Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 10 / a / Solution

Let $y$ be a stationary function and perturb it to $y_{\epsilon }=y+\epsilon \eta$, where $\eta$ is smooth and $\eta (a)=\eta (b)=0$. These perturbations preserve the prescribed endpoint values. Differentiate the functional at $\epsilon =0$:By integration by parts,The boundary term vanishes. The remaining integral must vanish for every such $\eta$, so the fundamental lemma of the calculus of variations gives the Euler-Lagrange equationIt is a necessary stationarity condition; deciding whether the stationary path is a minimum requires additional information.