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Principle of Least Time

Electromagnetic waves propagate through space with the shortest possible time, reflecting the fundamental principle of least action in physics.
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The statement of the theorem

The Principle of Least Time asserts that the path r⃗(s)\vec{r}(s) taken by an electromagnetic wave propagating from point AA to point BB in a medium characterized by a spatially dependent refractive index n(r⃗)n(\vec{r}) is the path that extremizes the travel time functional TT. This minimization is formulated via the calculus of variations:\n\nT[r⃗(s)]=∫ABn(r⃗)c(dxds)2+(dyds)2+(dzds)2dsT[\vec{r}(s)] = \int_{A}^{B} \frac{n(\vec{r})}{c} \sqrt{\left(\frac{dx}{ds}\right)^2 + \left(\frac{dy}{ds}\right)^2 + \left(\frac{dz}{ds}\right)^2} ds\n\nThe path r⃗(s)=(x(s),y(s),z(s))\vec{r}(s) = (x(s), y(s), z(s)) must satisfy the condition that the variation of the time functional vanishes: \n\nδT=δ∫ABn(r⃗)cds=0\delta T = \delta \int_{A}^{B} \frac{n(\vec{r})}{c} ds = 0\n\nThis leads to the Euler-Lagrange equations for the components of the path, where the Lagrangian density is L=n(r⃗)cx˙2+y˙2+z˙2L = \frac{n(\vec{r})}{c} \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2} (using x˙=dx/ds\dot{x} = dx/ds for simplicity in the functional form). The resulting differential equations define the trajectory of the wave front, which are equivalent to the eikonal equation derived from the wave equation in the paraxial approximation.