Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Symbolic dynamics for the anisotropic $N$-centre problem at negative
  energies   The planar $N$-centre problem describes the motion of a particle moving in
the plane under the action of the force fields of $N$ fixed attractive centres:
\[
  \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \]
  In this paper we prove symbolic dynamics at slightly negative energy for an
$N$-centre problem where the potentials $V_j$ are positive, anisotropic and
homogeneous of degree $-\alpha_j$: \[
V_j(x)=|x|^{-\alpha_j}V_j\left(\frac{x}{|x|}\right). \] The proof is based on a
broken geodesics argument and trajectories are extremals of the Maupertuis'
functional.
  Compared with the classical $N$-centre problem with Kepler potentials, a
major difficulty arises from the lack of a regularization of the singularities.
We will consider both the collisional dynamics and the non collision one.
Symbols describe geometric and topological features of the associated
trajectory.

34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph