Discussion on the equivalence of two relativistic point-particle Lagrangians

Liubin Wang, Xin Wu

In 2021, Lei et al. claimed the equivalence between the two Lagrangians ${\mathcal{L}}_1=-mc\sqrt{-g_{\hat{I}¼\hat{I}½}{\dot{x}}^{\hat{I}}¼{\dot{x}}^{\hat{I}}½}-V$ and $\mathcal{L}_2 = \frac{1}{2}mg_{μν} {\d…