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The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.^[a] Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

History

The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696).^[2] It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.^[3][4][b]

Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. Isaac Newton and Gottfried Leibniz also gave some early attention to the subject.^[5] To this discrimination Vincenzo Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development.^[5]

In the 20th century David Hilbert, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions.^[5] Marston Morse applied calculus of variations in what is now called Morse theory.^[6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory.^[6] The dynamic programming of Richard Bellman is an alternative to the calculus of variations.^[7][8][9][c]

Extrema

The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements ${\displaystyle y}$ of a given function space defined over a given domain. A functional ${\displaystyle J[y]}$ is said to have an extremum at the function ${\displaystyle f}$ if ${\displaystyle \Delta J=J[y]-J[f]}$ has the same sign for all ${\displaystyle y}$ in an arbitrarily small neighborhood of ${\displaystyle f.}$^[d] The function ${\displaystyle f}$ is called an extremal function or extremal.^[e] The extremum ${\displaystyle J[f]}$ is called a local maximum if ${\displaystyle \Delta J\leq 0}$ everywhere in an arbitrarily small neighborhood of ${\displaystyle f,}$ and a local minimum if ${\displaystyle \Delta J\geq 0}$ there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.^[11]

Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the converse may not hold. Finding strong extrema is more difficult than finding weak extrema.^[12] An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.^[13][f]

Euler–Lagrange equation

Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to zero. This leads to solving the associated Euler–Lagrange equation.^[g]

Consider the functional

$${\displaystyle J[y]=\int _{x_{1))^{x_{2))L\left(x,y(x),y'(x)\right)\,dx\,.}$$

• ${\displaystyle x_{1},x_{2))$ are constants,
• ${\displaystyle y(x)}$ is twice continuously differentiable,
• ${\displaystyle y'(x)={\frac {dy}{dx)),}$
• ${\displaystyle L\left(x,y(x),y'(x)\right)}$ is twice continuously differentiable with respect to its arguments ${\displaystyle x,y,}$ and ${\displaystyle y'.}$

If the functional ${\displaystyle J[y]}$ attains a local minimum at ${\displaystyle f,}$ and ${\displaystyle \eta (x)}$ is an arbitrary function that has at least one derivative and vanishes at the endpoints ${\displaystyle x_{1))$ and ${\displaystyle x_{2},}$ then for any number ${\displaystyle \varepsilon }$ close to 0,

$${\displaystyle J[f]\leq J[f+\varepsilon \eta ]\,.}$$

The term ${\displaystyle \varepsilon \eta }$ is called the variation of the function ${\displaystyle f}$ and is denoted by ${\displaystyle \delta f.}$^[1][h]

Substituting ${\displaystyle f+\varepsilon \eta }$ for ${\displaystyle y}$ in the functional ${\displaystyle J[y],}$ the result is a function of ${\displaystyle \varepsilon ,}$

$${\displaystyle \Phi (\varepsilon )=J[f+\varepsilon \eta ]\,.}$$

${\displaystyle J[y]}$

${\displaystyle y=f}$

${\displaystyle \Phi (\varepsilon )}$

${\displaystyle \varepsilon =0}$

$${\displaystyle \Phi '(0)\equiv \left.{\frac {d\Phi }{d\varepsilon ))\right|_{\varepsilon =0}=\int _{x_{1))^{x_{2))\left.{\frac {dL}{d\varepsilon ))\right|_{\varepsilon =0}dx=0\,.}$$

Taking the total derivative of ${\displaystyle L\left[x,y,y'\right],}$ where ${\displaystyle y=f+\varepsilon \eta }$ and ${\displaystyle y'=f'+\varepsilon \eta '}$ are considered as functions of ${\displaystyle \varepsilon }$ rather than ${\displaystyle x,}$ yields

$${\displaystyle {\frac {dL}{d\varepsilon ))={\frac {\partial L}{\partial y)){\frac {dy}{d\varepsilon ))+{\frac {\partial L}{\partial y')){\frac {dy'}{d\varepsilon ))}$$

${\displaystyle {\frac {dy}{d\varepsilon ))=\eta }$

${\displaystyle {\frac {dy'}{d\varepsilon ))=\eta ',}$

$${\displaystyle {\frac {dL}{d\varepsilon ))={\frac {\partial L}{\partial y))\eta +{\frac {\partial L}{\partial y'))\eta '.}$$

