# Which parameter to use in relativistic Lagrangian mechanics?

According to Wikipedia the formulation of the Relativistic Lagrangian is:

$$L = -mc\sqrt{g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}}+L_I(x,\dot{x}).$$

However, I have read that using both coordinate time and proper time would be correct and I understand that. The problem comes when we use the proper time when dealing with the Euler-Lagrange equations, because I am not sure about whether the quantity $$g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}$$ is conserved. I know that if we use the coordinate time $$x^0$$ as parameter of the Euler-Lagrange equations we can later manage to work out $$\frac{dx^0}{d\tau}$$ with the equation $$c^2 = g_{\alpha\beta}\frac{{dx}^{\alpha}}{d\tau}\frac{{dx}^{\beta}}{d\tau},$$ but I am not sure about whether this last equation would be met if we solved the equations with $$\tau$$ as our parameter, as the solutions would give us all the velocities with respect to $$\tau$$. So, can we use $$\tau$$ as out parameter so that $$c^2 = g_{\alpha\beta}\frac{{dx}^{\alpha}}{d\tau}\frac{{dx}^{\beta}}{d\tau}$$ is met? Supposing that there is an interaction term $$L_I$$.

• Read where? Which page? Commented Mar 10, 2022 at 9:04
• The reparameterization invariance is a non dynamical symmetry, which should be removed before solving E-L equation of motion. This note should be helpful: web.mit.edu/edbert/GR/gr5.pdf Commented Mar 10, 2022 at 15:37