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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 $L_I$.

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  • $\begingroup$ Read where? Which page? $\endgroup$
    – Qmechanic
    Mar 10, 2022 at 9:04
  • $\begingroup$ 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 $\endgroup$
    – KP99
    Mar 10, 2022 at 15:37

1 Answer 1

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  1. If the action is world-line (WL) reparametrization invariant, one can in principle use any WL parameter.

  2. Note that the notion of proper time (and more generally the notion of affine parametrization) refer to a specific curve/path. It does not make sense to use affine parametrization before applying the stationary action principle, only afterwards (because neighboring virtual paths may have a different notion of affine parametrization).

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  • $\begingroup$ When is an action world line reparametrisation invariable? I am supposing that the interacción term depends only on the coordinates (not velocuties). $\endgroup$ Mar 10, 2022 at 9:53
  • $\begingroup$ One might have to use an einbein. $\endgroup$
    – Qmechanic
    Mar 10, 2022 at 10:46

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