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

Roman R. Rafikov

arxiv: 2511.08709 · v3 · submitted 2025-11-11 · 🌌 astro-ph.EP

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

Roman R. Rafikov This is my paper

Pith reviewed 2026-05-17 23:05 UTC · model grok-4.3

classification 🌌 astro-ph.EP

keywords non-inertial framesindirect accelerationtwo-body problemconservation lawscelestial mechanicsvis viva integralrelative energyequivalence principle

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The pith

The energy of relative motion is not conserved in a non-inertial frame attached to one body because the indirect force does work on the system.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines motion of gravitationally interacting bodies when analyzed in a frame fixed to one member of the system, such as a central star. It proposes replacing the standard per-body definition of indirect acceleration with a single value that applies equally to every object. This uniform indirect acceleration behaves as an external force on the whole system, directly explaining why total momentum, angular momentum, and energy fail to be conserved. The author concludes that the classical vis viva relation cannot be read as a conservation law for relative energy in this frame, since the indirect force supplies or removes energy over time. The revised view applies particularly to interpreting interactions in planetary systems and discs.

Core claim

A uniform indirect acceleration, identical for every body, replaces the conventional per-object definition in a non-inertial frame fixed to one body. This acceleration acts as an external force, so linear momentum, angular momentum, and total energy of the system are not conserved. The vis viva integral of the two-body problem therefore does not express conservation of the energy of relative motion; that energy varies because the indirect force performs net work.

What carries the argument

The uniform indirect acceleration, defined identically for all bodies and acting as an external force that performs work in the non-inertial frame.

Load-bearing premise

Redefining the indirect acceleration to be identical for all bodies is both more physically motivated and consistent with the equivalence principle, while the conventional per-body definition is less so.

What would settle it

A numerical integration of the two-body equations in the non-inertial frame that tracks whether the specific energy of relative motion remains constant or changes exactly as the work done by the uniform indirect acceleration predicts.

read the original 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 a non-inertial frame attached to one of the bodies. The energy of relative motion is not conserved in this frame 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Rafikov redefines indirect acceleration as uniform across bodies and argues relative-motion energy is not conserved in the non-inertial frame because the indirect force does work, but the relative equations still admit the standard conserved vis viva quantity.

read the letter

The paper's main contribution is a uniform definition of indirect acceleration in the frame of one body, so that every other body feels the same indirect term. From there it follows that the indirect acceleration acts like an external force, which immediately explains why total momentum, angular momentum, and energy are not conserved for the system as a whole. That part is straightforward Newtonian mechanics and lines up with how people already treat non-inertial frames in practice. The second claim is that the vis viva integral should not be read as energy conservation for relative motion, because the indirect force performs work. This is presented as a clarification for disc-planet and similar calculations. It is new in the sense that the uniform definition and the explicit work argument are not the usual framing in the literature the abstract cites. The paper does a service by spelling out these points without extra machinery. The relative acceleration equation remains the usual two-body form either way, so the standard conserved quantity ½v_rel² − G(M+m)/r still drops out when you multiply by velocity and integrate. The non-conservation statement therefore depends on adopting a different energy expression whose time derivative equals the power injected by the indirect term. The abstract does not show that alternative functional explicitly, which leaves the interpretive step resting more on convention than on a dynamical contradiction. The argument that the uniform definition is more consistent with the equivalence principle is also presented as self-evident rather than derived from a concrete advantage in calculations or predictions. Readers who already work in star-centered frames for migration or disc torques will find the discussion useful for avoiding loose language about energy budgets. It is not a result that changes orbit integrations or torque formulas. The paper is worth sending to peer review so that the derivations and the choice of energy functional can be checked in detail.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a uniform definition of the indirect acceleration in non-inertial frames attached to one body in gravitational N-body systems. This definition makes the indirect acceleration the same for all objects, consistent with the equivalence principle, unlike the conventional per-body definition. The paper argues that this leads to natural non-conservation of total momentum, angular momentum, and energy in the non-inertial frame due to the indirect force acting as an external force. It further claims that the vis viva integral in the two-body problem should not be interpreted as energy conservation in the frame of one body, because the energy of relative motion is not conserved owing to work done by the indirect force. These ideas are suggested to be useful for astrophysical contexts such as disc-planet coupling.

