= Conserved transverse optical momentum
{title2=$p=\mu(x)y'/\sqrt{1+y'^2}$}
For a positive <refractive index> depending only on $x$, <Fermat principle> applied to $\int\mu(x)\sqrt{1+y'^2}\,dx$ has $y$ as a <cyclic coordinate>. The <Euler-Lagrange equation> therefore makes the displayed quantity constant along a stationary ray. When $\mu^2>p^2$, its slope is $y'=p/\sqrt{\mu^2-p^2}$. If $\theta$ is the angle to the $x$ axis, this is conservation of $\mu\sin\theta$, the stratified-medium form of <Snell's law>. For fixed endpoint values, the integrand has second derivative $\mu(1+y'^2)^{-3/2}>0$ with respect to $y'$. Its strict <convexity> and the constant momentum show, by the tangent-line inequality and integration, that any such ray is the unique minimum among competing graph paths with the same endpoints.
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