
$$
Suppose that $S$ is some hypersurface from transformation class
in $M$. In local coordinates $u^1,\,\ldots,\,u^{n-1}$ on $S$ it is
given by parametric equations \thetag{4.1}. Let's fix on $S$ (or
possibly on some smaller piece of $S$)) a smooth field of unitary
normal vectors $\bold n=\bold n(u^1,\ldots,u^{n-1})$. Vector $\bold n$
possess unit length in Finslerian metric, it is perpendicular to coordinate
tangent vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$
in the sense of definition~4.1. We shall use $S$ for to define data
that initiate a shift along trajectories of geodesic flow \thetag{5.3}.
Let's write them as follows:
$$
\aligned
&x^k(t)\,\hbox{\vrule height 8pt depth 9pt width 0.5pt}_{\,t=0}
=x^k(u^1,\ldots,u^{n-1}),\\
\vspace{3ex}
&v^k(t)\,\hbox{\vrule height 8pt depth 9pt width 0.5pt}_{\,t=0}=
n^k(u^1,\ldots,u^{n-1}).
\endaligned\hskip -2em
\tag5.4
$$
Solution of Cauchy problem \thetag{5.4} for the equations \thetag{5.3}
is given by functions
$$
\align
x^1&=x^1(u^1,\ldots,u^{n-1},t),\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .
\hskip -2em\tag5.5\\
x^n&=x^n(u^1,\ldots,u^{n-1},t).\hskip -2em
\endalign
$$
These functions define geodesic shift $f_t\!:\,S\to S_t$ of hypersurface
$S$. Here $t$ is a parameter of shift, variables $u^1,\,\ldots,\,u^{n-1}$
are local coordinates on $S$ for $t=0$, they can be used as local
coordinates on shifted hypersurfaces $S_t$ for sufficiently small
$t\neq 0$ as well. If we denote $t=u^n$, then complete set of variables
$u^1,\,\ldots,\,u^n$ form the set of local coordinates on $M$ in some
open neighborhood of $S$. These coordinates are called {\bf semigeodesic
coordinates}.
\proclaim{Theorem 5.1} Components of metric tensor for Finslerian metric
$\bold g$ in semigeodesic coordinates $u^1,\,\ldots,\,u^n$ satisfy the
following conditions:
$$
\align
&g_{kn}(u^1,\ldots,u^n,0,\ldots,0,v^n)=0\text{ \ for \ }k\neq n,
\hskip -2em
\tag5.6\\
\vspace{1ex}
&g_{nn}(u^1,\ldots,u^n,0,\ldots,0,v^n)=1\text{ \ for \ }k=n.
\hskip -2em
\tag5.7
\endalign
$$
\endproclaim
\demo{Proof} Due to the choice of semigeodesic coordinates by itself
geodesic lines, defined by the equations \thetag{5.3} and initial data
\thetag{5.4} on $S$, in semigeodesic coordinates are written in the
following form:
$$
u^1(t)=\const,\ \ldots,\ u^{n-1}(t)=\const,\qquad u^n(t)=t.\hskip -2em
\tag5.8
$$
Differentiating \thetag{5.8} in $t$ we find components of velocity
vector on geodesic lines:
$$
\pagebreak
v^1(t)=\ldots=v^{n-1}(t)=0,\qquad v^n(t)=1.\hskip -2em
\tag5.9
$$
The equations \thetag{5.3}, on account of \thetag{5.2}, are written as
$$
\dot v^i+\sum^n_{j=1}\sum^n_{k=1}\Gamma^i_{jk}(u^1,\ldots,u^n,v^1,
\ldots,v^n)\,v^j\,v^k=0.\hskip -3em
\tag5.10
$$
Let's substitute \thetag{5.8} and \thetag{5.9} into the equations
\thetag{5.10}, then for $\Gamma^i_{jk}$ we get 
$$
\Gamma^k_{nn}(u^1,\ldots,u^{n-1},t,0,\ldots,0,1)=0\text{ \ for \ }
k\leqslant n.\hskip -3em
\tag5.11
$$
In formula \thetag{5.11} we substitute parameter $t$ by $u^n$. Moreover,
we take into account that components of Cartan connection $\Gamma^i_{jk}$
are homogeneous functions of degree $0$ with respect to components of
velocity vector $\bold v$. This yields
$$
\Gamma^k_{nn}(u^1,\ldots,u^n,0,\ldots,0,v^n)=0\text{ \ for \ }
k\leqslant n.\hskip -3em
\tag5.12
$$
Now let's write the concordance condition for Cartan connection and
Finslerian metric $\bold g$ in semigeodesic coordinates. It looks like 
$$
\nabla_ng_{nn}=\frac{\partial g_{nn}}{\partial u^n}-\sum^n_{k=1}
2\,\Gamma^k_{nn}\,g_{kn}-\sum^n_{k=1}\Gamma^k_{nn}\,\frac{\partial
g_{nn}}{\partial v^k}\,v^n=0.
$$
Let's recall arguments of $\Gamma^k_{nn}$ from \thetag{5.12}. Then
from $\Gamma^k_{nn}=0$ we derive
$$
\frac{\partial g_{nn}(u^1,\ldots,u^n,0,\ldots,0,v^n)}{\partial
u^n}=0.\hskip -3em
\tag5.13
$$
Let's write \thetag{5.4} in semigeodesic coordinates:
$$
\xalignat 2
&u^k(t)\,\hbox{\vrule height 8pt depth 9pt width 0.5pt}_{\,t=0}
=u^k,
&&v^k(t)\,\hbox{\vrule height 8pt depth 9pt width 0.5pt}_{\,t=0}=
n^k.\hskip -3em
\tag5.14
\endxalignat
$$
Remember that $u^n=0$ for $t=0$ on $S$. Further, combining \thetag{5.14}
and \thetag{5.9}, we calculate the components of unit vector $\bold n$
on initial hypersurface $S$:
$$
\aligned
&n^k=n^k(u^1,\ldots,u^{n-1},0)=0\text{ \ for \ }k\neq n,\\
\vspace{1ex}
&n^k=n^k(u^1,\ldots,u^{n-1},0)=1\text{ \ for \ }k=n.
\endaligned\hskip -3em
\tag5.15
$$
Now let's write the condition that $|\bold n|=1$ on $S$:
$$
|\bold n|^2=\sum^n_{i=1}\sum^n_{i=1}g_{ij}(u^1,\ldots,u^{n-1},0,
\bold n)\,n^i\,n^j=1.
$$
Combining this equality with formulas \thetag{5.15} for components
of $\bold n$, we get
$$
g_{nn}(u^1,\ldots,u^{n-1},0,0,\ldots,0,1)=1.\hskip -3em
\tag5.16
$$
Components of metric tensor for Finslerian metric are homogeneous
functions of degree $0$ respective to $\bold v$. Therefore
\thetag{5.16} yields
$$
g_{nn}(u^1,\ldots,u^{n-1},0,0,\ldots,0,v^n)=1.\hskip -3em
\tag5.17
$$
Let's compare \thetag{5.17} and \thetag{5.13}. This comparison
shows that $u^n=0$ in \thetag{5.17} can be replaced by arbitrary
value of the variable $u^n$, i\.~e\. we get the relationships
coinciding with the relationships \thetag{5.7} in the statement of
theorem:
$$
g_{nn}(u^1,\ldots,u^n,0,\ldots,0,v^n)=1.
$$\par
    In order to prove the relationships \thetag{5.6} from theorem~5.1
let's apply the condition of concordance for Finslerian metric and
Cartan connection. We write $\nabla_kg_{nn}=0$ expressing covariant
derivative $\nabla_k$ through partial derivatives:
$$
\nabla_ig_{nn}=\frac{\partial g_{nn}}{\partial u^i}-\sum^n_{k=1}
2\,\Gamma^k_{in}\,g_{kn}-\sum^n_{s=1}\sum^n_{r=1}\Gamma^s_{ir}
\,\frac{\partial g_{nn}}{\partial v^s}\,v^r=0.
$$
Let's substitute $v^1=\ldots=v^{n-1}=0$ and $v^n=1$ into this formula.
Then
sum in $r$ reduces to one summand with $r=0$, while partial derivative 
$\partial g_{nn}/\partial u^i$ in the above equation vanishes due to the
relationship \thetag{5.7}, which we have already proved:
$$
\sum^n_{k=1}2\,\Gamma^k_{in}\,g_{kn}+\sum^n_{s=1}\Gamma^s_{in}
\,\frac{\partial g_{nn}}{\partial v^s}\,v^n=0.\hskip -3em
\tag5.18
$$
In calculating $\partial g_{nn}/\partial v^s$ we use formula \thetag{1.4}
and symmetry of Cartan tensor:
$$
\frac{\partial g_{nn}}{\partial v^s}=2\,C_{snn}=2\,C_{nsn}.\hskip -3em
\tag5.19
$$
Now let's write the relationship \thetag{1.5} substituting arguments
from \thetag{5.12} in it. The sum in $k$ thereby reduces to one term
with $k=n$:
$$
0=\sum^n_{k=1}C_{nsk}\,v^k=C_{nsn}(u^1,\ldots,u^n,0,\ldots,0,v^n)\,v^n.
\hskip -3em
\tag5.20
$$
The relationships \thetag{5.19} and \thetag{5.20} show that if we
substitute $v^1=\ldots=v^{n-1}=0$ and $v^n=1$ into \thetag{5.18},
sum in $s$ vanishes: 
$$
\aligned
\sum^n_{k=1}\Gamma^k_{in}&(u^1,\ldots,u^n,0,\ldots,0,v^n)\,\times\\
\vspace{1ex}
&\times\,g_{kn}(u^1,\ldots,u^n,0,\ldots,0,v^n)=0.
