Editing Calculus of variations (section)

== Applications ==

=== 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.

=== Mechanics ===
{{main|Action (physics)}}
In classical mechanics, the action, <math>S,</math> is defined as the time integral of the Lagrangian, <math>L.</math>  The Lagrangian is the difference of energies,
<math display="block">L = T - U, </math>
where <math>T</math> is the [[kinetic energy]] of a mechanical system and <math>U</math> its [[potential energy]]. [[Hamilton's principle]] (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral
<math display="block">S = \int_{t_0}^{t_1} L(x, \dot x, t) \, dt</math>
is stationary with respect to variations in the path <math>x(t).</math>
The Euler–Lagrange equations for this system are known as Lagrange's equations:
<math display="block">\frac{d}{dt} \frac{\partial L}{\partial \dot x} = \frac{\partial L}{\partial x}, </math>
and they are equivalent to Newton's equations of motion (for such systems).

The conjugate momenta <math>P</math> are defined by
<math display="block">p = \frac{\partial L}{\partial \dot x}. </math>
For example, if
<math display="block">T = \frac{1}{2} m \dot x^2, </math>
then <math display="block">p = m \dot x. </math>
[[Hamiltonian mechanics]] results if the conjugate momenta are introduced in place of <math>\dot x</math> by a Legendre transformation of the Lagrangian <math>L</math> into the Hamiltonian <math>H</math> defined by
<math display="block">H(x, p, t) = p \,\dot x - L(x,\dot x, t).</math>
The Hamiltonian is the total energy of the system: <math>H = T + U.</math>
Analogy with Fermat's principle suggests that solutions of Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function of <math>X.</math> This function is a solution of the [[Hamilton–Jacobi equation]]:
<math display="block">\frac{\partial \psi}{\partial t} + H\left(x,\frac{\partial \psi}{\partial x},t\right) = 0.</math>

=== Further applications ===

Further applications of the calculus of variations include the following:

* The derivation of the [[catenary]] shape
* Solution to [[Newton's minimal resistance problem]]
* Solution to the [[Brachistochrone curve|brachistochrone]] problem
* Solution to the [[Tautochrone curve|tautochrone problem]]
* Solution to [[isoperimetric]] problems
* Calculating [[geodesic]]s
* Finding [[minimal surface]]s and solving [[Plateau's problem]]
* [[Optimal control]]
* [[Analytical mechanics]], or reformulations of Newton's laws of motion, most notably [[Lagrangian mechanics|Lagrangian]] and [[Hamiltonian mechanics]];
* Geometric optics, especially Lagrangian and [[Hamiltonian optics]];
* [[Variational method (quantum mechanics)]], one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states;
* [[Variational Bayesian methods]], a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning;
* [[Variational methods in general relativity]], a family of techniques using calculus of variations to solve problems in Einstein's general theory of relativity;
* [[Finite element method]] is a variational method for finding numerical solutions to boundary-value problems in differential equations;
* [[Total variation denoising]], an [[image processing]] method for filtering high variance or noisy signals.