Editing Calculus of variations (section)

===== Connection with the wave equation =====
The [[wave equation]] for an inhomogeneous medium is
<math display="block">u_{tt} = c^2 \nabla \cdot \nabla u, </math>
where <math>c</math> is the velocity, which generally depends upon <math>X.</math> Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy
<math display="block">\varphi_t^2 = c(X)^2 \, \nabla \varphi \cdot \nabla \varphi. </math>

We may look for solutions in the form
<math display="block">\varphi(t,X) = t - \psi(X). </math>

In that case, <math>\psi</math> satisfies
<math display="block">\nabla \psi \cdot \nabla \psi = n^2, </math>
where <math>n=1/c.</math> According to the theory of [[first-order partial differential equation]]s, if <math>P = \nabla \psi,</math> then <math>P</math> satisfies
<math display="block">\frac{dP}{ds} = n \, \nabla n,</math>
along a system of curves ('''the light rays''') that are given by
<math display="block">\frac{dX}{ds} = P. </math>

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification
<math display="block">\frac{ds}{dt} = \frac{\sqrt{ \dot X \cdot \dot X} }{n}. </math>

We conclude that the function <math>\psi</math> is the value of the minimizing integral <math>A</math> as a function of the upper end point. That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem. This is the essential content of the [[Hamilton–Jacobi theory]], which applies to more general variational problems.