Fermat's principle

Remarks

Since light has speed $c/n$ in the medium, $\mathcal{L}[\gamma]/c$ is the travel time along $\gamma$, so the principle says the ray makes the travel time stationary. The ray need not minimise the time: a ray reflected by a concave mirror can be a saddle point or a maximum. When $n$ is smooth, the rays are the solutions of the Euler–Lagrange equation of $\mathcal{L}$, which in arc length $\sigma$ reads $\frac{d}{d\sigma}\big(n\, \frac{dx}{d\sigma}\big) = \nabla n$. Equivalently, the rays are the geodesics of the metric $n^2 \abs{dx}^2$. In a homogeneous medium ($n$ constant) the rays are straight lines. Take two homogeneous media with indices $n_1, n_2$ separated by a plane. Minimising $\mathcal{L}$ over broken lines gives Snell's law $n_1 \sin \theta_1 = n_2 \sin \theta_2$. Here $\theta_i$ is the angle between the ray and the normal of the plane. The principle is a statement of physics, not a theorem. In wave optics it is recovered in the limit of short wavelength, where the phase solves the eikonal equation and the rays are its characteristic curves.

Going deeper

For a student

Fermat's principle is the model for all variational principles of physics. The ray equations are the Euler–Lagrange equations of the optical length, and the rays are geodesics of the optical metric $n^2 \abs{dx}^2$. The same formalism appears in mechanics through Maupertuis's principle, whose Jacobi metric plays the role of $n^2 \abs{dx}^2$.