Significance. If substantiated, the results could offer a more consistent interpretive framework for dynamics in non-inertial frames, potentially aiding understanding of relative motions in planetary systems and discs. The approach is grounded in standard Newtonian mechanics without free parameters or ad-hoc assumptions. However, the significance is limited by the lack of explicit derivations for the energy non-conservation claim, which may not contradict the standard conserved relative energy quantity.

major comments (1)

[Analysis of the two-body problem and vis viva integral] The claim that the energy of relative motion is not conserved in the non-inertial frame due to the indirect force performing work requires an explicit alternative energy functional. The standard relative acceleration equation d²r/dt² = −G(M+m)/r² r̂ still admits the conserved quantity ½v_rel² − G(M+m)/r upon multiplication by dr/dt and integration. The manuscript should construct the specific energy expression (e.g., kinetic energy without reduced mass) and show its time derivative equals the power input by the indirect term to support the non-conservation assertion.

minor comments (1)

[Introduction or definitions] Clarify how the proposed uniform indirect acceleration differs quantitatively from the conventional definition, perhaps with an explicit equation comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We address the major comment below and have revised the paper to include the explicit derivation requested.

read point-by-point responses

Referee: The claim that the energy of relative motion is not conserved in the non-inertial frame due to the indirect force performing work requires an explicit alternative energy functional. The standard relative acceleration equation d²r/dt² = −G(M+m)/r² r̂ still admits the conserved quantity ½v_rel² − G(M+m)/r upon multiplication by dr/dt and integration. The manuscript should construct the specific energy expression (e.g., kinetic energy without reduced mass) and show its time derivative equals the power input by the indirect term to support the non-conservation assertion.

Authors: We appreciate the referee pointing out the need for an explicit derivation. The conserved quantity ½v_rel² − G(M+m)/r is indeed obtained from the relative acceleration equation and corresponds to the specific orbital energy in the inertial two-body problem. However, when interpreting the motion in the non-inertial frame attached to the primary (mass M), the appropriate energy functional for the relative motion is the kinetic energy of the secondary body alone together with the gravitational potential due solely to the primary: E = ½ m |dr/dt|² − G M m / r. Differentiating E with respect to time and substituting the non-inertial equation of motion d²r/dt² = −G(M+m)/r² r̂ (which incorporates the uniform indirect acceleration a_ind = −G m / r² r̂) yields dE/dt = m a_ind · (dr/dt). The right-hand side is precisely the power delivered by the indirect force. This demonstrates that E is not conserved because the indirect force performs work. We have added this explicit calculation, including the distinction from the reduced-mass formulation, as a new paragraph in the revised Section 3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained in Newtonian frame analysis

full rationale

The paper introduces a uniform indirect acceleration definition across bodies in a non-inertial frame attached to one mass and applies the work-energy theorem to show that the indirect force performs work, preventing conservation of relative-motion energy. This follows directly from the equations of motion under the redefined acceleration without reducing any prediction to a fitted input or self-citation chain. The vis viva integral is reinterpreted as not representing energy conservation in that frame, but the underlying relative acceleration equation remains the standard conservative form; the interpretive step does not loop back to the paper's own assumptions by construction. No self-definitional, fitted-prediction, or uniqueness-imported patterns appear. The analysis rests on external Newtonian gravity and equivalence-principle considerations that are independently verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard Newtonian mechanics and the equivalence principle without introducing new fitted parameters, ad-hoc constants, or postulated entities.

axioms (1)

standard math Newtonian gravity governs the motion and non-inertial frame transformations introduce fictitious accelerations
Invoked throughout the abstract when discussing indirect accelerations and conservation laws.

pith-pipeline@v0.9.0 · 5563 in / 1266 out tokens · 49700 ms · 2026-05-17T23:05:53.545692+00:00 · methodology

discussion (0)

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unclear
Relation between the paper passage and the cited Recognition theorem.

vis viva integral ... should not be interpreted as a statement of energy conservation

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Works this paper leans on

1 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Brouwer D., Clemence G. M., 1961, Methods of celestial mechanics Crida A., et al., 2025, The Open Journal of Astrophysics, 8, 84 KopeikinS.,EfroimskyM.,KaplanG.,2011,RelativisticCelestialMechanics of the Solar System, doi:10.1002/9783527634569. Murray C. D., Dermott S. F., 1999, Solar System Dynamics, doi:10.1017/CBO9781139174817. RafikovR.R.,CimermanN.P....

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1002/9783527634569 1961

[1] [1]

Brouwer D., Clemence G. M., 1961, Methods of celestial mechanics Crida A., et al., 2025, The Open Journal of Astrophysics, 8, 84 KopeikinS.,EfroimskyM.,KaplanG.,2011,RelativisticCelestialMechanics of the Solar System, doi:10.1002/9783527634569. Murray C. D., Dermott S. F., 1999, Solar System Dynamics, doi:10.1017/CBO9781139174817. RafikovR.R.,CimermanN.P....

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1002/9783527634569 1961