Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 9 / a / Solution

For fixed endpoint values and a smooth variation vanishing at the endpoints, the first variation isThe fundamental lemma of the calculus of variations gives the Euler-Lagrange equationIf $f$ is independent of $y$, then $y$ is a cyclic coordinate, and integration givesThis is conservation of the corresponding generalized momentum.