\endaligned\hskip -3em
\tag5.21
$$
We take into account this relationship \pagebreak and use the condition of
concordance of $\Gamma$ and $\bold g$ for the third time. We write
this condition $\nabla_ng_{ni}=0$ in semigeodesic coordinates
expressing $\nabla_n$ through partial derivatives:
$$
\frac{\partial g_{ni}}{\partial u^n}
-\sum^n_{k=1}\Gamma^k_{nn}\,g_{ki}-\sum^n_{k=1}\Gamma^k_{ni}\,g_{nk}
-\sum^n_{k=1}\sum^n_{r=1}\Gamma^k_{nr}\,\frac{\partial g_{ni}}
{\partial v^k}\,v^r=0.
$$
Second sum with respect to index $k$ in left hand side of resulting
equality is the same sum as in \thetag{5.21}. It vanishes if we
substitute $v^1=\ldots=v^{n-1}=0$ and $v^n=1$. First sum in $k$
vanishes due to \thetag{5.12}, while sum in $r$ reduces to one
summand with $r=n$. This summand vanishes due to relationships
analogous to \thetag{5.19} and \thetag{5.20}. Indeed, from the
identity \thetag {1.5} we derive
$$
\frac{\partial g_{ni}}{\partial v^k}\,v^n=2\,C_{ikn}(u^1,\ldots,u^n,0,
\ldots,0,v^n)\,v^n=\sum^n_{s=1}2\,C_{iks}\,v^s=0.
$$
Due to all above reductions the relationship $\nabla_ng_{ni}=0$ yields
$$
\frac{\partial g_{ni}(u^1,\ldots,u^n,0,\ldots,0,v^n)}{\partial u^n}=0.
\hskip -3em
\tag5.22
$$\par
    Let's consider coordinate tangent vectors \thetag{4.2} on the
hypersurface $S$. In semigeodesic coordinates this hypersurface
is described by the equation $u^n=0$, while coordinate tangent
vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ have
unitary components:
$$
\tau^k_i=\frac{\partial u^k}{\partial u^i}=\delta^k_i=
\cases 1&\text{for \ }k=i,\\
0&\text{for \ }k\neq i.\endcases
\hskip -3em
\tag5.23
$$
Let's write \thetag{4.3} in semigeodesic coordinates and let's take
into account \thetag{5.15} and \thetag{5.23}. For the components of
metric this yields 
$$
g_{ni}(u^1,\ldots,u^{n-1},0,0,\ldots,0,v^n)=0\text{ \ for \ }
i\neq n.\hskip -3em
\tag5.24
$$
Comparing \thetag{5.24} and \thetag{5.22}, we find that substituting
$u^n=0$ by nonzero value of this variable do not break the relationships
\thetag{5.24}. Therefore we have
$$
g_{ni}(u^1,\ldots,u^n,0,\ldots,0,v^n)=0\text{ \ for \ }i\neq n.
$$
They coincide with the equalities \thetag{5.6} in the statement of
theorem~5.1 that was to be proved. Proof is over.\qed\enddemo
    Comparing proof of the theorem~5.1 with proof of analogous
proposition in \S\,1 of Chapter \uppercase\expandafter{\romannumeral 5},
we conclude that they are based on the same idea. Additional obstacles
due to the dependence of $g_{ij}$ and $\Gamma^k_{ij}$ upon $\bold v$
are surmounted at the expense of \thetag{1.4} and \thetag{1.5}.\par
    Theorem~5.1 describes geodesic shift $f_t\!:\,S\to S_t$ defined by
the equations \thetag{5.3} and initial data \thetag{5.4}. The
relationships \thetag{5.6} show that this normal shift, i\.~e\.
vectors tangent to trajectories are perpendicular to hypersurfaces
$S_t$ in the sense of definition~4.1. We should stress that this
is true for {\bf any choice} of hypersurface $S$.
\head
\S\,6. Newtonian dynamical systems admitting the normal shift.
\endhead
\rightheadtext{\S\,6. Newtonian dynamical systems \dots}
     In this section we generalize the theory of Newtonian dynamical
systems admitting the normal shift to the background of Finslerian
geometry. Let's start with basic definitions. According to \S\,3
for the hypersurfaces of Finslerian manifold the concept of normal
vector is well defined. Therefore we can state initial data
$$
\xalignat 2
&\qquad x^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}
=x^k(p),
&&v^k(t)\,\hbox{\vrule height 8pt depth 8pt width 0.5pt}_{\,t=0}=
\nu(p)\cdot n^k(p)\hskip -3em
\tag6.1
\endxalignat
$$
in considering Newtonian dynamical systems \thetag{5.1}. Here $p$
is a point of hypersurface $S$ and $\nu(p)$ is some function that
defines modulus of initial velocity on trajectories of dynamical
system \thetag{5.1} coming out from $S$ in the direction of its
normal vector. Suppose that $f_t\!:\,S\to S_t$ is a shift of
hypersurface $S$ defined by dynamical system \thetag{5.1} and
initial data \thetag{6.1}. It is called the {\bf normal shift}, if
trajectories of shift are directed along normal vectors for all
hypersurfaces obtained by the shift $f_t\!:\,S\to S_t$. Dynamical
system \thetag{5.1} with force field $\bold F$ is called {\bf admitting
the normal shift}, if it is able to accomplish normal shift of any
hypersurface. More precisely, there is the definition including the
normalization condition
$$
\nu(p_0)=\nu_0\hskip -2em
\tag6.2
$$
with arbitrary number $\nu_0\neq 0$ at arbitrary point $p_0$ on $S$.
\definition{Definition 6.1} Newtonian dynamical system \thetag{5.1} on
Finslerian manifold $M$ is called the system {\bf admitting the normal
shift}, if for any hypersurface $S$ in $M$, for any point $p_0\in S$,
and for any real number $\nu_0\neq 0$ there is a piece $S'$ of
hypersurface $S$ belonging to transformation class and containing
$p_0$, and there exists a function $\nu(p)$ belonging to transformation
class on $S'$ and normalized by \thetag{6.2} such that the shift
$f_t\!:S'\to S'_t$, defined by this function in initial data
\thetag{6.1}, is a normal shift along the trajectories of dynamical
system.
\enddefinition
    Let $u^1,\,\ldots,\,u^{n-1}$ be local coordinates on $S$ in the
neighborhood of some marked point $p_0$. Shift $f_t\!:S'\to S'_t$
transfers these coordinates to all hypersurfaces $S_t$. Therefore
all these hypersurfaces can be represented in parametric form
$$
\align
x^1&=x^1(u^1,\ldots,u^{n-1},t),\hskip -2em\\
.\ .\ &.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .
\hskip -2em\tag6.3\\
x^n&=x^n(u^1,\ldots,u^{n-1},t),\hskip -2em
\endalign
$$
Differentiating these functions \thetag{6.3} with respect to
$u^1,\,\ldots,\,u^{n-1}$, we define tangent vectors $\boldsymbol\tau_1,
\,\ldots,\,\boldsymbol\tau_{n-1}$ to $S$ with components 
$$
\tau^i_k=\frac{\partial x^i}{\partial u^k},
$$
while differentiating them with respect to $t$, \pagebreak we get the
components of velocity vector on trajectories of shift: $v^i=\dot x^i
=\partial x^i/\partial t$. Functions of deviation are defined as scalar
products of vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$
with vector $\bold v$:
$$
\varphi_k=(\boldsymbol\tau_k\,|\,\bold v)=\sum^n_{i=1}\sum^n_{j=1}
\tau^i_k\,g_{ij}(x^1,\ldots,x^n,\bold v)\,v^j.\hskip -2em
\tag6.4
$$
Normality of shift $f_t\!:\,S\to S_t$ now is formulated as vanishing
condition for all functions of deviation \thetag{6.4} simultaneously:
$$
\varphi_k(t)=0\text{\ \ for \ \ }k=1,\,\ldots,\,n-1\hskip -3em
\tag6.5
$$
(compare with the relationship \thetag{4.3} that defines normal vector to
hypersurfaces in Finslerian geometry). From \thetag{6.5} for $t=0$
we get
$$
\xalignat 2
&\varphi_k(t)\,\hbox{\vrule height 8pt depth 8pt
width0.5pt}_{\,t=0}=0,
&&\dot\varphi_k(t)\,\hbox{\vrule height 8pt depth 8pt
width0.5pt}_{\,t=0}=0\hskip -2em
\tag6.6
\endxalignat
$$
The equalities of first series in \thetag{6.6} are fulfilled
unconditionally due to initial data \thetag{6.1} defining the
trajectories of shift. Second series of equalities in \thetag{6.6}
isn't a consequence of \thetag{6.1}. On a base of them we formulate
{\bf additional normality condition} for Newtonian dynamical
systems.
\definition{Definition 6.2} Suppose that Finslerian manifold $M$ of
the dimension $n\geqslant 2$ is equipped with some Newtonian dynamical
system, which is used to arrange the shift of hypersurfaces along its
trajectories according to the initial data \thetag{6.1}. We say that
this dynamical system satisfies {\bf additional normality} condition,
if for any hypersurface $S$ in $M$, for any point $p_0$ on $S$, and
for any real number $\nu_0\neq 0$ there exists some smaller piece $S'$
of hypersurface $S$ belonging to transformation class and containing
the point $p_0$, and there exists some function $\nu(p)$ from
transformation class on $S'$ normalized by the condition \thetag{6.2}
and such that, when substituted to \thetag{6.1}, it defines the
shift $f_t\!:\,S'\to S'_t$ such that 
$$
\dot\varphi_k(t)\,\hbox{\vrule height 8pt depth 8pt width
0.5pt}_{\,t=0}=0\text{, \ \ where \ \ \ }k=1,\,\ldots,\,n-1.
