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arxiv: 2604.10876 · v1 · submitted 2026-04-13 · 🌀 gr-qc · astro-ph.HE· hep-th

Discussion on the equivalence of two relativistic point-particle Lagrangians

Liubin Wang , Xin Wu This is my paper

Pith reviewed 2026-05-10 16:29 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords relativistic Lagrangiansequivalence of Lagrangiansmass shell constraintHamiltonian formulationchaotic dynamicsintegrable systemsSchwarzschild metricexternal potentials
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The pith

Two relativistic Lagrangians for particles in gravity plus external potentials agree only when the potential is electromagnetic or zero.

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

The paper tests whether two common Lagrangians describe identical motion for a relativistic particle moving in curved spacetime with an added external potential. Both produce the same Hamiltonian that automatically obeys the mass-shell condition precisely when the potential vanishes or takes the electromagnetic form. For any other potential the two Lagrangians generate distinct Hamiltonian pictures, one that enforces the mass constraint and one that does not. In a Schwarzschild background with a simple added potential, the first Lagrangian produces chaotic non-integrable orbits while the second produces integrable motion that can be solved analytically.

Core claim

The Lagrangians L1 = -mc sqrt(-g_mu nu dot x^mu dot x^nu) - V and L2 = (1/2)m g_mu nu dot x^mu dot x^nu - V are equivalent, yielding identical Hamiltonians that satisfy the mass-shell constraint, exactly when V is zero or electromagnetic. For generic external potentials they correspond to different Hamiltonian formulations; the Hamiltonian from L1 inherently enforces the constraint while the one from L2 does not. In the Schwarzschild metric with an artificial mechanical potential, numerical work shows L1 yields chaotic non-integrable dynamics whereas L2 yields integrable dynamics free of chaos.

What carries the argument

The two distinct Hamiltonian formulations obtained from each Lagrangian, and the presence or absence of automatic enforcement of the mass-shell constraint.

If this is right

  • When the potential is electromagnetic both Lagrangians can be used interchangeably without changing the physics.
  • For generic potentials the first Lagrangian is the theoretically consistent choice because it automatically respects the relativistic mass-shell condition.
  • The second Lagrangian remains useful for low-energy or weak-field approximations where computational simplicity matters more than strict constraint enforcement.
  • Near black holes the second Lagrangian can still be employed for charged or neutral particles if an extra constraint is added to its Hamiltonian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The integrability of the second formulation may allow closed-form solutions in spacetimes where the first requires only numerical integration.
  • Predictions for particle trajectories in strong gravity could change depending on which Lagrangian is chosen when the potential is neither electromagnetic nor zero.
  • The distinction may affect modeling of charged-particle motion in astrophysical environments that include both gravitational and non-electromagnetic forces.

Load-bearing premise

The Schwarzschild metric with one artificial mechanical potential is representative of the behavior that occurs for arbitrary external potentials.

What would settle it

A direct calculation showing that the Hamiltonian from the second Lagrangian also enforces the mass-shell constraint for a non-electromagnetic potential, or a numerical orbit integration in which the second Lagrangian produces chaos in the same toy model.

Figures

Figures reproduced from arXiv: 2604.10876 by Liubin Wang, Xin Wu.

Figure 1
Figure 1. Figure 1: Poincar´e section at the plane θ = π/2 with pθ > 0. The motion of particles is considered in the Hamiltonian K of Eq. (56) corresponding to the Lagrangian L2 of Eq. (55). The parameters are ω = 2, rc = 29, K = −1/2 and Φ = 1. All the plotted particle orbits are regular KAM torus. 6. 5 7. 0 7. 5 8. 0 8. 5 9. 0 9. 5 1 0. 0 1 0. 5 -0.1 5 -0.1 0 -0. 05 0. 00 0. 05 0.1 0 0.1 5 p r r        7 8 9 1 0 1 1 … view at source ↗
Figure 2
Figure 2. Figure 2: Poincar´e sections for three values of the energy [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as Fig. 2 but for the non-vanishing angular momentum [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

In 2021, Lei et al. claimed the equivalence between the two Lagrangians $\mathcal{L}_1 =-mc\sqrt{-g_{\mu\nu}{\dot{x}}^\mu{\dot{x}}^\nu}-V$ and $\mathcal{L}_2 = \frac{1}{2}mg_{\mu\nu} {\dot{x}}^\mu{\dot{x}}^\nu-V$ for describing particle dynamics in combined gravitational and matter fields. In the present work, we rigorously demonstrate that their equivalence depends critically on the external potential V. Both Lagrangians yield identical Hamiltonians that strictly satisfy the mass shell constraint, and are therefore equivalent when V vanishes or corresponds to an electromagnetic potential. However, they are generally not equivalent for generic external potentials excluding the electromagnetic ones. This discrepancy arises because L1 and L2 correspond to different Hamiltonian formulations. The Hamiltonian derived from L1 inherently enforces the mass shell constraint, whereas the Hamiltonian from L2 does not. When the Schwarzschild metric supplemented with an artificial mechanical potential is taken as a toy model, numerical investigations reveal that L1 leads to chaotic behavior, which signifies non-integrable dynamics. By contrast, L2 can be shown analytically to produce integrable dynamics free of chaos. In many scenarios, L1 is strongly recommended due to its theoretical superiority and universality. L2 is generally suitable for classical approximate problems involving low energy and weak gravity. Nevertheless, it is the preferred choice for strong field problems concerning the dynamics of charged (or neutral) particles near black holes with (or without) external electromagnetic fields, owing to its mathematical simplicity and computational efficiency. Moreover, it can still satisfy the mass shell constraint when an additional constraint is imposed on its corresponding Hamiltonian.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims that the two Lagrangians L1 = -mc sqrt(-g_μν ẋ^μ ẋ^ν) - V and L2 = (1/2)m g_μν ẋ^μ ẋ^ν - V are equivalent only when V=0 or V is an electromagnetic potential, since both then produce identical Hamiltonians obeying the mass-shell constraint. For generic external potentials they are inequivalent because the Hamiltonian from L1 enforces the constraint while that from L2 does not. Using the Schwarzschild metric plus an artificial mechanical potential as a toy model, numerical evidence shows chaotic (non-integrable) motion for L1 while L2 is shown analytically to be integrable; the authors recommend L1 for theoretical reasons and L2 for computational simplicity in certain regimes.

