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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^3$ be a region filled with an isotropic optical medium of refractive index $n \\colon \\Omega \\to (0, \\infty)$.\nFor a piecewise smooth curve $\\gamma \\colon [a, b] \\to \\Omega$ define the *optical path length*\n$$\\mathcal{L}[\\gamma] = \\int_a^b n(\\gamma(s))\\, \\abs{\\gamma'(s)} \\, ds.$$\nLet $A$ and $B$ be two points of $\\Omega$.\nIn geometric optics, the path of a light ray from $A$ to $B$ is a critical point of $\\mathcal{L}$ among curves from $A$ to $B$.\n\n## Remarks\n\nSince 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.\nThe ray need not minimise the time: a ray reflected by a concave mirror can be a saddle point or a maximum.\nWhen $n$ is smooth, the rays are the solutions of the [[Euler–Lagrange equation|PS0VV149ET]] of $\\mathcal{L}$, which in arc length $\\sigma$ reads $\\frac{d}{d\\sigma}\\big(n\\, \\frac{dx}{d\\sigma}\\big) = \\nabla n$.\nEquivalently, the rays are the [[geodesics|NSYMQMHPDN]] of the metric $n^2 \\abs{dx}^2$.\nIn a homogeneous medium ($n$ constant) the rays are straight lines.\nTake two homogeneous media with indices $n_1, n_2$ separated by a plane.\nMinimising $\\mathcal{L}$ over broken lines gives Snell's law $n_1 \\sin \\theta_1 = n_2 \\sin \\theta_2$.\nHere $\\theta_i$ is the angle between the ray and the normal of the plane.\nThe principle is a statement of physics, not a theorem.\nIn wave optics it is recovered in the limit of short wavelength, where the phase solves the [[eikonal equation|HGMX1QGTAZ]] and the rays are its characteristic curves.\n\n## Explanations\n\n### L1 human\n\nLight travelling from one point to another follows the path that takes the least time.\nMore exactly, its travel time does not change to first order under small changes of the path.\nThis explains why light goes straight in air, why it bends when it enters water, and why angles of incidence and reflection are equal at a mirror.\n\n### L2 human\n\nFermat's principle is the model for all variational principles of physics.\nThe ray equations are the Euler–Lagrange equations of the optical length, and the rays are geodesics of the optical metric $n^2 \\abs{dx}^2$.\nThe same formalism appears in mechanics through [[Maupertuis's principle|ZXY0AYKMXZ]], whose Jacobi metric plays the role of $n^2 \\abs{dx}^2$.\n",
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