Therefore,

$${\displaystyle {\begin{aligned}\int _{x_{1))^{x_{2))\left.{\frac {dL}{d\varepsilon ))\right|_{\varepsilon =0}dx&=\int _{x_{1))^{x_{2))\left({\frac {\partial L}{\partial f))\eta +{\frac {\partial L}{\partial f'))\eta '\right)\,dx\\&=\int _{x_{1))^{x_{2)){\frac {\partial L}{\partial f))\eta \,dx+\left.{\frac {\partial L}{\partial f'))\eta \right|_{x_{1))^{x_{2))-\int _{x_{1))^{x_{2))\eta {\frac {d}{dx)){\frac {\partial L}{\partial f'))\,dx\\&=\int _{x_{1))^{x_{2))\left({\frac {\partial L}{\partial f))\eta -\eta {\frac {d}{dx)){\frac {\partial L}{\partial f'))\right)\,dx\\\end{aligned))}$$

${\displaystyle L\left[x,y,y'\right]\to L\left[x,f,f'\right]}$

${\displaystyle \varepsilon =0}$

${\displaystyle \eta =0}$

${\displaystyle x_{1))$

${\displaystyle x_{2))$

$${\displaystyle \int _{x_{1))^{x_{2))\eta (x)\left({\frac {\partial L}{\partial f))-{\frac {d}{dx)){\frac {\partial L}{\partial f'))\right)\,dx=0\,.}$$

According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e.

$${\displaystyle {\frac {\partial L}{\partial f))-{\frac {d}{dx)){\frac {\partial L}{\partial f'))=0}$$

${\displaystyle J[f]}$

${\displaystyle \delta J/\delta f(x).}$

In general this gives a second-order ordinary differential equation which can be solved to obtain the extremal function ${\displaystyle f(x).}$ The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum ${\displaystyle J[f].}$ A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.

Example

In order to illustrate this process, consider the problem of finding the extremal function ${\displaystyle y=f(x),}$ which is the shortest curve that connects two points ${\displaystyle \left(x_{1},y_{1}\right)}$ and ${\displaystyle \left(x_{2},y_{2}\right).}$ The arc length of the curve is given by

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

$${\displaystyle y'(x)={\frac {dy}{dx))\,,\ \ y_{1}=f(x_{1})\,,\ \ y_{2}=f(x_{2})\,.}$$

The Euler–Lagrange equation will now be used to find the extremal function ${\displaystyle f(x)}$ that minimizes the functional ${\displaystyle A[y].}$

$${\displaystyle {\frac {\partial L}{\partial f))-{\frac {d}{dx)){\frac {\partial L}{\partial f'))=0}$$

$${\displaystyle L={\sqrt {1+[f'(x)]^{2))}\,.}$$

Since ${\displaystyle f}$ does not appear explicitly in ${\displaystyle L,}$ the first term in the Euler–Lagrange equation vanishes for all ${\displaystyle f(x)}$ and thus,

$${\displaystyle {\frac {d}{dx)){\frac {\partial L}{\partial f'))=0\,.}$$

${\displaystyle L}$

$${\displaystyle {\frac {d}{dx))\ {\frac {f'(x)}{\sqrt {1+[f'(x)]^{2))))\ =0\,.}$$

Thus

$${\displaystyle {\frac {f'(x)}{\sqrt {1+[f'(x)]^{2))))=c\,,}$$

${\displaystyle c.}$

$${\displaystyle {\frac {[f'(x)]^{2)){1+[f'(x)]^{2))}=c^{2}\,,}$$

$${\displaystyle 0\leq c^{2}<1.}$$

$${\displaystyle [f'(x)]^{2}={\frac {c^{2)){1-c^{2))))$$

$${\displaystyle f'(x)=m}$$

${\displaystyle \left(x_{1},y_{1}\right)}$

${\displaystyle \left(x_{2},y_{2}\right)}$

$${\displaystyle f(x)=mx+b\qquad {\text{with))\ \ m={\frac {y_{2}-y_{1)){x_{2}-x_{1))}\quad {\text{and))\quad b={\frac {x_{2}y_{1}-x_{1}y_{2)){x_{2}-x_{1))))$$

${\displaystyle f(x)}$

${\displaystyle A[y]}$

${\displaystyle A[f]}$

${\displaystyle y=f(x).}$

Beltrami's identity

In physics problems it may be the case that ${\displaystyle {\frac {\partial L}{\partial x))=0,}$ meaning the integrand is a function of ${\displaystyle f(x)}$ and ${\displaystyle f'(x)}$ but ${\displaystyle x}$ does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity^[16]

$${\displaystyle L-f'{\frac {\partial L}{\partial f'))=C\,,}$$

${\displaystyle C}$

${\displaystyle L}$

${\displaystyle f'(x).}$

The intuition behind this result is that, if the variable ${\displaystyle x}$ is actually time, then the statement ${\displaystyle {\frac {\partial L}{\partial x))=0}$ implies that the Lagrangian is time-independent. By Noether's theorem, there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.

