 Solution

ID: 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 is

δ T = \int_{a}^{b} (f_{y} δy + f_{y^{'}} δ y^{'}) d x = \int_{a}^{b} (f_{y} - \frac{d}{d x} f_{y^{'}}) δy d x .

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

f

is independent of

y

, then

y

is a cyclic coordinate, and integration gives

\frac{\partial f}{\partial y ^{'}} = C .

This is conservation of the corresponding generalized momentum.

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