jbergman said:
The square root action is reparametrization invariant. The energy functional is not.
See,
https://physics.stackexchange.com/q...e-squared-lagrangian-equivalent/149204#149204
The square-root action is indeed reparametrization invariant by construction, and this is the reason, why it describes only 3 independent degrees of freedom and not 4 as it might look at the first glance. It's a case of "gauge symmetry", and this can be used to get to the "square form", i.e., you can partially fix the gauge by imposing a constraint.
From a geometrical point of view it's suggestive to use the constraint that the parameter is an affine parameter, i.e., to impose the constraint ##g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=\text{const}##. This you can do using a Lagrange parameter, but it's easier to directly observe that the Lagrangian
$$L=\frac{1}{2} g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}$$
automatically does the job. First it obeys the "affinity constraint" automatically. Since ##L## is not explicitly ##\lambda## determined (where ##\lambda## is the arbitrary parameter of the curves), you have
$$p_{\mu} \dot{x}^{\mu} -L=L=\text{const},$$
and it's easy to see that the Euler-Lagrange equations of motion coincide with the Euler-Lagrange equations of motion for the square-root form in the case where you choose an affine parameter, fulfilling the constraint condition. That's why the square form and the square-root form lead to the same equations of motion, i.e., the geodesics in spacetime you are looking for.
The physical cases are of course time-like geodesics for the description of massive point particles moving through a given gravitational field or the light-like geodesics for the calculation of "light rays" (in the literature usually called "photons", which I find a bit misleading, but that's maybe sementics) in a given gravitational field.