Euler–Poisson equation

If ${\displaystyle S}$ depends on higher-derivatives of ${\displaystyle y(x),}$ that is, if

$${\displaystyle S=\int _{a}^{b}f(x,y(x),y'(x),\dots ,y^{(n)}(x))dx,}$$

${\displaystyle y}$

$${\displaystyle {\frac {\partial f}{\partial y))-{\frac {d}{dx))\left({\frac {\partial f}{\partial y'))\right)+\dots +(-1)^{n}{\frac {d^{n)){dx^{n))}\left[{\frac {\partial f}{\partial y^{(n)))}\right]=0.}$$

Du Bois-Reymond's theorem

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral ${\displaystyle J}$ requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a weak form of the Euler–Lagrange equation. The theorem of Du Bois-Reymond asserts that this weak form implies the strong form. If ${\displaystyle L}$ has continuous first and second derivatives with respect to all of its arguments, and if

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

${\displaystyle f}$

Lavrentiev phenomenon

Hilbert was the first to give good conditions for the Euler–Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler–Lagrange equations in the interior.

However Lavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. The Lavrentiev Phenomenon identifies a difference in the infimum of a minimization problem across different classes of admissible functions. For instance the following problem, presented by Manià in 1934:^[18]

$${\displaystyle L[x]=\int _{0}^{1}(x^{3}-t)^{2}x'^{6},}$$

$${\displaystyle {A}=\{x\in W^{1,1}(0,1):x(0)=0,\ x(1)=1\}.}$$

Clearly, ${\displaystyle x(t)=t^{\frac {1}{3))}$minimizes the functional, but we find any function ${\displaystyle x\in W^{1,\infty ))$ gives a value bounded away from the infimum.

Examples (in one-dimension) are traditionally manifested across ${\displaystyle W^{1,1))$ and ${\displaystyle W^{1,\infty },}$ but Ball and Mizel^[19] procured the first functional that displayed Lavrentiev's Phenomenon across ${\displaystyle W^{1,p))$ and ${\displaystyle W^{1,q))$ for ${\displaystyle 1\leq p<q<\infty .}$ There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals.

Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.^[20]

Functions of several variables

For example, if ${\displaystyle \varphi (x,y)}$ denotes the displacement of a membrane above the domain ${\displaystyle D}$ in the ${\displaystyle x,y}$ plane, then its potential energy is proportional to its surface area:

$${\displaystyle U[\varphi ]=\iint _{D}{\sqrt {1+\nabla \varphi \cdot \nabla \varphi ))\,dx\,dy.}$$

${\displaystyle D}$

$${\displaystyle \varphi _{xx}(1+\varphi _{y}^{2})+\varphi _{yy}(1+\varphi _{x}^{2})-2\varphi _{x}\varphi _{y}\varphi _{xy}=0.}$$

Dirichlet's principle

It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by

$${\displaystyle V[\varphi ]={\frac {1}{2))\iint _{D}\nabla \varphi \cdot \nabla \varphi \,dx\,dy.}$$

${\displaystyle V}$

${\displaystyle \varphi }$

${\displaystyle D.}$

${\displaystyle u}$

${\displaystyle v}$

${\displaystyle D,}$

${\displaystyle V[u+\varepsilon v]}$

$${\displaystyle \left.{\frac {d}{d\varepsilon ))V[u+\varepsilon v]\right|_{\varepsilon =0}=\iint _{D}\nabla u\cdot \nabla v\,dx\,dy=0.}$$

$${\displaystyle \iint _{D}\nabla \cdot (v\nabla u)\,dx\,dy=\iint _{D}\nabla u\cdot \nabla v+v\nabla \cdot \nabla u\,dx\,dy=\int _{C}v{\frac {\partial u}{\partial n))\,ds,}$$