\hskip -2em
\tag6.7
$$
Here $\varphi_k=(\boldsymbol\tau_k\,|\,\bold v)$ are the functions of
deviation defined by some choice of local coordinates $u^1,\,\ldots,
\,u^{n-1}$ on $S$ and corresponding vectors of variation $\boldsymbol
\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$.
\enddefinition
     As well as in Riemannian geometry, here the definition~6.2 formulates
a condition, which is additional to the condition of {\bf weak normality}.
The latter is formulated as follows.
\definition{Definition 6.3} We say that Newtonian dynamical system
on Finslerian manifold $M$ of the dimension $n\geqslant 2$ satisfies
{\bf weak normality} condition, if for each its trajectory there exists
some ordinary differential equation
$$
\pagebreak
\ddot\varphi=\Cal A(t)\,\dot\varphi+\Cal B(t)\,\varphi\hskip -2em
\tag6.8
$$
such that any function of deviation $\varphi(t)$ corresponding to any
choice of variation vector $\boldsymbol\tau$ on that trajectory is
the solution of this equation.
\enddefinition
\noindent Let's recall that variation vector $\boldsymbol\tau$ appears
in considering some family of trajectories parameterized by some variable
$u$. If in local coordinates $x^1,\,\ldots,\,x^n$ in $M$ trajectories
are given by functions $x^i(t,u)$, then components of $\boldsymbol\tau$
are calculated as derivatives $\partial x^i/\partial u$. Vectors
$\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ that arise in
considering shift $f_t\!:\,S\to S_t$ are special cases of this general
construction. In case of Newtonian dynamical systems on Finslerian
manifold $M$ arbitrary vector of variation $\boldsymbol\tau$ for its
trajectories satisfies ordinary differential equation of the form
$$
\aligned
\nabla_{tt}&\boldsymbol\tau=C(\nabla_t\boldsymbol\tau\otimes
\tilde\nabla\bold F)+C(\boldsymbol\tau\otimes\nabla\bold F)-\\
\vspace{0.5ex}
&-\bold R(\boldsymbol\tau,\bold v)\bold v-\bold D(\boldsymbol\tau,
\bold F)\bold v+\bold D(\bold v,\nabla_t\boldsymbol\tau)
\bold v.
\endaligned\hskip -2em
\tag6.9
$$
Derivation of these equations \thetag{6.9} does not differ from
derivation of the equations \thetag{3.10} in Chapter
\uppercase\expandafter {\romannumeral 5}. Thereby we use formulas
\thetag{5.14} and \thetag{5.15} in Chapter
\uppercase\expandafter {\romannumeral 4}. In local coordinates
the equation \thetag{6.9} is written as 
$$
\gathered
\nabla_{tt}\tau^k=\sum^n_{m=1}\nabla_t\tau^m\,\tilde\nabla_mF^k
+\sum^n_{m=1}\tau^m\,\nabla_mF^k-\\
-\sum^n_{m=1}\sum^n_{i=1}\sum^n_{j=1}\bigl(R^k_{mij}\,\tau^i\,v^j
+D^k_{mij}\bigl(\tau^i\,F^j-v^i\,\nabla_t\tau^j\bigr)\bigr)v^m.
\endgathered\hskip -2em
\tag6.10
$$\par
    Each vector of  variation  $\boldsymbol\tau$  determines  the 
function
of deviation $\varphi$, which is formed as scalar product of two vectors
$\boldsymbol\tau$ and $\bold v$:
$$
\varphi=(\boldsymbol\tau\,|\,\bold v).\hskip -3em
\tag6.11
$$
Scalar product \thetag{6.11} is linear with respect to $\boldsymbol\tau$,
but it isn't linear with respect to its second argument $\bold v$, since
metric tensor by itself depends on $\bold v$:
$$
\varphi=\sum^n_{i=1}\sum^n_{j=1}\tau^i\,g_{ij}(x^1,\ldots,x^n,\bold v)
\,v^j.\hskip -3em
\tag6.12
$$
Formula \thetag{6.12} is in a complete agreement with formula \thetag{6.4}
above, and with definition of normal vector to hypersurfaces in Finslerian
manifolds  (see definition~4.1). In differentiating with \thetag{6.12}
with respect to $t$ here we get
$$
\dot\varphi=(\bold F\,|\,\boldsymbol\tau)+(\bold v\,|\,
\nabla_t\boldsymbol\tau)+\sum^n_{i=1}\sum^n_{j=1}
\sum^n_{k=1}\tilde\nabla_kg_{ij}\,\tau^i\,v^j\,F^k.
$$
This formula contain additional term that differs it from corresponding
formula \thetag{4.7} in Chapter \uppercase\expandafter {\romannumeral 5}.
But due to \thetag{1.4} and \thetag{1.5} this term appears to vanish.
Therefore
$$
\dot\varphi=(\boldsymbol\tau\,|\,\bold F)+(\nabla_t\boldsymbol\tau
\,|\,\bold v).\hskip -3em
\tag6.13
$$
In calculating second derivative of deviation function $\ddot\varphi$
we also get additional terms:
$$
\aligned
\ddot\varphi&=(\boldsymbol\tau\,|\,\nabla_t\bold F)
+2\,(\nabla_t\boldsymbol\tau\,|\,\bold F)+(\nabla_{tt}\boldsymbol\tau\,
|\,\bold v)\,+\\
\vspace{1ex}
&+\,\sum^n_{i=1}\sum^n_{j=1}\sum^n_{k=1}\tilde\nabla_kg_{ij}\,
\bigl(\tau^i\,F^j+\nabla_t\tau^i\,v^j\bigr)\,F^k,
\endaligned
\hskip -3em
\tag6.14
$$
Such terms differ \thetag{6.14} from corresponding formula \thetag{4.8}
in Chapter \uppercase\expandafter {\romannumeral 5}. Some of them
vanish de to the equalities \thetag{1.4} and \thetag{1.5}. Therefore
\thetag{6.14} is rewritten as
$$
\ddot\varphi=(\boldsymbol\tau\,|\,\nabla_t\bold F)
+2\,(\nabla_t\boldsymbol\tau\,|\,\bold F)
+(\nabla_{tt}\boldsymbol\tau\,|\,\bold v)\,+
2\,\bold C(\boldsymbol\tau,\bold F,\bold F).
\hskip -2em
\tag6.15
$$
Here $\bold C(\boldsymbol\tau,\bold F,\bold F)$ is a value of
completely symmetric trilinear form defined by Cartan tensor
$\bold C$.\par
    In order to calculate second derivative of variation vector
$\nabla_{tt}\boldsymbol\tau$ in \thetag{6.15} we use formula
\thetag{6.9}, while to calculate $\nabla_t\bold F$ we apply formula
\thetag{4.10} from Chapter \uppercase\expandafter {\romannumeral 5}.
This formula is a special case of formula \thetag{4.3} from
Chapter \uppercase\expandafter {\romannumeral 4}, it is true
irrespective to the metric, in Riemannian geometry and in
Finslerian geometry as well. Taking into account these formulas
we transform \thetag{6.15} to
$$
\gather
\ddot\varphi=(C(\bold v\otimes\nabla\bold F)\,|\,\boldsymbol\tau)+
(C(\bold F\otimes\tilde\nabla\bold F)\,|\,\boldsymbol\tau)
+2\,(\bold F\,|\,\nabla_t\boldsymbol\tau)\,+\\
+\,(\bold v\,|\,C(\boldsymbol\tau\otimes\nabla\bold F))
+(\bold v\,|\,C(\nabla_t\boldsymbol\tau\otimes\tilde\nabla\bold F))
-(\bold v\,|\,\bold R(\boldsymbol\tau,\bold v)\bold v)\,-\\
-\,(\bold v\,|\,\bold D(\boldsymbol\tau,\bold F)\bold v)
+(\bold v\,|\,\bold D(\bold v,\nabla_t\boldsymbol\tau)\bold v)
+2\,\bold C(\boldsymbol\tau,\bold F,\bold F).
\endgather
$$
This formula differs from corresponding formula \thetag{4.10}
in Chapter \uppercase\expandafter {\romannumeral 5} by last
three terms. Two terms $(\bold v\,|\,\bold D(\boldsymbol\tau,
\bold F)\bold v)$ and $(\bold v\,|\,\bold D(\bold v,\nabla_t
\boldsymbol\tau)\bold v)$ in it are zero due to theorem~3.1.
Term $(\bold v\,|\,\bold R(\boldsymbol\tau,\bold v)\bold v)$
is zero due to theorem~3.2. Therefore 
$$
\gathered
\ddot\varphi=(C(\bold v\otimes\nabla\bold F)\,|\,\boldsymbol\tau)+
(C(\bold F\otimes\tilde\nabla\bold F)\,|\,\boldsymbol\tau)\,+\\
+\,2\,(\bold F\,|\,\nabla_t\boldsymbol\tau)
+(\bold v\,|\,C(\boldsymbol\tau\otimes\nabla\bold F))\,+\\
+\,(\bold v\,|\,C(\nabla_t\boldsymbol\tau\otimes\tilde\nabla\bold F))
+2\,\bold C(\boldsymbol\tau,\bold F,\bold F).
