Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 9 / ii / a

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- Solution a

Solution

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

has no explicit dependence on

y

, so part (i) gives

\frac{n ( x ) y ^{'}}{1 + ( y ^{'} ) ^{2}} = c .

For positive refractive index and a regular graph with

n (x) > ∣ c ∣

, squaring and then restoring the sign of

y^{'}

yields

\frac{d y}{d x} = \frac{c}{n ( x ) ^{2} - c ^{2}} .

It is the continuous-medium version of Snell's law, expressing conserved horizontal optical momentum. The Euler-Lagrange equation establishes stationarity; on a monotone-height segment, convexity of the integrand in

y^{'}

also gives a minimum among graphs with fixed endpoints.

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