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Two-body central force problem

Two bodies of masses m and m with position vectors r and r are in orbit about each other due to an attractive central potential V. We may write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce the Jacobi coordinates; the separation of the bodies r = r − r and the location of the center of mass R = (m r + m r)/(m + m). The Lagrangian is then L = 1 2 M R ˙ 2 ⏟ L cm + 1 2 μ r ˙ 2 − V (| r |) ⏟ L rel {\displaystyle L=\underbrace {{\frac {1}{2}}M{\dot {\mathbf {R} }}^{2}} _{L_{\text{cm}}}+\underbrace {{\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}-V(|\mathbf {r} |)} _{L_{\text{rel}}}} where M = m + m is the total mass, μ = m m /(m + m) is the reduced mass, and V the potential of the radial force, which depends only on the magnitude of the separation | r | = | r − r |. The Lagrangian splits into a center-of-mass term L and a relative motion term L.

The Euler–Lagrange equation for R is simply M R ¨ = 0, {\displaystyle M{\ddot {\mathbf {R} }}=0,} which states the center of mass moves in a straight line at constant velocity.

Since the relative motion only depends on the magnitude of the separation, it is ideal to use polar coordinates (r, θ) and take r = | r |, L rel = 1 2 μ (r ˙ 2 + r 2 θ ˙ 2) − V (r), {\displaystyle L_{\text{rel}}={\frac {1}{2}}\mu \left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)-V(r),} so θ is a cyclic coordinate with the corresponding conserved (angular) momentum p θ = ∂ L rel ∂ θ ˙ = μ r 2 θ ˙ = ℓ. {\displaystyle p_{\theta }={\frac {\partial L_{\text{rel}}}{\partial {\dot {\theta }}}}=\mu r^{2}{\dot {\theta }}=\ell.}

The radial coordinate r and angular velocity d θ /d t can vary with time, but only in such a way that ℓ is constant. The Lagrange equation for r is μ r θ ˙ 2 − d V d r = μ r ¨. {\displaystyle \mu r{\dot {\theta }}^{2}-{\frac {dV}{dr}}=\mu {\ddot {r}}.}

This equation is identical to the radial equation obtained using Newton's laws in a co-rotating reference frame, that is, a frame rotating with the reduced mass so it appears stationary. Eliminating the angular velocity d θ /d t from this radial equation, μ r ¨ = − d V d r + ℓ 2 μ r 3. {\displaystyle \mu {\ddot {r}}=-{\frac {\mathrm {d} V}{\mathrm {d} r}}+{\frac {\ell ^{2}}{\mu r^{3}}}.} which is the equation of motion for a one-dimensional problem in which a particle of mass μ is subjected to the inward central force −d V /d r and a second outward force, called in this context the (Lagrangian) centrifugal force (see centrifugal force#Other uses of the term): F c f = μ r θ ˙ 2 = ℓ 2 μ r 3. {\displaystyle F_{\mathrm {cf} }=\mu r{\dot {\theta }}^{2}={\frac {\ell ^{2}}{\mu r^{3}}}.}

Of course, if one remains entirely within the one-dimensional formulation, ℓ enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general two-dimensional problem from which the one-dimensional problem originated.

If one arrives at this equation using Newtonian mechanics in a co-rotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation of the frame itself. If one arrives at this equation directly by using the generalized coordinates (r, θ) and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth of using polar coordinates. As Hildebrand says:

"Since such quantities are not true physical forces, they are often called inertia forces. Their presence or absence depends, not upon the particular problem at hand, but upon the coordinate system chosen." In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides the centripetal force for a curved motion.

This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates. For example, see for a comparison of Lagrangians in an inertial and in a noninertial frame of reference. See also the discussion of "total" and "updated" Lagrangian formulations in. Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not an inertial frame of reference), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces as generalized inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we deal always with generalized forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" will be omitted frequently."

It is known that the Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the symmetry of the system.