\endgathered\hskip -3em
\tag6.16
$$
When written in local coordinates, formula \thetag{6.16} looks like
$$
\pagebreak
\align
\ddot\varphi&=\sum^n_{i=1}\sum^n_{j=1}\left(2\,g_{ij}\,F^j
+\shave{\sum^n_{k=1}}v^j\,g_{jk}\tilde\nabla_iF^k\right)
\nabla_t\tau^i\,+\\
\vspace{1.5ex}
&+\,\sum^n_{i=1}\sum^n_{j=1}\sum^n_{k=1}\left(v^k
\vphantom{\,F^k\,\tilde\nabla_kF^j\,g_{ij}+2\,C_{ijk}\,F^k\,F^j}
\left(\nabla_kF^j\,g_{ij}+\nabla_iF^j\,g_{jk}\right)\,+\right.\\
&\left.\hskip 7em+\,F^k\,\tilde\nabla_kF^j\,g_{ij}+2\,C_{ijk}
\,F^j\,F^k\right)\tau^i.
\endalign
$$
Contracting with metric tensor we lower indices (this correspond to
transfer from vectors to covectors). Such operation commutates with
covariant differentiation $\nabla$. But in Finslerian geometry it
doesn't commutate with second covariant differentiation $\tilde\nabla$,
since $\tilde\nabla\bold g\neq 0$. Therefore 
$$
\gather
\sum^n_{k=1}F^k\,\tilde\nabla_kF_i=\sum^n_{j=1}\sum^n_{k=1}
\left(F^k\,\tilde\nabla_kF^j\,g_{ij}+2\,F^k\,C_{ijk}\,F^j\right),\\
\vspace{1ex}
\sum^n_{j=1}v^j\,\tilde\nabla_iF_j=\sum^n_{j=1}\sum^n_{k=1}
\left(v^j\,g_{jk}\,\tilde\nabla_iF^k+2\,F^k\,C_{ijk}\,v^j\right).
\endgather
$$
Taking into account these formulas and taking into account the
equality \thetag{1.5} for Cartan tensor, we transform formula
for $\ddot\varphi$ to 
$$
\gathered
\ddot\varphi=\sum^n_{i=1}\left(2\,F_i+\shave{\sum^n_{j=1}} v^j\,
\tilde\nabla_iF_j\right)\nabla_t\tau^i\,+\hskip -2em\\
+\,\sum^n_{i=1}\left(\,\shave{\sum^n_{j=1}}v^j\left(\nabla_jF_i+
\nabla_iF_j\right)+\shave{\sum^n_{j=1}} F^j\,\tilde\nabla_jF_i
\right)\tau^i.\hskip -2em
\endgathered
\tag6.17
$$
This form of formula \thetag{6.15} is identical to corresponding
formula \thetag{4.16} in Chapter
\uppercase\expandafter {\romannumeral 5}. Further we introduce
covector fields $\boldsymbol\alpha$ and $\boldsymbol\beta$ with
components 
$$
\aligned
\alpha_i&=2\,F_i+\sum^n_{j=1} v^j\,\tilde\nabla_iF_j,\hskip -2em\\
\beta_i&=\sum^n_{j=1}v^j\left(\nabla_jF_i+
\nabla_iF_j\right)+\sum^n_{j=1} F^j\,\tilde\nabla_jF_i.\hskip -2em
\endaligned
\tag6.18
$$
Then formulas for derivatives $\dot\varphi$ and $\ddot\varphi$
are transformed to 
$$
\align
&\dot\varphi=(\bold F\,|\,\bold P\,\boldsymbol\tau)+
\frac{(\bold F\,|\,\bold N)}{v}\,(\bold v\,|\,\boldsymbol\tau)+
(\bold v\,|\,\nabla_t\boldsymbol\tau).\hskip -3em
\tag6.19\\
\vspace{2ex}
&\ddot\varphi=\boldsymbol\alpha(\bold P\nabla_t\boldsymbol\tau)
+\boldsymbol\beta(\bold P\boldsymbol\tau)
+\frac{\boldsymbol\alpha(\bold N)}{v}\,(\bold v\,|\,
\nabla_t\boldsymbol\tau)+\frac{\boldsymbol\beta(\bold N)}{v}\,
(\bold v\,|\,\boldsymbol\tau).
\hskip -3em
\tag6.20
\endalign
$$
Further calculations, which are used to derive {\bf weak normality}
equations, do not differ from corresponding calculations in
Riemannian geometry (see \S\,6 in Chapter
\uppercase\expandafter {\romannumeral 5}). We reproduce them here
omitting details. Substituting \thetag{6.11}, \thetag{6.19}, and
\thetag{6.20} into the differential equation \thetag{6.8}, we get
$$
\gathered
\left(\frac{\boldsymbol\alpha(\bold N)}{v}-\Cal A\right)\,
(\bold v\,|\,\nabla_t\boldsymbol\tau)+
\boldsymbol\alpha(\bold P\nabla_t\boldsymbol\tau)
+\boldsymbol\beta(\bold P\boldsymbol\tau)\,-\\
\vspace{1ex}
-\,\Cal A\,(\bold F\,|\,\bold P\,\boldsymbol\tau)
+\left(\frac{\boldsymbol\beta(\bold N)}{v}-\Cal A\,
\frac{(\bold F\,|\,\bold N)}{v}-\Cal B\right)\,
(\bold v\,|\,\boldsymbol\tau)=0.
\endgathered\hskip -1.5em
\tag6.21
$$
From \thetag{6.21} one can derive four equations, two of them
determine coefficients $\Cal A$ and $\Cal B$ in ordinary
differential equation \thetag{6.8}:
$$
\xalignat 2
&\quad\Cal A=\frac{\boldsymbol\alpha(\bold N)}{v},
&&\Cal B=\frac{\boldsymbol\beta(\bold N)}{v}-\Cal A\,
\frac{(\bold F\,|\,\bold N)}{v}.\hskip -3em
\tag6.22
\endxalignat
$$
Two other equations have the following form:
$$
\xalignat 2
&\quad\boldsymbol\alpha(\bold P\nabla_t\boldsymbol\tau)=0,
&&\boldsymbol\beta(\bold P\boldsymbol\tau)=\Cal A\,(\bold F\,|\,
\bold P\,\boldsymbol\tau).\hskip -3em
\tag6.23
\endxalignat
$$
If we take into account that $\boldsymbol\tau$ and
$\nabla_t\boldsymbol\tau$ are two arbitrary vectors, then
from \thetag{6.23} we derive two weak normality equations.
In local coordinates they are written as
$$
\align
&\sum^n_{i=1}\left(v^{-1}\,F_i+\shave{\sum^n_{j=1}}
\tilde\nabla_i\left(N^j\,F_j\right)\right)P^i_k=0,\hskip -2em
\tag6.24\\
\vspace{1.5ex}
&\aligned
&\sum^n_{i=1}\sum^n_{j=1}\left(\nabla_iF_j+\nabla_jF_i-2\,v^{-1}
\,F_i\,F_j\right)N^j\,P^i_k\,+\\
\vspace{0.5ex}
&+\sum^n_{i=1}\sum^n_{j=1}\left(\frac{F^j\,\tilde\nabla_jF_i}{v}
-\sum^n_{r=1}\frac{N^r\,N^j\,\tilde\nabla_jF_r}{v}\,F_i\right)P^i_k=0.
\endaligned\hskip -3em
\tag6.25
\endalign
$$
The equations \thetag{6.24} and \thetag{6.25} are identical to weak
normality condition \thetag{6.16} and \thetag{6.17} in Chapter
\uppercase\expandafter {\romannumeral 5}. Here we have theorem
analogous to theorem~6.2 in Chapter
\uppercase\expandafter {\romannumeral 5}.
\proclaim{Theorem 6.1} Newtonian dynamical system on Finslerian
manifold $M$ of the dimension $n\geqslant 2$ satisfies weak
normality condition if and only if its force field satisfies
the equations \thetag{6.24} and \thetag{6.25} at all points
$q=(p,\bold v)$ of tangent bundle $TM$, where $\bold v\neq 0$.
\endproclaim
    Analysis of additional normality condition in Finslerian geometry
is more complicated than analysis of weak normality condition (though
this is true in Riemannian geometry too). Let's begin with applying
formula \thetag{6.13} for to transform the equation \thetag{6.7}.
Thereby we take into account the relationship \thetag{7.8} from
Chapter \uppercase\expandafter {\romannumeral 5}:
$$
\frac{\partial\nu}{\partial u^k}=-\nu^{-1}\,(\boldsymbol\tau_k\,|
\,\bold F)-\nu\,(\nabla_{u^k}\bold n\,|\,\bold n).\hskip -3em
\tag6.26
$$
Here $\nu$ is a function on hypersurface $S$ that determines modulus
of velocity vector at initial instant of time, $\bold n$ is a normal
vector to hypersurface $S$, $u^1,\,\ldots,\,u^{n-1}$ are local
coordinates on $S$ and $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol
\tau_{n-1}$ are corresponding coordinate tangent vectors (vectors
of variations). Covariant derivative $\nabla_{u^k}\bold n$ is a
derivative of vector $\bold n$ with respect to parameter $\bold u^k$
along coordinate lines on hypersurface $S$ (see \S\,5 in Chapter
\uppercase\expandafter {\romannumeral 4}). Components of metric tensor
and components of Cartan connection in Finslerian geometry depend on
velocity vector $\bold v$. Therefore in order to calculate
covariant derivative $\nabla_{u^k}\bold n$ and in order to calculate
scalar products in \thetag{6.26} we should fix some lift of curves
from $M$ to $TM$. Due to the origin of the equations \thetag{6.26}
such lift is defined by velocity vector on trajectories of shift
$f_t\!:\,S\to S_t$, on initial hypersurface $S$ this lift is
determined by normal vector $\bold n$, since $\bold v=\nu\,\bold n$.
