Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 9 / b / Solution

The total derivative contributes only the boundary termThis is fixed by the endpoint data, so its first variation vanishes. It cannot change the Euler-Lagrange equations or which path minimizes $F$.

For a direct check, let $h=g_x\dot{x}+g_y\dot{y}$. ThenThe equality uses commuting mixed partial derivatives. Thus $\boxed{dg/dt\text{ contributes zero to both Euler-Lagrange equations}}$, as in total time derivatives preserve fixed-endpoint Euler-Lagrange equations.