Calculus of variations

Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. Such functionals can for example be formed as integrals involving an unknown function and its derivatives. The interest is in extremal functions: those making the functional attain a maximum or minimum value. Perhaps the simplest example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is obviously a straight line, but if the curve is constrained to lie on a surface in space, then it is less obvious. The solutions of the latter problem are called geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. A corresponding idea in mechanics is the principle of stationary action. The theory of optimal control is a generalization of the calculus of variations.

The preceding examples have all involved unknown functions of a single variable, which my be identified with a time variable. Other important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle: they minimize the potential energy of a membrane. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: the solution or solutions may be found by dipping a wire frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

The Euler-Lagrange equation

The maxima and minima of a given function may be located by finding the points where its derivative vanishes. In analogy, solutions of variational problems may be obtained by solving the associated Euler-Lagrange equation. In order to illustrate this process, consider the problem of finding the shortest curve in the plane that connects two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$. The arc length is given by

$A[f]=\int _{x_{1}}^{x_{2}}{\sqrt {1+\left({\frac {df}{dx}}\right)^{2}}}\,dx,$

where $y=f(x)$, $f(x_{1})=y_{1}$ and $f(x_{2})=y_{2}.$ The function f should have at least one derivative in order to be admitted to the competition. If $f_{0}$ is a minimizing function and $f_{1}$ is any function with at least one derivative that vanishes at the endpoints $x_{1}$ and $x_{2}$, then we must have

$A[f_{0}]\leq A[f_{0}+\epsilon f_{1}]$

for any number ε. Therefore, the derivative of $L[f_{0}+\epsilon f_{1}]$ with respect to ε (the first variation of A) must vanish at ε=0. Thus

$\int _{x_{1}}^{x_{2}}{\frac {{\frac {df_{0}}{dx}}{\frac {df_{1}}{dx}}}{\sqrt {1+\left({\frac {df_{0}}{dx}}\right)^{2}}}}=0,\,$

for any choice of the function $f_{1}$. We may interpret this condition as the vanishing of all directional derivatives of $A[f_{0}]$ in the space of differentiable functions. If we assume that $f_{0}$ has two continuous derivatives, then it follows from integration by parts that

$\int _{x_{1}}^{x_{2}}f_{1}(x){\frac {d}{dx}}\left[{\frac {\frac {df_{0}}{dx}}{\sqrt {1+\left({\frac {df_{0}}{dx}}\right)^{2}}}}\right]\,dx=0,$

for any choice of $f_{1}$ with two derivatives that vanishes at the endpoints of the interval. This is a special case of the following situation:

$I=\int _{x_{1}}^{x_{2}}f_{1}(x)H(x)dx=0,\,$

for any $f_{1}(x)$ with one derivative that vanishes at the endpoints on the interval. If $H(x)>0$ at a point ${\hat {x}},$ then there is an interval surrounding ${\hat {x}}$ where H is positive. We may choose $f_{1}$ to vanish outside that interval, and to be non-negative inside. With this choice, $I>0,$ which is a contradiction. There is a similar argument if $H(x)<0$ at a point ${\hat {x}}$. We conclude that

${\frac {d}{dx}}\left[{\frac {\frac {df_{0}}{dx}}{\sqrt {1+\left({\frac {df_{0}}{dx}}\right)^{2}}}}\right]=0.\,$

It follows from this equation that

${\frac {d^{2}f_{0}}{dx^{2}}}=0,$

and hence the extremals are straight lines.

A similar calculation holds in the general case where

$A[f]=\int _{x_{1}}^{x_{2}}L(x,f,f')\,dx\,$

with

$f'(x)={\frac {df}{dx}},\,$

and f is required to have two continuous derivatives. In that case, an extremal $f_{0}$ will satisfy the Euler-Lagrange equation

$-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}+{\frac {\partial L}{\partial f}}=0.\,$

The Euler-Lagrange equation is a necessary condition for an extremal. but its satisfaction does not guarantee that the solution is in fact an extremal. Sufficient conditions for an extremal are discussed in the references.

du Bois Raymond's theorem

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral A requires only first derivatives of trial functions. The condition that the first variation vanish at a extremal may be regarded as a weak form of the Euler-Lagrange equation. The theorem of du Bois Raymond asserts that this weak form implies the strong form. If L has continuous first and second derivatives with respect to all of its arguments, and if

${\frac {\partial ^{2}L}{(\partial f')^{2}}}\neq 0,$

then $f_{0}$ has two continuous derivatives, and it satisfies the Euler-Lagrange equation.

Fermat's principle

Fermat's principle states that light takes a path that minimizes the optical length between its endpoints. If the x-coordinate is chosen as the parameter along the path, and $y=f(x)$ along the path, then the optical length is given by

$A[f]=\int _{x=x_{0}}^{x_{1}}n(x,f(x)){\sqrt {1+f'(x)^{2}}}dx,\,$

where the refractive index $n(x,y)$ depends upon the material. If we try $f(x)=f_{0}(x)+\epsilon f_{1}(x)$ then the first variation of A (the derivative of A with respect to ε) is

$\delta A[f_{0},f_{1}]=\int _{x=x_{0}}^{x_{1}}\left[{\frac {n(x,f_{0})f_{0}'(x)f_{1}'(x)}{\sqrt {1+f_{0}'(x)^{2}}}}+n_{y}(x,f_{0})f_{1}\right]dx\,$

After integration by parts of the first term within brackets, we obtain the Euler-Lagrange equation

$-{\frac {d}{dx}}\left[{\frac {n(x,f_{0})f_{0}'}{\sqrt {1+f_{0}'^{2}}}}\right]+n_{y}(x,f_{0})=0.\,$

The light rays may be determined by integrating this equation.