${\displaystyle C}$

${\displaystyle D,}$

${\displaystyle s}$

${\displaystyle C}$

${\displaystyle \partial u/\partial n}$

${\displaystyle u}$

${\displaystyle C.}$

${\displaystyle v}$

${\displaystyle C}$

$${\displaystyle \iint _{D}v\nabla \cdot \nabla u\,dx\,dy=0}$$

${\displaystyle D.}$

$${\displaystyle \nabla \cdot \nabla u=0}$$

${\displaystyle D.}$

The difficulty with this reasoning is the assumption that the minimizing function u must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize

$${\displaystyle W[\varphi ]=\int _{-1}^{1}(x\varphi ')^{2}\,dx}$$

${\displaystyle \varphi }$

${\displaystyle \varphi (-1)=-1}$

${\displaystyle \varphi (1)=1.}$

${\displaystyle W}$

${\displaystyle W=0.}$

Generalization to other boundary value problems

A more general expression for the potential energy of a membrane is

$${\displaystyle V[\varphi ]=\iint _{D}\left[{\frac {1}{2))\nabla \varphi \cdot \nabla \varphi +f(x,y)\varphi \right]\,dx\,dy\,+\int _{C}\left[{\frac {1}{2))\sigma (s)\varphi ^{2}+g(s)\varphi \right]\,ds.}$$

${\displaystyle f(x,y)}$

${\displaystyle D,}$

${\displaystyle g(s)}$

${\displaystyle C,}$

${\displaystyle \sigma (s)}$

${\displaystyle C.}$

${\displaystyle u.}$

${\displaystyle f}$

${\displaystyle g}$

${\displaystyle u}$

${\displaystyle v.}$

${\displaystyle V[u+\varepsilon v]}$

$${\displaystyle \iint _{D}\left[\nabla u\cdot \nabla v+fv\right]\,dx\,dy+\int _{C}\left[\sigma uv+gv\right]\,ds=0.}$$

$${\displaystyle \iint _{D}\left[-v\nabla \cdot \nabla u+vf\right]\,dx\,dy+\int _{C}v\left[{\frac {\partial u}{\partial n))+\sigma u+g\right]\,ds=0.}$$

${\displaystyle v=0}$

${\displaystyle C,}$

$${\displaystyle -\nabla \cdot \nabla u+f=0}$$

${\displaystyle D.}$

${\displaystyle v}$

${\displaystyle u}$

$${\displaystyle {\frac {\partial u}{\partial n))+\sigma u+g=0,}$$

${\displaystyle C.}$

${\displaystyle u}$

The preceding reasoning is not valid if ${\displaystyle \sigma }$ vanishes identically on ${\displaystyle C.}$ In such a case, we could allow a trial function ${\displaystyle \varphi \equiv c,}$ where ${\displaystyle c}$ is a constant. For such a trial function,

$${\displaystyle V[c]=c\left[\iint _{D}f\,dx\,dy+\int _{C}g\,ds\right].}$$

${\displaystyle c,}$

${\displaystyle V}$

$${\displaystyle \iint _{D}f\,dx\,dy+\int _{C}g\,ds=0.}$$

Eigenvalue problems

Both one-dimensional and multi-dimensional eigenvalue problems can be formulated as variational problems.

Sturm–Liouville problems

The Sturm–Liouville eigenvalue problem involves a general quadratic form

$${\displaystyle Q[y]=\int _{x_{1))^{x_{2))\left[p(x)y'(x)^{2}+q(x)y(x)^{2}\right]\,dx,}$$

${\displaystyle y}$

$${\displaystyle y(x_{1})=0,\quad y(x_{2})=0.}$$

${\displaystyle R}$

$${\displaystyle R[y]=\int _{x_{1))^{x_{2))r(x)y(x)^{2}\,dx.}$$

${\displaystyle p(x)}$

${\displaystyle r(x)}$

${\displaystyle Q/R}$

${\displaystyle y}$

${\displaystyle Q[y]}$

${\displaystyle R[y]}$

${\displaystyle u}$

$${\displaystyle -(pu')'+qu-\lambda ru=0,}$$

${\displaystyle \lambda }$

$${\displaystyle \lambda ={\frac {Q[u]}{R[u])).}$$

${\displaystyle u}$

${\displaystyle \lambda }$

${\displaystyle \lambda _{1))$

${\displaystyle u_{1}(x).}$

${\displaystyle u}$

The next smallest eigenvalue and eigenfunction can be obtained by minimizing ${\displaystyle Q}$ under the additional constraint

$${\displaystyle \int _{x_{1))^{x_{2))r(x)u_{1}(x)y(x)\,dx=0.}$$

The variational problem also applies to more general boundary conditions. Instead of requiring that ${\displaystyle y}$ vanish at the endpoints, we may not impose any condition at the endpoints, and set

$${\displaystyle Q[y]=\int _{x_{1))^{x_{2))\left[p(x)y'(x)^{2}+q(x)y(x)^{2}\right]\,dx+a_{1}y(x_{1})^{2}+a_{2}y(x_{2})^{2},}$$

