In Wikipedia Fermat's Principle is stated as:

A ray of light prefers the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the same time to traverse.

So how does the principle for shortest time come from this statement? Is there any mathematical derivation or any assumption to make?


Consider two points $a$ and $b$ and ray traveling between these two points. Now imagine two nearby paths $p({\bf x})$ and $p({\bf x}) + \epsilon q({\bf x})$, connecting those points. Here $\epsilon$ indicates a small number and $q(x)$ is an arbitrary (well behaved) function. Clearly the traversing time for a ray depends on the path $t = t(p)$

The version above just indicates that

$$ t(p) \approx t(p + \epsilon q) $$

Or in other words

$$ \lim_{\epsilon \to 0}\frac{t(p) - t(p + \epsilon q)}{\epsilon} = 0 = \int_a^b \frac{\delta t}{\delta p}q ~{\rm d}x = \delta \int_a^b t(p){\rm d}x $$

where I used the functional derivative $\delta$ in the last step. The last equation is just the variational formulation of the principle

  • $\begingroup$ Can you explain the last bit please? I couldn't understand that mathematical part well. $\endgroup$ – Theoretical Dec 28 '18 at 13:42
  • $\begingroup$ @Theoretical The first identity comes from the fact that the times are pretty similar, so the difference is zero. The second identity is just the definition of functional derivative, see the linked Wiki $\endgroup$ – caverac Dec 28 '18 at 13:48

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