Snell's law

There is a discontinuity of the refractive index when light enters or leaves a lens. Let

$n(x,y)=n_{-}\quad {\hbox{if}}\quad x<0,\,$

$n(x,y)=n_{+}\quad {\hbox{if}}\quad x>0,\,$

where $n_{-}$ and $n_{+}$ are constants. Then the Euler-Lagrange equation holds as before in the region where x<0 or x>0, and in fact the path is a straight line there, since the refractive index is constant. At the x=0, f must be continuous, but f' may be discontinuous. After integration by parts in the separate regions and using the Euler-Lagrange equations, the first variation takes the form

$\delta A[f_{0},f_{1}]=f_{1}(0)\left[n_{-}{\frac {f_{0}'(0_{-})}{\sqrt {1+f_{0}'(0_{-})^{2}}}}-n_{+}{\frac {f_{0}'(0_{+})}{\sqrt {1+f_{0}'(0_{+})^{2}}}}\right].\,$

The factor multiplying $n_{-}$ is the sine of angle of the incident ray with the x axis, and the factor multiplying $n_{+}$ is the sine of angle of the refracted ray with the x axis. This is Snell's law for refraction. 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 $X=(x_{1},x_{2},x_{3}),$ let t be a parameter, let $X(t)$ be the parametric representation of a curve C, and let ${\dot {X}}(t)$ be its tangent vector. The optical length of the curve is given by

$A[C]=\int _{t=t_{0}}^{t_{1}}n(X){\sqrt {{\dot {X}}\cdot {\dot {X}}}}dt.\,$

Note that this integral is invariant with respect to changes in the parametric representation of C. The Euler-Lagrange equations for a minimizing curve have the symmetric form

${\frac {d}{dt}}P={\sqrt {{\dot {X}}\cdot {\dot {X}}}}\nabla n,\,$

where

$P={\frac {n(x){\dot {X}}}{\sqrt {{\dot {X}}\cdot {\dot {X}}}}}.\,$

It follows from the definition that P satisfies

$P\cdot P=n(X)^{2}.\,$

Therefore the integral may also be written as

$A[C]=\int _{t=t_{0}}^{t_{1}}P\cdot {\dot {X}}\,dt.\,$

Connection with the wave equation

The wave equation for an inhomogeneous medium is

$u_{tt}=c^{2}\nabla \cdot \nabla u,\,$

where c is the propagation velocity, which generally depends upon X. Wave fronts are characteristic surfaces for this partial differential equation: they satisfy

$\varphi _{t}^{2}=c(X)^{2}\nabla \varphi \cdot \nabla \varphi .\,$

We may look for solutions in the form

$\varphi (t,X)=t-\psi (X).\,$

In that case, ψ satisfies

$\nabla \psi \cdot \nabla \psi =n^{2},\,$

where $n=1/c.$ According to the theory of first order partial differential equations, if $P=\nabla \psi ,$ then P satisfies

${\frac {dP}{ds}}=2n\nabla n,\,$

along a system of curves (the rays) that are given by

${\frac {dX}{ds}}=P.\,$

These equations are identical to the Euler-Lagrange equations if we make the identification

${\frac {ds}{dt}}={\frac {\sqrt {{\dot {X}}\cdot {\dot {X}}}}{n}}.\,$

We conclude that the function ψ is the value of the minimizing integral A 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 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 general variational problems.

See also

• Isoperimetric inequality
• Variational principle
• Fermat's principle
• Principle of least action
• Infinite-dimensional optimization
• Functional analysis
• Perturbation methods

Reference books

• Fomin, S.V. and Gelfand, I.M.: Calculus of Variations, Dover Publ., 2000
• Lebedev, L.P. and Cloud, M.J.: The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics, World Scientific, 2003, pages 1-98
• Charles Fox: An Introduction to the Calculus of Variations, Dover Publ., 1987
• Forsyth, A.R.: Calculus of Variations, Dover, 1960
• Sagan, Hans: Introduction to the Calculus of Variations, Dover, 1992
• Weinstock, Robert: Calculus of Variations with Applications to Physics and Engineering, Dover, 1974
• Clegg, J.C.: Calculus of Variations, Interscience Publishers Inc., 1968
• Elsgolc, L.E.: Calculus of Variations, Pergamon Press Ltd., 1962

External links

• Chapter III: Introduction to the calculus of variations by Johan Byström, Lars-Erik Persson, and Fredrik Strömberg
• PlanetMath.org: Calculus of variations
• Wolfram Research's MathWorld: Calculus of Variations
• Example problems in the calculus of variations