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There have been a lot of duplicates for this question , but my question asks a bit something else ..

1.Does the Fermat's principle have any intuitive/ mathematical explanation , If so , it would very helpful if you attach a link or simply post one .

  1. Secondly, I am familiar that the Snell's Law can be derived by it and the lifeguard - water-sand situation, but none answer my doubt , which is why does light take the shortest path, where the path is between 2 points , how does the light knows those 2 points and accordingly takes the shortest path ?

Thanks

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Snell's law of refraction is essentially equivalent to Fermat's principle. Look at the picture below, it shows two neighboring rays starting at point $S$ in a medium of refractive index $\eta_1$ and ending at point $F$ in a medium $\eta_2$, the path length from $S$ to $F$ then $$\eta_1 r +\eta_2 \rho = \eta_1 (r+dr) + \eta_2 (\rho-d\rho) \tag{1}\label{1}$$ to 1st order differentials. This can be written as $$\eta_1 dr = \eta_2 d\rho \tag{2}\label{2}.$$ Now if you divide both sides of $\eqref{2}$ by $ds$ you get Snell's law. Of course, the steps can be taken in reverse and you can get from $\eqref{2}$ to $\eqref{1}$ and get the differential from of Fermat's law of stationarity from Snell's law of refraction. The equation $\eqref{1}$ means that within an error of 2nd order, i.e, equality within a 1st order, the optical path length (time of flight) between fixed points of neighboring rays are equal.


The argument and the picture are taken from this excellent article by Cornbleet, Geometrical Optics Reviewed: A New Light on an Old Subject, PROCEEDINGS OF THE IEEE, VOL. 71, NO. 4, APRIL 1983. law of refraction

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