Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 9 / a / Solution

For fixed endpoint values, a stationary differentiable path obeys the Euler-Lagrange equationIt follows by taking the first variation of the functional and integrating the term in $\delta y^{\prime }$ by parts; $\delta y$ vanishes at the endpoints.