The calculus of variations (or Variational Calculus) is a field of

mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

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 number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s. Functionals are often expressed as definite integrals involving functions and their derivative
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

s. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action (physics), action functional. The equations ...

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
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...

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 ''geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...

s''. 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
Mechanics (from Ancient Greek: wikt:μηχανική#Ancient_Greek, μηχανική, ''mēkhanikḗ'', "of machine, machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among Ph ...

is the principle of least/stationary action.
Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...

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
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

.
History

The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by thebrachistochrone curve
In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...

problem raised by Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating ...

(1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

first elaborated the subject, beginning in 1733. Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLegendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima.

"Weak Lower Semicontinuity of Integral Functionals and Applications"

''SIAM Review'' 59(4) (2017), 703–766. * Bolza, O.

Lectures on the Calculus of Variations

Chelsea Publishing Company, 1904, available on Digital Mathematics library. 2nd edition republished in 1961, paperback in 2005, . * Cassel, Kevin W.

Variational Methods with Applications in Science and Engineering

Cambridge University Press, 2013. * Clegg, J.C.

Interscience Publishers Inc., 1968. * Courant, R.

Dirichlet's principle, conformal mapping and minimal surfaces

Interscience, 1950. * Dacorogna, Bernard:

Introduction

Introduction to the Calculus of Variations

', 3rd edition. 2014, World Scientific Publishing, . * Elsgolc, L.E.

Calculus of Variations

Pergamon Press Ltd., 1962. * Forsyth, A.R.

Calculus of Variations

Dover, 1960. * Fox, Charles

Dover Publ., 1987. * Giaquinta, Mariano; Hildebrandt, Stefan: Calculus of Variations I and II, Springer-Verlag, and * Jost, J. and X. Li-Jost

Calculus of Variations

Cambridge University Press, 1998. * 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. * Logan, J. David

Applied Mathematics

3rd edition. Wiley-Interscience, 2006 * * Roubicek, T.:

Calculus of variations

. Chap.17 in:

Mathematical Tools for Physicists

'. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, , pp. 551–588. * Sagan, Hans

Dover, 1992. * Weinstock, Robert

Calculus of Variations with Applications to Physics and Engineering

Dover, 1974 (reprint of 1952 ed.).

Variational calculus

''

calculus of variations

'' PlanetMath''.

Calculus of Variations

''

Calculus of variations

Example problems.

Mathematics - Calculus of Variations and Integral Equations

Lectures on

Part I

Part II

{{Authority control Optimization in vector spaces

Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...

and Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...

also gave some early attention to the subject. To this discrimination Vincenzo Brunacci (1810), Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

(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
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...

(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
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...

. 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.
In the 20th century David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether
Amalie Emmy NoetherEmmy (given name), Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promot ...

, Leonida Tonelli, Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are con ...

and Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ...

among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory. Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. The dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
I ...

of Richard Bellman is an alternative to the calculus of variations.
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 $y$ of a givenfunction space
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

defined over a given domain. A functional $J;\; href="/html/ALL/s/.html"\; ;"title="">$sign
A sign is an Physical object, object, quality (philosophy), quality, event, or Non-physical entity, entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to ...

for all $y$ in an arbitrarily small neighborhood of $f.$ The function $f$ is called an extremal function or extremal. The extremum $J;\; href="/html/ALL/s/.html"\; ;"title="">$Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action (physics), action functional. The equations ...

.
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 associatedEuler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action (physics), action functional. The equations ...

.
Consider the functional
$$J;\; href="/html/ALL/s/.html"\; ;"title="">$$
where
*$x\_1,\; x\_2$ are constants,
*$y(x)$ is twice continuously differentiable,
*$y\text{'}(x)\; =\; \backslash frac,$
*$L\backslash left(x,\; y(x),\; y\text{'}(x)\backslash right)$ is twice continuously differentiable with respect to its arguments $x,\; y,$ and $y\text{'}.$
If the functional $J;\; href="/html/ALL/s/.html"\; ;"title="">$local minimum
In mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, ...

at $f,$ and $\backslash eta(x)$ is an arbitrary function that has at least one derivative and vanishes at the endpoints $x\_1$ and $x\_2,$ then for any number $\backslash varepsilon$ close to 0,
$$J;\; href="/html/ALL/s/.html"\; ;"title="">$$
The term $\backslash varepsilon\; \backslash eta$ is called the variation of the function $f$ and is denoted by $\backslash delta\; f.$
Substituting $f\; +\; \backslash varepsilon\; \backslash eta$ for $y$ in the functional $J;\; href="/html/ALL/s/.html"\; ;"title="">$ the result is a function of $\backslash varepsilon,$
$$\backslash Phi(\backslash varepsilon)\; =\; J;\; href="/html/ALL/s/+\backslash varepsilon\backslash eta.html"\; ;"title="+\backslash varepsilon\backslash eta">+\backslash varepsilon\backslash eta$$
Since the functional $J;\; href="/html/ALL/s/.html"\; ;"title="">$integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral (mathematics), integral of a product (mathematics), product of Function (mathematics), functions in terms o ...

