Discussion on the equivalence of two relativistic point-particle Lagrangians
In 2021, Lei et al. claimed the equivalence between the two Lagrangians $\mathcal{L}_1 =-mc\sqrt{-g_{μν}{\dot{x}}^μ{\dot{x}}^ν}-V$ and $\mathcal{L}_2 = \frac{1}{2}mg_{μν} {\dot{x}}^μ{\dot{x}}^ν-V$ 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.