 Solution

ID: 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 equation

\frac{\partial f}{\partial y} - \frac{d}{d x} (\frac{\partial f}{\partial y ^{'}}) = 0.

It follows by taking the first variation of the functional and integrating the term in

δ y^{'}

by parts;

δy

vanishes at the endpoints.

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