\par
    In two-dimensional case $n=2$ the dimension of hypersurface $S$
is equal to unity. The number of equations \thetag{6.26} is also
equal to unity, while partial derivative $\partial\nu/\partial u^1$
turns to ordinary derivative. Therefore we have theorem similar
to theorem~9.1 in Chapter \uppercase\expandafter {\romannumeral 5}.
\proclaim{Theorem 6.2} In two-dimensional case $\dim M=2$ additional
normality condition is fulfilled unconditionally for arbitrary
Newtonian dynamical system on Finslerian manifold $M$.
\endproclaim
Analysis of multidimensional case $\dim M\geqslant 3$ requires
the use of theory of hypersurfaces in Finslerian manifolds.
\head
\S\,7. Induced Riemannian metric\\and derivational formulas of Weingarten.
\endhead
\rightheadtext{\S\,7. Induced Riemannian metric \dots}
     Let $S$ be hypersurface in Finslerian manifold of the dimension
$\dim M\geqslant 3$ and let $u^1,\,\ldots,\,u^{n-1}$ be local coordinates
on $S$. Local coordinates $u^1,\,\ldots,\,u^{n-1}$ define coordinate
network on $S$, coordinate tangent vectors $\boldsymbol\tau_1,\,\ldots,
\,\boldsymbol\tau_{n-1}$ being tangent to the lines of this network.
In \S\,5 above we showed that there exists a normal vector $\bold n$
perpendicular to vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol
\tau_{n-1}$. It is determined uniquely up to a change of sign
$\bold n\to\pm\,\bold n$, this arbitrariness is eliminated by
fixing orientation on $S$.\par
    Completing the set of vectors $\boldsymbol\tau_1,\,\ldots,\,
\boldsymbol\tau_{n-1}$ by unitary normal vector $\bold n$, we get
moving frame of hypersurface of $S$. Vectors of such frame are
defined on the lines of coordinate network of $S$. Therefore we can
differentiate them with respect to parameters $u^1,\,\ldots,\,u^{n-1}$
along coordinate lines. In Chapter
\uppercase\expandafter {\romannumeral 4} we defined {\bf two types} of
covariant differentiations with respect to parameter along the curves.
{\bf First type} of differentiation is determined by canonical lift of
curves from $M$ to $TM$. Here parameter $t$ is interpreted as time
variable, tangent vector to curve is interpreted as velocity vector.
{\bf Second type} of covariant differentiation corresponds to
deformations of curves, when, apart from main parameter $t$, we have
parameter of deformation $u$. This is the very type of covariant
differentiation naturally arises in the equation \thetag{6.26}. In
considering separate hypersurface $S$, which is not included into the
scheme of normal shift, parameter $t$ doesn't arise. But here we
have normal vector $\bold n$, which determines the direction of
velocity vector in case if we want to initiate normal shift:
$$
\bold v=\nu\cdot\bold n.\hskip -2em
\tag7.1
$$
Components of metric tensor of Finslerian metric $\bold g$ and
components of Cartan connection are homogeneous functions of degree
$0$ respective to the vector $\bold v$. Therefore from \thetag{7.1}
we get
$$
\align
g_{ij}(x^1,\ldots,x^n,\bold v)&=g_{ij}(x^1,\ldots,x^n,\bold n).
\hskip -2em
\tag7.2\\
\vspace{0.5ex}
\Gamma^k_{ij}(x^1,\ldots,x^n,\bold v)&=\Gamma^k_{ij}(x^1,\ldots,x^n,
\bold n).\hskip -2em
\tag7.3
\endalign
$$
Therefore we can replace canonical lift of curves by {\bf normal lift}
in calculating covariant derivatives $\nabla_u$ for the families of
curves on $S$.\par
     Let $\bold X$ be smooth tensor field of the type $(r,s)$ defined
at the points of hypersurface $S$. This means that to each point
$p\in S$ one puts into correspondence some tensor $\bold X(p)\in
T^r_s(p,M)$. Suppose that we have some parametric curve with parameter
$u$ on $S$, and let $\boldsymbol\tau$ be a tangent vector to this curve.
Restriction of the field $\bold X$ to this curve we denote by $\bold
X(u)$. Applying differentiation $\nabla_u$ to $\bold X(u)$ we get
tensor field $\nabla_u\bold X(u)$of the type $(r,s)$ on the curve.
Its components are calculated by formula
$$
\align
\nabla_u&X^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}=
\frac{dX^{i_1\ldots\,i_r}_{j_1\ldots\,j_s}}{du}+\\
\vspace{1ex}
&+\sum^r_{k=1}\sum^n_{m=1}\sum^n_{a_k=1}\tau^m\,\Gamma^{i_k}_{m\,a_k}
\,X^{i_1\ldots\,a_k\ldots\,i_r}_{j_1\ldots\,\ldots\,\ldots\,j_s}-
\hskip -2em
\tag7.4\\
\vspace{1ex}
&-\sum^s_{k=1}\sum^n_{m=1}\sum^n_{b_k=1}\tau^m\,\Gamma^{b_k}_{m\,j_k}
X^{i_1\ldots\,\ldots\,\ldots\,i_r}_{j_1\ldots\,b_k\ldots\,j_s},
\hskip -2em
\endalign
$$
where $\Gamma^k_{ij}=\Gamma^k_{ij}(x^1,\ldots,x^n,\bold n)$, and
$\tau^1,\,\ldots,\,\tau^n$ are components of tangent vector
$\boldsymbol\tau$ corresponding to parameter $u$.\par
    Let $\bold X=\bold X(q)$ be a tensor field of type $(r,s)$
from extended algebra of tensor fields on $M$. If to each point
$p\in S$ we put into correspondence point $q=(p,\bold n)$ from
tangent bundle $TM$, then we get {\bf normal lift} of hypersurface
$S$ to $TM$. This allows us to define the restriction of extended
tensor field $\bold X$ to $S$. The derivative $\nabla_u\bold X$
for such restriction can be calculated by formula
$$
\nabla_u\bold X=C(\tau\otimes\nabla\bold X)+C(\nabla_u\bold n\otimes
\tilde\nabla\bold X).\hskip -2em
\tag7.5
$$
Formula \thetag{7.5} is proved by direct calculation in local coordinates
on a base of formula \thetag{7.4} and formulas \thetag{7.3} and \thetag{7.4}
from Chapter
\uppercase\expandafter {\romannumeral 3}.\par
     Suppose that on hypersurface $S$ some local coordinates $u^1,\,\ldots,
\,u^{n-1}$ are chosen. Let's apply differentiation $\nabla_{u^k}$ to
coordinate tangent vector field $\boldsymbol\tau_r$. As a result we get
vector field on $S$, it can be expanded in the base of vectors
$\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ and $\bold n$:
$$
\nabla_{u^k}\boldsymbol\tau_r=\sum^{n-1}_{m=1}\theta^m_{kr}\,
\boldsymbol\tau_m+b_{kr}\,\bold n,\hskip -2em
\tag7.6
$$
Formula \thetag{7.6} is analog of derivational formula \thetag{7.9},
which we used in Chapter \uppercase\expandafter {\romannumeral 5}.
Let's find geometrical interpretation for the quantities $\theta^m_{kr}$
and $b_{kr}$ in right hand side of \thetag{7.6}. Changing local
coordinates $u^1,\,\ldots,\,u^{n-1}$ by $\tilde u^1,\,\ldots,\,
\tilde u^{n-1}$ and taking into account corresponding change of
base vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$
by $\tilde{\boldsymbol\tau}_1,\,\ldots,\,\tilde{\boldsymbol\tau}_{n-1}$,
one easily finds that $\theta^m_{kr}$ are components of affine
connection, and $b_{kr}$ are components of tensor field in $S$. From
symmetry of $\Gamma^k_{ij}$ in \thetag{7.4} we derive symmetry
of $\theta^m_{kr}$ and $b_{kr}$:
$$
\xalignat 2
&\theta^m_{kr}=\theta^m_{rk},&&b_{kr}=b_{rk}.\hskip -2em
\tag7.7
\endxalignat
$$
Finslerian metric $\bold g$ is an extended tensor field of type
$(0,2)$ on $M$. Using the construction of normal lift of $S$ to
$TM$, we restrict $\bold g$ to $S$; and by means of such restriction
we define scalar product of coordinate vector fields on $S$:
$$
\rho_{kr}=(\boldsymbol\tau_k\,|\,\boldsymbol\tau_r)=
\sum^n_{i=1}\sum^n_{j=1}g_{ij}(x^1,\ldots,x^n,\bold n)
\,\tau^i_k\,\tau^j_r.\hskip -2em
\tag7.8
$$
Quantities $\rho_{kr}$ in \thetag{7.8} are components of Riemannian
metric $\boldsymbol\rho$ in inner geometry of the manifold $S$.
The condition $|\bold n|=1$ is written in terms of scalar product:
$$
(\bold n\,|\,\bold n)=\sum^n_{i=1}\sum^n_{j=1}g_{ij}(x^1,\ldots,x^n,
\bold n)\,n^i\,n^j=1.\hskip -2em
\tag7.9
$$
Let's differentiate the relationship \thetag{7.9} with respect to $u^k$:
$$
\gather
\frac{\partial (\bold n\,|\,\bold n)}{\partial u^k}=\nabla_{u^k}
(\bold n\,|\,\bold n)=\nabla_{u^k}C(\bold n\otimes\bold n\otimes
\bold g)=\\
\vspace{1ex}
=(\nabla_{u^k}\bold n\,|\,\bold n)+(\bold n\,|\,\nabla_{u^k}\bold n)+
C(\bold n\otimes\bold n\otimes\nabla_{u^k}\bold g)=0.
