 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/9/a/solution

S = \int n d s

, or equivalently travel time

S / c_{0}

, is stationary under small variations of the ray with fixed endpoints. Stationarity need not mean a global minimum.

For a plane ray written as

y (x)

, use

S [y] = \int F (x, y, y^{'}) d x

, with

F = n (x, y) 1 + y^{'2}

. Its stationary paths obey 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, integrating the term in

δ y^{'}

by parts, and using zero endpoint variations.

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