By <Fermat principle>, a <light ray> makes optical travel time stationary. Multiplication by the constant vacuum <speed of light> does not change the stationary paths, so the functional can be taken to be the <refractive index> integrated over <arc length>. The constant denoted $c$ below is an integration constant, not that vacuum speed.

Solution

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 / Solution

 Solution (source code)