\endgather
$$
In order to calculate $\nabla_{u^k}\bold g$ in this equality we use
formula \thetag{7.5} and take into account that $\nabla\bold g=0$
and $\tilde\nabla\bold g=\bold C$ (see formula \thetag{1.4} above). 
As a result we get 
$$
2\,(\nabla_{u^k}\bold n\,|\,\bold n)+2\,\bold C(\bold n,\bold n,
\nabla_{u^k}\bold n)=0.\hskip -3em
\tag7.10
$$
Second term in left hand side of \thetag{7.10} is a result of
contracting Cartan tensor with two copies of vector $n$ and with
vector $\nabla_{u^k}\bold n$. It is zero due to the relationship
\thetag{1.5}:
$$
\bold C(\bold n,\bold n,\nabla_{u^k}\bold n)
=\sum^n_{i=1}\sum^n_{j=1}\sum^n_{s=1}
C_{ijs}(x^1,\ldots,x^n,\bold n)\,n^i\,n^j\,\nabla_{u^k}n^s=0.
$$
Then from \thetag{7.10} we derive $(\nabla_{u^k}\bold n\,|\,\bold n)=0$.
This means that vector $\nabla_{u^k}\bold n$ is perpendicular to
$\bold n$, i\.~e\. it is in tangent hyperplane to $S$. Therefore
it is given by some linear combination of tangent vectors $\boldsymbol
\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$:
$$
\nabla_{u^k}\bold n=\sum^{n-1}_{m=1}d^m_k\,\boldsymbol\tau_m.
\hskip -3em
\tag7.11
$$
Coefficients $d^m_k$ in linear combination \thetag{7.11} are components
of tensor field of type $(1,1)$ in inner geometry of manifold $M$. In
order to calculate these coefficients let's write the equality
$(\boldsymbol\tau_r\,|\,\bold n)=0$, which means $\boldsymbol\tau_r
\perp\bold n$, and let's differentiate this equality with respect to
the parameter $u^k$:
$$
\gather
\frac{\partial (\boldsymbol\tau_r\,|\,\bold n)}{\partial u^k}=
\nabla_{u^k}(\boldsymbol\tau_r\,|\,\bold n)=\nabla_{u^k}
C(\boldsymbol\tau_r\otimes\bold n\otimes\bold g)=\\
\displaybreak
=(\nabla_{u^k}\boldsymbol\tau_r\,|\,\bold n)+(\boldsymbol\tau_r\,|
\,\nabla_{u^k}\bold n)+C(\boldsymbol\tau_r\otimes\bold n\otimes
\nabla_{u^k}\bold g)=\\
\vspace{1ex}
=(\nabla_{u^k}\boldsymbol\tau_r\,|\,\bold n)+(\boldsymbol\tau_r\,|
\,\nabla_{u^k}\bold n)+2\,\bold C(\boldsymbol\tau_r,\bold n,
\nabla_{u^k}\bold n)=0.
\endgather
$$
Last term in left hand side of this equality is zero due to
the identity \thetag{1.5}:
$$
\bold C(\boldsymbol\tau_r,\bold n,\nabla_{u^k}\bold n)
=\sum^n_{i=1}\sum^n_{j=1}\sum^n_{s=1}
C_{ijs}(x^1,\ldots,x^n,\bold n)\,n^i\,\tau^j_r\,\nabla_{u^k}n^s=0.
$$
Therefore $(\nabla_{u^k}\boldsymbol\tau_r\,|\,\bold n)+(\boldsymbol
\tau_r\,|\,\nabla_{u^k}\bold n)=0$. Now by means of derivational
formulas \thetag{7.6} and \thetag{7.11} for $\nabla_{u^k}\boldsymbol
\tau_r$ and $\nabla_{u^k}\bold n$ we get
$$
b_{kr}=-\sum^{n-1}_{m=1}\rho_{rm}\,d^m_k.\hskip -3em
\tag7.12
$$
Formula \thetag{7.12} can be inverted. Then coefficients $d^m_k$
from \thetag{7.11} appear to be expressed through components of
second fundamental form:
$$
d^m_k=-b^m_k=-\sum^{n-1}_{r=1}\rho^{mr}\,b_{rk}.\hskip -3em
\tag7.13
$$
Finally derivational formulas \thetag{7.6} and \thetag{7.11}
are written as 
$$
\align
\nabla_{u^k}\boldsymbol\tau_r&=\sum^{n-1}_{m=1}\theta^m_{kr}\,
\boldsymbol\tau_m+b_{kr}\,\bold n,\hskip -3em
\tag7.14\\
\nabla_{u^k}\bold n&=-\sum^{n-1}_{m=1}b^m_k\,\boldsymbol\tau_m.
\hskip -3em
\tag7.15
\endalign
$$
Now let's study how Riemannian metric with components \thetag{7.8} and
symmetric affine connection $\theta^m_{kr}$ in \thetag{7.14} are
bound with each other. Let's calculate derivatives
$$
\gather
\frac{\partial\rho_{kr}}{\partial u^s}=
\nabla_{u^s}(\boldsymbol\tau_k\,|\,\boldsymbol\tau_r)=
\nabla_{u^s}C(\boldsymbol\tau_k\otimes\boldsymbol\tau_r\otimes
\bold g)=\\
\vspace{1ex}
=(\nabla_{u^s}\boldsymbol\tau_k\,|\,\boldsymbol\tau_r)
+(\boldsymbol\tau_k\,|\,\nabla_{u^s}\boldsymbol\tau_r)
+C(\boldsymbol\tau_k\otimes\boldsymbol\tau_r\otimes
\nabla_{u^s}\bold g).
\endgather
$$
We use derivational formula \thetag{7.14} for to calculate
$\nabla_{u^s}\boldsymbol\tau_k$ and $\nabla_{u^s}\boldsymbol\tau_r$, and
we use formula \thetag{7.5} to transform $\nabla_{u^s}\bold g$. Then we
obtain 
$$
\frac{\partial\rho_{kr}}{\partial u^s}-\sum^{n-1}_{m=1}
\theta^m_{sk}\rho_{mr}-\sum^{n-1}_{m=1}\theta^m_{sr}\rho_{km}=
-\sum^{n-1}_{m=1}2\,c_{krm}\,b^m_s.\hskip -3em
\tag7.16
$$
The quantities $c_{krm}$ in \thetag{7.16} \pagebreak are obtained by
contracting Cartan tensor $\bold C$ with three coordinate tangent
vectors $\boldsymbol\tau_k$, $\boldsymbol\tau_r$, and $\boldsymbol\tau_m$:
$$
c_{krm}=\bold C(\boldsymbol\tau_k,\boldsymbol\tau_r,\boldsymbol\tau_m)=
\sum^n_{i=1}\sum^n_{j=1}\sum^n_{s=1}C_{ijs}(x^1,\ldots,x^n,\bold n)
\,\tau^i_k\,\tau^j_r\,\tau^s_m.
$$
They are components of tensor field $\bold c$ of type $(0,3)$ in inner
geometry of submanifold $S$. The relationship \thetag{7.16} in
coordinate free form is written as
$$
\nabla\boldsymbol\rho=-2\,C(\bold c\otimes\bold B).\hskip -3em
\tag7.17
$$
In  general case right hand side of \thetag{7.17} is nonzero, i\.~e\.
Riemannian metric $\boldsymbol\rho$ from \thetag{7.8} and connection
$\theta$ from \thetag{7.14} are not concordant. Tensor
$\boldsymbol\beta$ with components 
$$
\beta_{ijs}=\sum^{n-1}_{m=1}c_{ijm}\,b^m_s
$$
is a measure of such disconcordance. We can calculate $\theta^k_{ij}$
explicitly:
$$
\aligned
\theta^k_{ij}=\frac{1}{2}&\sum^{n-1}_{s=1}\rho^{ks}\left(
\frac{\partial\rho_{sj}}{\partial u^i}+\frac{\partial\rho_{is}}
{\partial u^j}-\frac{\partial\rho_{ij}}{\partial u^s}\right)\,+\\
&+\,\sum^{n-1}_{s=1}\rho^{ks}\left(\beta_{jsi}+\beta_{sij}-\beta_{ijs}
\right).
\endaligned
\hskip -3em
\tag7.18
$$
Formula \thetag{7.18} is an analog of well known formula for components
of metric connection. It generalizes formula \thetag{7.11} from
Chapter \uppercase\expandafter {\romannumeral 5} to the present situation.
\head
\S\,8. The additional normality equations.
\endhead
     Now let's apply the result of \S\,7 for further analysis of the
equations \thetag{6.26} derived from additional normality condition.
Scalar product $(\nabla_{u^k}\bold n\,|\,\bold n)$ in right hand side
of \thetag{6.26} vanishes due to derivational formulas \thetag{7.15}.
Therefore 
$$
\frac{\partial\nu}{\partial u^k}=-\nu^{-1}\,(\boldsymbol\tau_k
\,|\,\bold F).\hskip -2em
\tag8.1
$$
Vector field $\bold F$ can be represented by covector field $\bold F$
obtained by lowering index procedure. Here is the formula for components
of such field:
$$
F_i=\sum^n_{j=1}g_{ij}(x^1,\ldots,x^n,\bold v)\,F^j.
$$
Using covector field $\bold F$, we can eliminate explicit entry of
metric in right hand side of the equation \thetag{8.1}. Now this
equation is written as
$$
\frac{\partial\nu}{\partial u^k}=-\nu^{-1}\,
C(\boldsymbol\tau_k\otimes\bold F).\hskip -2em
\tag8.2
$$
In multidimensional case $\dim M\geqslant 3$ the equations \thetag{8.2}
form complete system of Pfaff equations with respect to scalar function
$\nu(u^1,\ldots,u^{n-1})$. Brief sketch of this theory is given in
\S\,8 of Chapter \uppercase\expandafter {\romannumeral 5}. Normalization
condition \thetag{6.2} with arbitrary constant $\nu_0$ means that
system of Pfaff equations \thetag{8.2} is compatible in the sense of
definition~8.1 from Chapter \uppercase\expandafter {\romannumeral 5}.