Significance. If the claimed non-equivalence for generic V holds, the work would clarify Lagrangian choice for relativistic particles in combined gravitational and external fields, with direct implications for integrability and chaos near black holes. The analytic demonstration of integrability for L2 and the numerical contrast with L1 are concrete strengths; however, the generality of the non-equivalence rests on a single artificial-potential example rather than a general derivation.

major comments (2)
  1. [Abstract and Hamiltonian derivation section] Abstract and the section deriving the Hamiltonians: the claim that the Lagrangians are 'generally not equivalent for generic external potentials' is not backed by an analytic proof that the Legendre transforms differ for arbitrary V; the distinction is shown only for the specific Schwarzschild-plus-artificial-potential toy model.
  2. [Toy-model numerical section] Toy-model section: the numerical evidence for chaos (non-integrability) in L1 is obtained with an unspecified 'artificial mechanical potential'; without its explicit functional form, the parameter values used, and the integration scheme, it is impossible to assess whether the observed chaos is generic or an artifact of that particular choice.
minor comments (2)
  1. [Abstract] The abstract states that L2 'can still satisfy the mass shell constraint when an additional constraint is imposed'; the precise form of that constraint and how it is implemented should be stated explicitly.
  2. [Introduction] Notation for the metric signature and the definition of the four-velocity normalization should be stated once at the beginning to avoid ambiguity when comparing the two Lagrangians.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment point by point below. Where the comments identify areas needing greater clarity or detail, we have revised the manuscript accordingly while preserving the core arguments.

read point-by-point responses

  1. Referee: [Abstract and Hamiltonian derivation section] Abstract and the section deriving the Hamiltonians: the claim that the Lagrangians are 'generally not equivalent for generic external potentials' is not backed by an analytic proof that the Legendre transforms differ for arbitrary V; the distinction is shown only for the specific Schwarzschild-plus-artificial-potential toy model.

    Authors: We appreciate the referee highlighting the need for a clearer separation between the general analytic argument and the illustrative example. In the Hamiltonian derivation section we show that the Legendre transform of L1, owing to its square-root structure and reparametrization invariance, always produces a Hamiltonian that enforces the mass-shell constraint independently of the explicit form of the scalar potential V. By contrast, the Legendre transform of L2 yields an unconstrained Hamiltonian whose level sets coincide with the mass shell only for special choices of V (V=0 or electromagnetic-type potentials that effectively restore the constraint). This general distinction is analytic and does not rely on the toy model; the Schwarzschild-plus-artificial-potential example is used solely to exhibit the resulting dynamical difference (chaos versus integrability). We have added an explicit paragraph in the revised Section 2 that writes the two Hamiltonians side-by-side for arbitrary V, making the proof independent of any specific metric or potential. revision: yes

  2. Referee: [Toy-model numerical section] Toy-model section: the numerical evidence for chaos (non-integrability) in L1 is obtained with an unspecified 'artificial mechanical potential'; without its explicit functional form, the parameter values used, and the integration scheme, it is impossible to assess whether the observed chaos is generic or an artifact of that particular choice.

    Authors: We agree that the original presentation of the toy model lacked sufficient technical detail. In the revised manuscript we now specify the explicit functional form of the artificial mechanical potential, the numerical values of all parameters (including the Schwarzschild mass and the potential strength), and the integration algorithm together with its step-size control and accuracy tolerances. We have also added a short convergence study showing that the reported chaotic indicators remain robust under moderate variations of these parameters, indicating that the non-integrability is not an artifact of the particular choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations use standard Legendre transforms independent of claims

full rationale

The paper's core steps consist of applying the standard Legendre transform to obtain Hamiltonians from L1 and L2, then comparing the resulting expressions for different classes of V. These transforms follow directly from the given Lagrangian definitions without any fitted parameters, self-referential definitions, or renaming of known results. Equivalence is demonstrated by explicit algebraic identity when V=0 or V is electromagnetic (both yield the same mass-shell-constrained Hamiltonian). Non-equivalence for generic V is asserted from the structural difference in the Hamiltonian formulations (constraint enforcement vs. non-enforcement), with the Schwarzschild-plus-artificial-potential case serving only as a numerical illustration of integrability differences rather than a load-bearing proof or fitted input. The citation to Lei et al. (2021) is to the claim being critiqued and carries no self-citation load. No uniqueness theorems, ansatzes smuggled via prior work, or self-definitional loops appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard relativistic Lagrangian mechanics and Hamiltonian formalism in general relativity without introducing new free parameters or entities. The toy model potential is introduced for illustration but not as a fundamental addition.

axioms (2)
  • standard math Standard derivation of Hamiltonian from Lagrangian via Legendre transform in relativistic mechanics
    Used to obtain and compare the Hamiltonians from L1 and L2.
  • domain assumption Mass shell constraint p^2 = m^2 for relativistic particles
    Central to the equivalence discussion and enforcement in L1.

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