Conservation laws in non-inertial frames and non-conservation of energy of relative motion in two-body problem

TL;DR

The paper analyzes dynamics in a non-inertial frame attached to a reference body and introduces a reformulation where the indirect acceleration $\mathbf{a}_{\mathrm{id}}$ is uniform across all objects. This leads to simple conservation-law forms for total momentum $\mathbf{P}$, angular momentum $\mathbf{L}$, and energy $E$ as they evolve under the external reflex force, clarifying the interpretation of non-inertial dynamics. In the two-body problem, the vis viva integral becomes an energy-like invariant rather than true energy of relative motion, since the true energy changes due to the work done by the indirect force. The approach has implications for interpreting disc–planet coupling and secular dynamics in astrophysical systems, highlighting the distinction between conventional energy concepts and the effects of non-inertial forces.

Abstract

The dynamics of systems of multiple gravitationally interacting bodies is often studied in a frame attached to one of the objects (e.g. a central star in a planetary system). As this frame is generally non-inertial, indirect forces appear in the equations describing the motion of bodies relative to the reference object. According to the convention adopted in celestial mechanics, the associated indirect acceleration is defined to be different for every object under consideration, whereas the gravitational coupling between each body and the reference object is described via the effective two-body potential, which does not obey the equivalence principle. Here we point out that a slightly different and more physically motivated definition of the indirect acceleration provides significant benefits when interpreting relative motion in a non-inertial frame. First, the indirect acceleration ends up being the same for all objects in the system. Second, the non-conservation of momentum, angular momentum, and energy of the whole system in a non-inertial frame naturally follow from the action of the indirect acceleration on the system as an external force. We also argue that the vis viva integral of the classical two-body problem should not be interpreted as a statement of energy conservation. In fact, the energy of relative motion is not conserved due to the work done on the two-body system by the indirect force. These results can be useful for interpreting dynamics in various astrophysical contexts, in particular the physics of disc-planet coupling.