Hence for \thetag{8.2} compatibility equations are fulfilled (see equations
\thetag{8.7} in Chapter
\uppercase\expandafter {\romannumeral 5}). It order to write these
equations let's calculate second order derivatives of $\nu$ by virtue
of the equations \thetag{8.2}:
$$
\gather
\frac{\partial^2\nu}{\partial u^k\,\partial u^r}=-\nabla_{u^r}
C(\boldsymbol\tau_k\otimes\bold F)
=\frac{C(\boldsymbol\tau_k\otimes\bold F)}{\nu^2}\,\frac{\partial\nu}
{\partial u^r}\,-\\
\vspace{2ex}
-\,\nu^{-1}\,C(\nabla_{u^r}\boldsymbol\tau_k\otimes\bold F)
-\nu^{-1}\,C(\boldsymbol\tau_k\otimes\nabla_{u^r}\bold F)=\\
\vspace{2ex}
=-\frac{C(\boldsymbol\tau_k\otimes\bold F)\,C(\boldsymbol\tau_r\otimes
\bold F)}{\nu^3}-\frac{C(\nabla_{u^r}\boldsymbol\tau_k\otimes\bold F)}
{\nu}-\frac{C(\boldsymbol\tau_k\otimes\nabla_{u^r}\bold F)}{\nu}.
\endgather
$$
Let's transpose indices $r$ and $k$. Then we have 
$$
\gather
\frac{\partial^2\nu}{\partial u^r\,\partial u^k}=-\nabla_{u^k}
C(\boldsymbol\tau_r\otimes\bold F)
=\frac{C(\boldsymbol\tau_r\otimes\bold F)}{\nu^2}\,\frac{\partial\nu}
{\partial u^k}\,-\\
\vspace{2ex}
-\,\nu^{-1}\,C(\nabla_{u^k}\boldsymbol\tau_r\otimes\bold F)
-\nu^{-1}\,C(\boldsymbol\tau_r\otimes\nabla_{u^k}\bold F)=\\
\vspace{2ex}
=-\frac{C(\boldsymbol\tau_r\otimes\bold F)\,C(\boldsymbol\tau_k\otimes
\bold F)}{\nu^3}-\frac{C(\nabla_{u^k}\boldsymbol\tau_r\otimes\bold F)}
{\nu}-\frac{C(\boldsymbol\tau_r\otimes\nabla_{u^k}\bold F)}{\nu}.
\endgather
$$
Let's subtract one of these equations from another. Thereby we take into
account the relationship $\nabla_{u^k}\boldsymbol\tau_r=\nabla_{u^r}
\boldsymbol\tau_k$. This equality holds, since for to calculate covariant
both derivatives $\nabla_{u^k}$ and $\nabla_{u^r}$ we use the same lift of
curves given by the velocity vector on shift trajectories (or, what is
equivalent, by normal vector on initial hypersurface $S$). Then we get
$$
C(\boldsymbol\tau_k\otimes\nabla_{u^r}\bold F)=C(\boldsymbol
\tau_r\otimes\nabla_{u^k}\bold F).\hskip -2em
\tag8.3
$$
Compatibility equations \thetag{8.7} from Chapter
\uppercase\expandafter {\romannumeral 5} for the system of Pfaff
equations \thetag{8.2} mean that the relationships \thetag{8.3}
are fulfilled identically with respect to $n$ variables
$u^1,\,\ldots,\,u^{n-1}$ and $\nu$. Its important to note
that the value of function $\nu$ in \thetag{8.3} is understood
as independent variable. For covariant derivatives $\nabla_{u^k}
\bold F$ and $\nabla_{u^r}\bold F$ in \thetag{8.3} we use
formula \thetag{8.18} from Chapter
\uppercase\expandafter {\romannumeral 5}, which, in turn, is derived
from formula \thetag{5.19} in Chapter
\uppercase\expandafter {\romannumeral 4}: $\nabla_{u^k}\bold F
=C(\nabla_{u^k}\bold v\otimes\tilde\nabla\bold F)+C(\boldsymbol\tau_k
\otimes\nabla\bold F)$. In order to calculate covariant derivative
$\nabla_{u^k}\bold v$ on $S$ we use formula $\bold v=\nu\cdot\bold n$,
which holds on initial hypersurface $S$. And for covariant derivative
$\nabla_{u^k}\bold n$ we use derivational formula \thetag{7.15}. As
a result for $\nabla_{u^k}\bold v$ we get the relationship 
$$
\nabla_{u^k}\bold v=-\frac{C(\boldsymbol\tau_k\otimes\bold F)}{\nu}
\cdot\bold n-\nu\cdot\sum^{n-1}_{m=1}b^m_k\,\boldsymbol\tau_m
$$
coinciding with \thetag{8.19} in Chapter
\uppercase\expandafter {\romannumeral 5}. From \thetag{8.3} we obtain
$$
\gather
\frac{C(\boldsymbol\tau_k\otimes\bold F)\,C(\bold n\otimes\tilde\nabla
\bold F\otimes\boldsymbol\tau_r)}{\nu}
-\frac{C(\boldsymbol\tau_r\otimes\bold F)\,C(\bold n\otimes\tilde\nabla
\bold F\otimes\boldsymbol\tau_k)}{\nu}+C(\boldsymbol\tau_r\otimes\nabla
\bold F\otimes\boldsymbol\tau_k)\,-\\
\vspace{1ex}
-\,C(\boldsymbol\tau_k\otimes\nabla\bold F\otimes\boldsymbol\tau_r)
=\sum^{n-1}_{m=1}\nu\left(b^m_r\,C(\boldsymbol\tau_m\otimes\tilde\nabla
\bold F\otimes\boldsymbol\tau_k)-b^m_k\,C(\boldsymbol\tau_m\otimes\tilde
\nabla\bold F\otimes\boldsymbol\tau_r)\right).
\endgather
$$
Remember that here, as in \thetag{8.2}, by $\bold F$ we denote covector
field obtained from vector field $\bold F$ as a result of lowering index
by means of Finslerian metric $\bold g$. In coordinate form this equation
takes the form
$$
\gathered
\sum^n_{i=1}\sum^n_{j=1}\tau^i_k\,\tau^j_r\left(\,\shave{\sum^n_{m=1}}
n^m\,\frac{F_i\,\tilde\nabla_mF_j-F_j\,\tilde\nabla_mF_i}{\nu}+
\nabla_jF_i-\nabla_iF_j\right)=\\
\vspace{2ex}
=\sum^n_{i=1}\sum^n_{j=1}\sum^{n-1}_{m=1}\nu\left(b^m_r\,
\tau^j_m\tilde\nabla_jF_i\,\tau^i_k-b^m_k\,\tau^j_m\tilde\nabla_jF_i
\,\tau^i_r\right).
\endgathered\hskip -1.5em
\tag8.4
$$
which coincides with corresponding equation \thetag{8.21} in
Chapter \uppercase\expandafter {\romannumeral 5}. In what follows
the equations \thetag{8.4} should be subdivided into two parts.
For this goal we introduce two operators $\bold P$ and $\bold H$
with components
$$
\gather
P^i_\varepsilon=\sum^n_{s=1}\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}
g_{\varepsilon s}\,\tau^s_a\,\rho^{ak}\,\tau^i_k,\hskip -2em
\tag8.5\\
\vspace{1ex}
H^j_\varepsilon=\sum^n_{s=1}\sum^{n-1}_{a=1}\sum^{n-1}_{k=1}
\sum^{n-1}_{m=1}g_{\varepsilon s}\,\tau^s_a\,\rho^{ak}\,b^m_k\,
\tau^j_m.\hskip -2em
\tag8.6
\endgather
$$
Here $\rho^{ak}$ is determined by induced Riemannian metric
$\boldsymbol\rho$ on $S$ (see \S\,7 above), $b^m_k$ are components
of second fundamental form from \thetag{7.15}. Formulas \thetag{8.5}
and \thetag{8.6} are analogous to formulas \thetag{9.6} and \thetag{9.11}
in Chapter \uppercase\expandafter {\romannumeral 5}. The only difference
is that quantities $g_{\varepsilon s}$ depend on normal vector
$$
g_{\varepsilon s}=g_{\varepsilon s}(x^1,\ldots,x^n,\bold v)=
g_{\varepsilon s}(x^1,\ldots,x^n,\bold n).
