Discussion on the equivalence of two relativistic point-particle Lagrangians

Abstract In 2021, Lei et al. claimed the equivalence between the two Lagrangians $$\mathcal {L}_1 = -mc\sqrt{-g_{\mu \nu }{\dot{x}}^\mu {\dot{x}}^\nu }-V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>m</mml:mi> <mml:mi>c</mml:mi> <mml:msqrt> <mml:mrow> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>μ</mml:mi> </mml:msup> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>ν</mml:mi> </mml:msup> </mml:mrow> </mml:msqrt> <mml:mo>-</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> and $$\mathcal {L}_2 = \frac{1}{2}mg_{\mu \nu }{\dot{x}}^\mu {\dot{x}}^\nu -V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mi>m</mml:mi> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>μ</mml:mi> </mml:msup> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>ν</mml:mi> </mml:msup> <mml:mo>-</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> 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 $$\mathcal {L}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and $$\mathcal {L}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> correspond to different Hamiltonian formulations. The Hamiltonian derived from $$\mathcal {L}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> inherently enforces the mass shell constraint, whereas the Hamiltonian from $$\mathcal {L}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> does not. When the Schwarzschild metric supplemented with an artificial mechanical potential is taken as a toy model, numerical investigations reveal that $$\mathcal {L}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> leads to chaotic behavior, which signifies non-integrable dynamics. By contrast, $$\mathcal {L}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> can be shown analytically to produce integrable dynamics free of chaos. In many scenarios, $$\mathcal {L}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> is strongly recommended due to its theoretical superiority and universality. $$\mathcal {L}_2$$