Fermat principle

A variational principle that enables one to find rays, that is, curves, along which a wave process propagates. Let $x=x(\sigma)$, $x=(x_1,\ldots,x_m)$, $\sigma_0\leq\sigma\leq\sigma_1$, be the equation of a curve $l$ joining two points $M_0$ and $M_1$, and let $c=c(x)>0$ be the velocity of wave propagation at $x$. The Fermat principle asserts that $\delta\int ds/c=0$ for the ray joining $M_0$ and $M_1$. Here $\delta$ is the variation symbol, and $ds=(\sum(dx_i)^2)^{1/2}$ is the arc differential. The physical meaning of $\int_{M_0}^{M_1}ds/c$ is the time of motion from $M_0$ to $M_1$ along $l$ with velocity $c(x)$. The Fermat principle implies the classical laws of reflection, refraction and straightness of rays for $c=\textrm{const}$. Diffraction rays, rays propagated from the edges of screens, and rays of leading waves can also be found using the Fermat principle. Rays determined by the Fermat principle are characteristics of the eikonal equation. The integral $\int ds/c$ gives a Riemannian metric of a particular type. The rays are the geodesics corresponding to this metric. The Fermat principle can be generalized to the case of a velocity depending on the direction (an anisotropic medium). The rays in this case are the geodesics of some Finsler metric.

The Fermat principle for the problem of the refraction of light was first stated by P. Fermat in about 1660.

For references see Ray method.