# Why does light take the shortest path?

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

• Possibly enlightening: Feynman QED en.wikipedia.org/wiki/… (Watch @ 17m30s in the video at vega.org.uk/video/programme/46 ) Mar 6 at 15:24
• One may be able to skip to 28m40s in vega.org.uk/video/programme/46 . Mar 6 at 15:40
• You asked, "how does the light know those 2 points...?" But, why do you think that a light ray "knows" where it is going? Mar 6 at 18:25

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.