Worldline variation before proper-time normalization

Use metric signature $(-,+,+,+)$ and an arbitrary increasing parameter $\lambda$ on a massive particle's timelike curve. Write $v^a=d{x}^a/d\lambda$ without yet imposing proper-time normalization. The free relativistic particle action isIts derivative with respect to $v^a$ is $p_a=mc{g}_{ab}{v}^b/\sqrt{-v^2}$. The square root makes the action independent of the chosen increasing parameter. Varying it first preserves the nontrivial velocity dependence that produces the geodesic equation.^[ref]

After variation, choose $\lambda =\tau$, so $v^a=U^a$ is the four-velocity, $U^2=-c^2$ and $p_a=m{U}_a$. Substituting $\sqrt{-U^2}=c$ into the integrand before an unconstrained variation would instead make it look constant and lose the momentum equation. The choice of proper time must therefore follow the variation, or be handled by an equivalent constrained action.