${\displaystyle a_{1))$

${\displaystyle a_{2))$

${\displaystyle y=u+\varepsilon v}$

${\displaystyle Q/R}$

$${\displaystyle V_{1}={\frac {2}{R[u]))\left(\int _{x_{1))^{x_{2))\left[p(x)u'(x)v'(x)+q(x)u(x)v(x)-\lambda r(x)u(x)v(x)\right]\,dx+a_{1}u(x_{1})v(x_{1})+a_{2}u(x_{2})v(x_{2})\right),}$$

${\displaystyle Q[u]/R[u]}$

$${\displaystyle {\frac {R[u]}{2))V_{1}=\int _{x_{1))^{x_{2))v(x)\left[-(pu')'+qu-\lambda ru\right]\,dx+v(x_{1})[-p(x_{1})u'(x_{1})+a_{1}u(x_{1})]+v(x_{2})[p(x_{2})u'(x_{2})+a_{2}u(x_{2})].}$$

${\displaystyle v}$

${\displaystyle v}$

$${\displaystyle -(pu')'+qu-\lambda ru=0\quad {\hbox{for))\quad x_{1}<x<x_{2}.}$$

${\displaystyle u}$

${\displaystyle v}$

$${\displaystyle -p(x_{1})u'(x_{1})+a_{1}u(x_{1})=0,\quad {\hbox{and))\quad p(x_{2})u'(x_{2})+a_{2}u(x_{2})=0.}$$

Eigenvalue problems in several dimensions

Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain ${\displaystyle D}$ with boundary ${\displaystyle B}$ in three dimensions we may define

$${\displaystyle Q[\varphi ]=\iiint _{D}p(X)\nabla \varphi \cdot \nabla \varphi +q(X)\varphi ^{2}\,dx\,dy\,dz+\iint _{B}\sigma (S)\varphi ^{2}\,dS,}$$

$${\displaystyle R[\varphi ]=\iiint _{D}r(X)\varphi (X)^{2}\,dx\,dy\,dz.}$$

${\displaystyle u}$

${\displaystyle Q[\varphi ]/R[\varphi ],}$

${\displaystyle B.}$

${\displaystyle u}$

$${\displaystyle -\nabla \cdot (p(X)\nabla u)+q(x)u-\lambda r(x)u=0,}$$

$${\displaystyle \lambda ={\frac {Q[u]}{R[u])).}$$

${\displaystyle u}$

$${\displaystyle p(S){\frac {\partial u}{\partial n))+\sigma (S)u=0,}$$

${\displaystyle B.}$

Applications

Optics

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

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

${\displaystyle n(x,y)}$

${\displaystyle f(x)=f_{0}(x)+\varepsilon f_{1}(x)}$

${\displaystyle A}$

${\displaystyle A}$

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

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

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

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

$${\displaystyle n(x,y)={\begin{cases}n_{(-)}&{\text{if))\quad x<0,\\n_{(+)}&{\text{if))\quad x>0,\end{cases))}$$

${\displaystyle n_{(-)))$

${\displaystyle n_{(+)))$

${\displaystyle x<0}$

${\displaystyle x>0,}$

${\displaystyle x=0,}$

${\displaystyle f}$

${\displaystyle f'}$

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

The factor multiplying ${\displaystyle n_{(-)))$ is the sine of angle of the incident ray with the ${\displaystyle x}$ axis, and the factor multiplying ${\displaystyle n_{(+)))$ is the sine of angle of the refracted ray with the ${\displaystyle x}$ 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 ${\displaystyle X=(x_{1},x_{2},x_{3}),}$ let ${\displaystyle t}$ be a parameter, let ${\displaystyle X(t)}$ be the parametric representation of a curve ${\displaystyle C,}$ and let ${\displaystyle {\dot {X))(t)}$ be its tangent vector. The optical length of the curve is given by

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

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

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

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

It follows from the definition that ${\displaystyle P}$ satisfies

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

Therefore, the integral may also be written as

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

This form suggests that if we can find a function ${\displaystyle \psi }$ whose gradient is given by ${\displaystyle P,}$ then the integral ${\displaystyle A}$ is given by the difference of ${\displaystyle \psi }$ 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 ${\displaystyle \psi .}$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