$$
This is the feature of Finslerian metric. But this feature doesn't
affect further transformation of the equations \thetag{8.4}. First
they are brought to the form 
$$
\gathered
\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(\,\shave{
\sum^n_{m=1}}n^m\,\frac{F_i\,\tilde\nabla_mF_j-F_j\,\tilde\nabla_mF_i}
{\nu^2}\right.\,+\\
\vspace{2ex}
+\,\left.\frac{\nabla_jF_i-\nabla_iF_j}{\nu}\vphantom{\shave{\sum^n_{m=1}}
n^m\,\frac{F_i\,\tilde\nabla_mF_j-F_j\,\tilde\nabla_mF_i}{\nu}}\right)=
\sum^n_{i=1}\sum^n_{j=1}(P^i_\varepsilon\,H^j_\sigma-
P^i_\sigma\,H^j_\varepsilon)\,\tilde\nabla_jF_i
\endgathered\hskip -2em
\tag8.7
$$
coinciding with formula \thetag{9.12} from Chapter
\uppercase\expandafter {\romannumeral 5}. Then some properties
of the operators $\bold P$ and $\bold H$ defined by \thetag{8.5}
and \thetag{8.6} are used. We use the following ones:\par
\vskip 1ex
\parshape 2 0cm 12.8cm 0.9cm 11.9cm\noindent
\,\,(1) \ operators $\bold P$ and $\bold H$ are symmetric with respect
to the metric with components $g_{ij}=g_{ij}(x^1,\ldots,x^n,\bold n)$,
they are bound by the relationship
$$
\bold P\compos\bold H=\bold H\compos\bold P=\bold H,\hskip -2em
\tag8.8
$$\par
\parshape 2 0cm 12.8cm 0.9cm 11.9cm\noindent
\,\,(2) \ operator $\bold P$ is an orthogonal projector to the
hyperplane tangent to $S$.\par
\vskip 1ex
    Operators $\bold P$ and $\bold H$ with the properties (1) and (2)
above are determined by hypersurface $S$. But this hypersurface by
itself in definitions~6.1 and 6.2 is quite arbitrary. It can be passing
through any preassigned point $p$ on $M$, and it can have any
preassigned direction of normal vector $\bold n$. Moreover, it can have
any  preassigned  symmetric  matrix  as  a   matrix   of   second 
fundamental
form at this point. The latter is expressed by theorem which is
analogous to theorem~9.4 in Chapter
\uppercase\expandafter {\romannumeral 5}.
\proclaim{Theorem 8.1 ({\bf on the second fundamental form})}
Let $\bold v_0$ be nonzero vector at some point $p_0$ on Finslerian
manifold $M$, and let $\bold P$ be the operator of orthogonal projection
onto the hyperplane $\alpha$ perpendicular to $\bold v_0$. Then for
any linear operator $\bold H$ in $T_{p_0}(M)$ symmetric in metric
$\bold g(p_0,\bold v_0)$ on $M$ and satisfying the relationships
\thetag{8.8} one can find hypersurface $S$ passing through the point
$p_0$ and perpendicular to $\bold v_0$ such that matrix elements of
the operator $\bold H$ are determined by second fundamental form of
$S$ according to the formula \thetag{8.6}.
\endproclaim
    Theorem~8.1 allows us to identify $\bold P$ with projector field
of orthogonal projectors to hyperplane $\alpha$ perpendicular to the
vector of velocity. Operator $\bold H$ is identified with arbitrary
operator field from extended algebra of tensor fields symmetric with
respect to Finslerian metric $\bold g$ and bound with $\bold P$ by
\thetag{8.8}. This arbitrariness of $\bold H$ allows us to subdivide
\thetag{8.7} into two parts. Resulting equations have the form:
$$
\gather
\gathered
\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_i\,\tilde\nabla_mF_j}{v}
-\nabla_iF_j\right)=\\
\vspace{1ex}
=\sum^n_{i=1}\sum^n_{j=1}P^i_\varepsilon\,P^j_\sigma\left(
\,\shave{\sum^n_{m=1}}N^m\,\frac{F_j\,\tilde\nabla_mF_i}{v}
-\nabla_jF_i\right),
\endgathered\hskip -2em
\tag8.9\\
\vspace{2ex}
\sum^n_{m=1}\sum^n_{i=1}\sum^n_{j=1}g^{\sigma m}\,P^j_m\,
\tilde\nabla_jF_i\,P^i_\varepsilon=
\sum^n_{i=1}\sum^n_{j=1}\frac{P^{ij}\,\tilde\nabla_jF_i}
{n-1}\,P^\sigma_\varepsilon.
\hskip -3em
\tag8.10
\endgather
$$
These are the required {\bf additional normality} equations
for the force field of Newtonian dynamical system. Their
derivation above is based on the same arguments as the derivation
of the equations \thetag{9.20} and \thetag{9.42} in \S\,9
of Chapter \uppercase\expandafter {\romannumeral 5}. The
difference between Riemannian and Finslerian cases appears
to be not essential. Therefore we have a theorem similar to
the theorem~9.5 in Chapter
\uppercase\expandafter {\romannumeral 5}.
\proclaim{Theorem 8.5} Newtonian dynamical system on Finslerian
manifold of the dimension $n\geqslant 3$ satisfies additional
normality condition if and only if its force field satisfies the
equations \thetag{8.9} and \thetag{8.10} at all points $q=(p,\bold v)$
of tangent bundle $TM$, except for those, where $\bold v=0$.
\endproclaim
     Proof of the auxiliary theorem~9.4 on second fundamental form
is given in \S\,10 of Chapter \uppercase\expandafter {\romannumeral 5}.
This proof can be easily modified for the case of Finslerian manifolds.
Here we shall not give the proof of theorem~8.1 in details. We shall
sketch it and mark out the differences occurring due to difference between
Riemannian and Finslerian cases. In the first step theorem~8.1 is reduced
to the following one that coincides with theorem~10.1 from Chapter
\uppercase\expandafter {\romannumeral 5} almost literally.
\proclaim{Theorem 8.3} Suppose that in tangent space $T_{p_0}(M)$
at some point $p_0$ on Finslerian manifold of the dimension $n\geqslant 3$
we choose some nonzero vector $\bold v_0$, mark hyperplane
$\alpha$ perpendicular to $\bold v_0$, mark some base $\boldsymbol
\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$ in $\alpha$, and choose some
symmetric $(n-1)\times (n-1)$ matrix $b$. Then there exists some
hypersurface $S$ passing through the point $p_0$ tangent to $\alpha$
and there exist some local coordinates $u^1,\,\ldots,\,u^{n-1}$ on $S$
such that vectors $\boldsymbol\tau_1,\,\ldots,\,\boldsymbol\tau_{n-1}$
are coordinate tangent vectors to $S$ at the point $p_0$, while $b$
coincides with the matrix of second fundamental form for $S$ at this
point.
\endproclaim
\proclaim{Lemma 8.1} For any point $p_0$ on Finslerian manifold $M$
and for any vector $\bold v_0\neq 0$ at this point there exist local
coordinates $x^1,\,\ldots,\,x^n$ such that values of all components
of Cartan connection $\Gamma^k_{ij}(x^1,\ldots,x^1,v^1_0,\ldots,v^n_0)$
are zero at the point $q_0=(p_0,\bold v_0)$ of tangent bundle $TM$.
\endproclaim
     Lemma~8.1 differs from corresponding lemma~10.1 in Chapter
\uppercase\expandafter {\romannumeral 5}, since Cartan connection is
extended one. Its components depend not only on the point of $M$, but
on velocity vector $\bold v$ at this point. Therefore formula
\thetag{10.6} from Chapter \uppercase\expandafter {\romannumeral 5}
here in Finslerian geometry looks like
$$
x^k=\tx^k+\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1}\tilde\Gamma^k_{ij}
(p_0,\bold v_0)\,\tx^i\,\tx^j.\hskip -3em
\tag8.11
$$
In other details proof of lemma~8.1 is a copy of the proof of
corresponding lemma~10.1 in Chapter \uppercase\expandafter
{\romannumeral 5}.\par
    Proof of lemma~8.1 is a second step in proving theorem on second
fundamental form. Proof of theorem~8.3 is a last step. It doesn't
differ from proof of corresponding theorem~10.1 in Chapter
\uppercase\expandafter {\romannumeral 5}.
\head
\S\,9. Complete and strong normality conditions.
\endhead
     Theorems~6.1 and 8.2 reduce {\bf weak normality} condition
from definition~6.3 and {\bf additional normality} condition
from definition~6.2 to differential equations \thetag{6.24},
\thetag{6.25}, \thetag{8.9}, and \thetag{8.10} for the force
field $\bold F$ of Newtonian dynamical system. These equations
form complete system of {\bf normality equations}. Therefore
the couple of two conditions (of weak normality and of additional
normality) is called {\bf complete normality} condition. \pagebreak
While the condition given by definition~6.1 is called {\bf strong
normality} condition. In Finslerian geometry we have a theorem
establishing equivalence of last two conditions.
\proclaim{Theorem 9.1} Complete and strong normality conditions
for Newtonian dynamical systems on Finslerian manifolds are
equivalent to each other.
\endproclaim
     This is analog of theorem~12.1 in Chapter
\uppercase\expandafter {\romannumeral 5}. Direct assertion
of this theorem follows from the simple fact that the solution
of homogeneous linear ordinary differential equation $\ddot\varphi
=\Cal A(t)\,\dot\varphi+\Cal B(t)$ with zero initial data
\thetag{6.6} is identically zero: $\varphi=0$. Proof of
inverse assertion is more complicated. But it reproduces, almost
literally, all arguments that we used in the proof of theorem~12.1
in Chapter \uppercase\expandafter {\romannumeral 5}.
\enddocument
\end

ediss-09.tex
\input amstex
\documentstyle{amsppt}
%
% Author:   Ruslan Abdulovich Sharipov
% Title:    Dynamical systems admitting the normal shift
% Comment:  Thesis for the degree of Doctor of Sciences in Russia
% Address:  Rabochaya street 5, Ufa, 450003, Russia
% E-mail:   R_Sharipov@ic.bashedu.ru
%           ruslan-sharipov@usa.net
% Web-page: http://www.geocities.com/CapeCanaveral/Lab/5341
%
\nopagenumbers
\catcode`@=11
\redefine\logo@{\relax}
\catcode`\@=\active
\def\startpage#1{\pageno=#1}
\pagewidth{12.8cm}
\pageheight{20cm}
\startpage{215}
\Monograph
\topmatter
\title References
\endtitle
\endtopmatter
\document
\Refs\nofrills{}
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\end