on the second term. The second term on the second line vanishes because $\backslash eta\; =\; 0$ at $x\_1$ and $x\_2$ by definition. Also, as previously mentioned the left side of the equation is zero so that
$$\backslash int\_^\; \backslash eta\; (x)\; \backslash left(\backslash frac\; -\; \backslash frac\backslash frac\; \backslash right)\; \backslash ,\; dx\; =\; 0\; \backslash ,\; .$$
According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e.
$$\backslash frac\; -\backslash frac\; \backslash frac=0$$
which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of $J;\; href="/html/ALL/s/.html"\; ;"title="">$ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable (mathematics), variable and involves the derivatives of those functions. The term ''ordinary ...

which can be solved to obtain the extremal function $f(x).$ The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum $J;\; href="/html/ALL/s/.html"\; ;"title="">$ 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 $y\; =\; f(x),$ which is the shortest curve that connects two points $\backslash left(x\_1,\; y\_1\backslash right)$ and $\backslash left(x\_2,\; y\_2\backslash right).$ Thearc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...

of the curve is given by
$$A;\; href="/html/ALL/s/.html"\; ;"title="">$$
with
$$y\text{'}(x)\; =\; \backslash frac\; \backslash ,\; ,\; \backslash \; \backslash \; y\_1=f(x\_1)\; \backslash ,\; ,\; \backslash \; \backslash \; y\_2=f(x\_2)\; \backslash ,\; .$$
Note that assuming is a function of loses generality; ideally both should be a function of some other parameter. This approach is good solely for instructive purposes.
The Euler–Lagrange equation will now be used to find the extremal function $f(x)$ that minimizes the functional $A;\; href="/html/ALL/s/.html"\; ;"title="">$
$$\backslash frac\; -\backslash frac\; \backslash frac=0$$
with
$$L\; =\; \backslash sqrt\; \backslash ,\; .$$
Since $f$ does not appear explicitly in $L,$ the first term in the Euler–Lagrange equation vanishes for all $f(x)$ and thus,
$$\backslash frac\; \backslash frac\; =\; 0\; \backslash ,\; .$$
Substituting for $L$ and taking the derivative,
$$\backslash frac\; \backslash \; \backslash frac\; \backslash \; =\; 0\; \backslash ,\; .$$
Thus
$$\backslash frac\; =\; c\; \backslash ,\; ,$$
for some constant $c.$ Then
$$\backslash frac\; =\; c^2\; \backslash ,\; ,$$
where
$$0\; \backslash le\; c^2<1.$$
Solving, we get
$$;\; href="/html/ALL/s/\text{\'}(x).html"\; ;"title="\text{\'}(x)">\text{\'}(x)$$
which implies that
$$f\text{'}(x)=m$$
is a constant and therefore that the shortest curve that connects two points $\backslash left(x\_1,\; y\_1\backslash right)$ and $\backslash left(x\_2,\; y\_2\backslash right)$ is
$$f(x)\; =\; m\; x\; +\; b\; \backslash qquad\; \backslash text\; \backslash \; \backslash \; m\; =\; \backslash frac\; \backslash quad\; \backslash text\; \backslash quad\; b\; =\; \backslash frac$$
and we have thus found the extremal function $f(x)$ that minimizes the functional $A;\; href="/html/ALL/s/.html"\; ;"title="">$Beltrami's identity

In physics problems it may be the case that $\backslash frac\; =\; 0,$ meaning the integrand is a function of $f(x)$ and $f\text{'}(x)$ but $x$ does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity $$L\; -\; f\text{'}\; \backslash frac\; =\; C\; \backslash ,\; ,$$ where $C$ is a constant. The left hand side is theLegendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involution (mathematics), involutive List of transforms, transformation on real number, real-valued convex functions of one real variable ...

of $L$ with respect to $f\text{'}(x).$
The intuition behind this result is that, if the variable $x$ is actually time, then the statement $\backslash frac\; =\; 0$ 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 $S$ depends on higher-derivatives of $y(x),$ that is, if $$S\; =\; \backslash int\_^\; f(x,\; y(x),\; y\text{'}(x),\; \backslash dots,\; y^(x))\; dx,$$ then $y$ must satisfy the Euler– Poisson equation, $$\backslash frac\; -\; \backslash frac\; \backslash left(\; \backslash frac\; \backslash right)\; +\; \backslash dots\; +\; (-1)^\; \backslash frac\; \backslash left;\; href="/html/ALL/s/\backslash frac\_\backslash right.html"\; ;"title="\backslash frac\; \backslash right">\backslash frac\; \backslash right$$Du Bois-Reymond's theorem