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

${\displaystyle c}$

${\displaystyle X.}$

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

We may look for solutions in the form

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

In that case, ${\displaystyle \psi }$ satisfies

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

${\displaystyle n=1/c.}$

${\displaystyle P=\nabla \psi ,}$

${\displaystyle P}$

$${\displaystyle {\frac {dP}{ds))=n\,\nabla n,}$$

$${\displaystyle {\frac {dX}{ds))=P.}$$

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification

$${\displaystyle {\frac {ds}{dt))={\frac {\sqrt ((\dot {X))\cdot {\dot {X)))){n)).}$$

We conclude that the function ${\displaystyle \psi }$ is the value of the minimizing integral ${\displaystyle A}$ 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 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

In classical mechanics, the action, ${\displaystyle S,}$ is defined as the time integral of the Lagrangian, ${\displaystyle L.}$ The Lagrangian is the difference of energies,

$${\displaystyle L=T-U,}$$

${\displaystyle T}$

${\displaystyle U}$

$${\displaystyle S=\int _{t_{0))^{t_{1))L(x,{\dot {x)),t)\,dt}$$

${\displaystyle x(t).}$

$${\displaystyle {\frac {d}{dt)){\frac {\partial L}{\partial {\dot {x))))={\frac {\partial L}{\partial x)),}$$

The conjugate momenta ${\displaystyle P}$ are defined by

$${\displaystyle p={\frac {\partial L}{\partial {\dot {x)))).}$$

$${\displaystyle T={\frac {1}{2))m{\dot {x))^{2},}$$

$${\displaystyle p=m{\dot {x)).}$$

${\displaystyle {\dot {x))}$

${\displaystyle L}$

${\displaystyle H}$

$${\displaystyle H(x,p,t)=p\,{\dot {x))-L(x,{\dot {x)),t).}$$

${\displaystyle H=T+U.}$

${\displaystyle X.}$

$${\displaystyle {\frac {\partial \psi }{\partial t))+H\left(x,{\frac {\partial \psi }{\partial x)),t\right)=0.}$$

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 problem
• Solution to the tautochrone problem
• Solution to isoperimetric problems
• Calculating geodesics
• Finding minimal surfaces and solving Plateau's problem
• Optimal control
• Analytical mechanics, or reformulations of Newton's laws of motion, most notably 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.

Variations and sufficient condition for a minimum

Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation^[l] is defined as the linear part of the change in the functional, and the second variation^[m] is defined as the quadratic part.^[22]

For example, if ${\displaystyle J[y]}$ is a functional with the function ${\displaystyle y=y(x)}$ as its argument, and there is a small change in its argument from ${\displaystyle y}$ to ${\displaystyle y+h,}$ where ${\displaystyle h=h(x)}$ is a function in the same function space as ${\displaystyle y,}$ then the corresponding change in the functional is^[n]

$${\displaystyle \Delta J[h]=J[y+h]-J[y].}$$

The functional ${\displaystyle J[y]}$ is said to be differentiable if

$${\displaystyle \Delta J[h]=\varphi [h]+\varepsilon \|h\|,}$$

${\displaystyle \varphi [h]}$

${\displaystyle \|h\|}$

${\displaystyle h,}$

${\displaystyle \varepsilon \to 0}$

${\displaystyle \|h\|\to 0.}$

${\displaystyle \varphi [h]}$

${\displaystyle J[y]}$

$${\displaystyle \delta J[h]=\varphi [h].}$$

The functional ${\displaystyle J[y]}$ is said to be twice differentiable if

$${\displaystyle \Delta J[h]=\varphi _{1}[h]+\varphi _{2}[h]+\varepsilon \|h\|^{2},}$$

${\displaystyle \varphi _{1}[h]}$

${\displaystyle \varphi _{2}[h]}$

${\displaystyle \varepsilon \to 0}$

${\displaystyle \|h\|\to 0.}$

${\displaystyle \varphi _{2}[h]}$

${\displaystyle J[y]}$

$${\displaystyle \delta ^{2}J[h]=\varphi _{2}[h].}$$

The second variation ${\displaystyle \delta ^{2}J[h]}$ is said to be strongly positive if

$${\displaystyle \delta ^{2}J[h]\geq k\|h\|^{2},}$$

${\displaystyle h}$

${\displaystyle k>0}$

Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.