Editing Calculus of variations (section)

=== Optics ===
[[Fermat's principle]] states that light takes a path that (locally) minimizes the optical length between its endpoints. If the <math>x</math>-coordinate is chosen as the parameter along the path, and <math>y=f(x)</math> along the path, then the optical length is given by
<math display="block">A[f] = \int_{x_0}^{x_1} n(x,f(x)) \sqrt{1 + f'(x)^2} dx, </math>
where the refractive index <math>n(x,y)</math> depends upon the material.
If we try <math>f(x) = f_0 (x) + \varepsilon f_1 (x)</math> then the [[first variation]] of <math>A</math> (the derivative of <math>A</math> with respect to ε) is
<math display="block">\delta A[f_0,f_1] = \int_{x_0}^{x_1} \left[ \frac{ n(x,f_0) f_0'(x) f_1'(x)}{\sqrt{1 + f_0'(x)^2}} + n_y (x,f_0) f_1 \sqrt{1 + f_0'(x)^2} \right] dx.</math>

After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation
<math display="block">-\frac{d}{dx} \left[\frac{ n(x,f_0) f_0'}{\sqrt{1 + f_0'^2}} \right] + n_y (x,f_0) \sqrt{1 + f_0'(x)^2} = 0. </math>

The light rays may be determined by integrating this equation. This formalism is used in the context of [[Lagrangian optics]] and [[Hamiltonian optics]].

==== Snell's law ====
There is a discontinuity of the refractive index when light enters or leaves a lens. Let
<math display="block">n(x,y) = \begin{cases}
n_{(-)} & \text{if} \quad x<0, \\
n_{(+)} & \text{if} \quad x>0,
\end{cases}</math>
where <math>n_{(-)}</math> and <math>n_{(+)}</math> are constants. Then the Euler–Lagrange equation holds as before in the region where <math>x < 0</math> or <math>x > 0,</math> and in fact the path is a straight line there, since the refractive index is constant. At the <math>x = 0,</math> <math>f</math> must be continuous, but <math>f'</math> may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form
<math display="block">\delta A[f_0,f_1] = f_1(0)\left[ n_{(-)}\frac{f_0'(0^-)}{\sqrt{1 + f_0'(0^-)^2}} - n_{(+)}\frac{f_0'(0^+)}{\sqrt{1 + f_0'(0^+)^2}} \right].</math>

The factor multiplying <math>n_{(-)}</math> is the sine of angle of the incident ray with the <math>x</math> axis, and the factor multiplying <math>n_{(+)}</math> is the sine of angle of the refracted ray with the <math>x</math> axis.  [[Snell's law]] for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.

==== Fermat's principle in three dimensions ====
It is expedient to use vector notation: let <math>X = (x_1,x_2,x_3),</math> let <math>t</math> be a parameter,  let <math>X(t)</math> be the parametric representation of a curve <math>C,</math> and let <math>\dot X(t)</math> be its tangent vector. The optical length of the curve is given by
<math display="block">A[C] = \int_{t_0}^{t_1} n(X) \sqrt{ \dot X \cdot \dot X} \, dt. </math>

Note that this integral is invariant with respect to changes in the parametric representation of <math>C.</math> The Euler–Lagrange equations for a minimizing curve have the symmetric form
<math display="block">\frac{d}{dt} P = \sqrt{ \dot X \cdot \dot X} \, \nabla n, </math>
where
<math display="block">P = \frac{n(X) \dot X}{\sqrt{\dot X \cdot \dot X} }.</math>

It follows from the definition that <math>P</math> satisfies
<math display="block">P \cdot P = n(X)^2. </math>

Therefore, the integral may also be written as
<math display="block">A[C] = \int_{t_0}^{t_1} P \cdot \dot X \, dt.</math>

This form suggests that if we can find a function <math>\psi</math> whose gradient is given by <math>P,</math> then the integral <math>A</math> is given by the difference of <math>\psi</math> at the endpoints of the interval of integration. Thus the problem of studying the curves that make the integral stationary can be related to the study of the level surfaces of <math>\psi.</math>In order to find such a function, we turn to the wave equation, which governs the propagation of light. This formalism is used in the context of [[Lagrangian optics]] and [[Hamiltonian optics]].

===== 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.