Abstract In 2021, Lei et al. claimed the equivalence between the two Lagrangians L₁ = -mc-g ₗ̇^ {ẋ^ }-V L 1 = - m c - g μ ν x ˙ μ x ˙ ν - V and L₂ = 12mg ẋ^ ẋ^ -V L 2 = 1 2 m g μ ν x ˙ μ 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 L 1 and L₂ L 2 correspond to different Hamiltonian formulations. The Hamiltonian derived from L₁ L 1 inherently enforces the mass shell constraint, whereas the Hamiltonian from L₂ L 2 does not. When the Schwarzschild metric supplemented with an artificial mechanical potential is taken as a toy model, numerical investigations reveal that L₁ L 1 leads to chaotic behavior, which signifies non-integrable dynamics. By contrast, L₂ L 2 can be shown analytically to produce integrable dynamics free of chaos. In many scenarios, L₁ L 1 is strongly recommended due to its theoretical superiority and universality. L₂
Wang et al. (Fri,) studied this question.