How is Fermat's least time principle proven?

Question

How is Fermat's least time principle proven? Or it is what usually is observed and is basis for the theories?

Answer 1

For geometrical optics we can introduce eikonal $\Psi$ by relation $$ f = Ae^{-ik_{\mu}r^{\mu} + i\varphi} = Ae^{i\Psi}. \tag 1 $$ For small time interval and space lengths eikonal can be expanded in a form $$ \Psi = \Psi_{0} + \mathbf r \frac{\partial \Psi}{\partial \mathbf r} + t \frac{\partial \Psi}{\partial t}, $$ so, by comparing with left side $(1)$ it can be directly show that $$ \mathbf k = \frac{\partial \Psi}{\partial \mathbf r}, \quad \omega = -\frac{\partial \Psi}{\partial t}. \tag 2 $$ So by comparing $(2)$ with Hamilton-Jacobi equations we have clear analogy: wave vector acts rule of classical momentum and frequency acts rule of Hamiltonian in geometrical optics. So it is possible to introduce the analogy of principle of the least action for rays. For light it can be done by Maupertuis principle: $$ \delta S = \delta \int \mathbf p \,\mathrm d \mathbf l = 0 \to \delta \Psi = \delta \int \mathbf k\,\mathrm d \mathbf l = 0. $$ For example, for optically isotropic and homogeneous space $\mathbf k = \textrm{const} \cdot \mathbf n $, and $$ \delta \int \mathrm dl = 0, $$

which leads to Fermat's least time principle.