Worldline variation before proper-time normalization
= Worldline variation before proper-time normalization
In the reparametrization-invariant massive-particle action, vary the square-root kinetic term before choosing proper time. Its velocity derivative gives $mc\,u_a/\sqrt{-u^2}$, which becomes $mu_a$ after imposing $u^2=-c^2$. Replacing the square root by a constant before variation would incorrectly remove the inertial momentum.