The Lagrangians L1 = -mc sqrt(-g dot x dot x) - V and L2 = (1/2)m g dot x dot x - V are equivalent only for V=0 or electromagnetic V; otherwise L1 enforces mass shell while L2 does not, leading to chaotic vs integrable dynamics in toy models.
Mikkola, Practical Symplectic Methods with Time Transformation for the Few-Body Problem, Celestial Me- chanics and Dynamical Astronomy67, 145 (1997)
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Discussion on the equivalence of two relativistic point-particle Lagrangians
The Lagrangians L1 = -mc sqrt(-g dot x dot x) - V and L2 = (1/2)m g dot x dot x - V are equivalent only for V=0 or electromagnetic V; otherwise L1 enforces mass shell while L2 does not, leading to chaotic vs integrable dynamics in toy models.