 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/9/b/i/solution

The monotonic horizontal coordinate allows the path to be written as a graph. Its arc length element is

d s = 1 + y^{'2} d x

, so the variational density is

f (y, y^{'}) = g (y) 1 + y^{'2}

. It has no explicit horizontal-coordinate dependence. The Beltrami identity becomes

f - y^{'} f_{y^{'}} = g 1 + y^{'2} - \frac{g y ^{'2}}{1 + y ^{'2}} = \frac{g}{1 + y ^{'2}} .

Consequently every smooth minimizing path satisfies

\frac{d}{d x} (\frac{g ( y )}{1 + y ^{'2}}) = 0.

This is a first integral of the weighted arc-length problem; fixed endpoint values remove the endpoint variation terms.

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