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Every proof I've seen of Snell's law from Fermat's principle uses some sort of variational argument, mostly involving variational calculus. Niven's wonderful book, Maxima and Minima Without Calculus has a geometric derivation, but also has to rely on a variational argument (e.g. one that treats the point of refraction R in a broken line segment PR+RQ, representing a luminal trajectory, as a perimeter and showing that the path's optical length is minimum iff R satisfies Snell's law).

However, is there a simple derivation that does not rely on a variational argument. I feel that there should. The transition from path length to optical length seems to be a piecewise geodesic diffeomorphism of the Euclidean plane (minus the line of refraction) such that straight lines transform to optical geodesics, which is done by some sort of scaling transformations on the half-planes (homothety about R?). Is there a derivation of Snell's law that approximately follows this line of thinking?

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The geodesic equation itself comes from a variational argument, so in the end you can't really get away from it ... –  Neal Feb 9 '13 at 0:45
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Can you explain what a non variational argument based on a variational principle would look like? For example, what is an argument for straight paths in homogeneous media based on Fermat which doesn't use any "variational argument"? –  Sharkos Jun 18 '13 at 18:24

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Snall's Law may be derived from Maxwell's equations, specifically, by the requirement that the normal component of the displacement vector $\vec{D} = \epsilon \vec{E}$ be continuous across a boundary between different media. While this does not rely on Fermat directly, and is really a statement of the directions of plane waves rather than rays ( which are really just normals to wavefronts in non-catastrophic situations), there is an equivalence in the limit of small wavelength.

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