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral $J$ 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 $L$ has continuous first and second derivatives with respect to all of its arguments, and if $$\backslash frac\; \backslash ne\; 0,$$ then $f$ has two continuous derivatives, and it satisfies the Euler–Lagrange equation.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: $$L;\; href="/html/ALL/s/.html"\; ;"title="">$$ $$=\; \backslash .$$ Clearly, $x(t)\; =\; t^$minimizes the functional, but we find any function $x\; \backslash in\; W^$ gives a value bounded away from the infimum. Examples (in one-dimension) are traditionally manifested across $W^$ and $W^,$ but Ball and Mizel procured the first functional that displayed Lavrentiev's Phenomenon across $W^$ and $W^$ for $1\; \backslash leq\; p\; <\; q\; <\; \backslash infty.$ 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.Functions of several variables

For example, if $\backslash varphi(x,\; y)$ denotes the displacement of a membrane above the domain $D$ in the $x,y$ plane, then its potential energy is proportional to its surface area: $$U;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$$ Plateau's problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of $D$; the solutions are called minimal surfaces. The Euler–Lagrange equation for this problem is nonlinear: $$\backslash varphi\_(1\; +\; \backslash varphi\_y^2)\; +\; \backslash varphi\_(1\; +\; \backslash varphi\_x^2)\; -\; 2\backslash varphi\_x\; \backslash varphi\_y\; \backslash varphi\_\; =\; 0.$$ See Courant (1950) for details.Dirichlet's principle

It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by $$V;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$$ The functional $V$ is to be minimized among all trial functions $\backslash varphi$ that assume prescribed values on the boundary of $D.$ If $u$ is the minimizing function and $v$ is an arbitrary smooth function that vanishes on the boundary of $D,$ then the first variation of $V;\; href="/html/ALL/s/\_+\_\backslash varepsilon\_v.html"\; ;"title="\; +\; \backslash varepsilon\; v">\; +\; \backslash varepsilon\; v$Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...

. However Weierstrass gave an example of a variational problem with no solution: minimize
$$W;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$$
among all functions $\backslash varphi$ that satisfy $\backslash varphi(-1)=-1$ and $\backslash varphi(1)=1.$
$W$ can be made arbitrarily small by choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makes $W=0.$ Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for elliptic partial differential equation
Second-order linear differential equation, linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic partial differential equation, hyperbolic, or parabolic partial differential equation, parabolic. Any second-or ...

s; see Jost and Li–Jost (1998).
Generalization to other boundary value problems

A more general expression for the potential energy of a membrane is $$V;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$$ This corresponds to an external force density $f(x,y)$ in $D,$ an external force $g(s)$ on the boundary $C,$ and elastic forces with modulus $\backslash sigma(s)$acting on $C.$ The function that minimizes the potential energy with no restriction on its boundary values will be denoted by $u.$ Provided that $f$ and $g$ are continuous, regularity theory implies that the minimizing function $u$ will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment $v.$ The first variation of $V;\; href="/html/ALL/s/\_+\_\backslash varepsilon\_v.html"\; ;"title="\; +\; \backslash varepsilon\; v">\; +\; \backslash varepsilon\; v$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 $$Q;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$$ where $\backslash varphi$is restricted to functions that satisfy the boundary conditions $$\backslash varphi(x\_1)=0,\; \backslash quad\; \backslash varphi(x\_2)=0.$$ Let $R$ be a normalization integral $$R;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$$ The functions $p(x)$ and $r(x)$ are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio $Q/R$ among all $\backslash varphi$ satisfying the endpoint conditions. It is shown below that the Euler–Lagrange equation for the minimizing $u$ is $$-(p\; u\text{'})\text{'}\; +q\; u\; -\backslash lambda\; r\; u\; =\; 0,$$ where $\backslash lambda$ is the quotient $$\backslash lambda\; =\; \backslash frac.$$ It can be shown (see Gelfand and Fomin 1963) that the minimizing $u$ has two derivatives and satisfies the Euler–Lagrange equation. The associated $\backslash lambda$ will be denoted by $\backslash lambda\_1$; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by $u\_1(x).$ This variational characterization of eigenvalues leads to theRayleigh–Ritz method The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues and eigenvectors, eigenvalues, originated in the context of solving physical Boundary value problem, boundary value problems and named after Lord Rayleigh and Walth ...

: choose an approximating $u$ as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.
The next smallest eigenvalue and eigenfunction can be obtained by minimizing $Q$ under the additional constraint
$$\backslash int\_^\; r(x)\; u\_1(x)\; \backslash varphi(x)\; \backslash ,\; dx\; =\; 0.$$
This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem.
The variational problem also applies to more general boundary conditions. Instead of requiring that $\backslash varphi$ vanish at the endpoints, we may not impose any condition at the endpoints, and set
$$Q;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$$
where $a\_1$ and $a\_2$ are arbitrary. If we set $\backslash varphi\; =\; u\; +\; \backslash varepsilon\; v$the first variation for the ratio $Q/R$ is
$$V\_1\; =\; \backslash frac\; \backslash left(\; \backslash int\_^\; \backslash left;\; href="/html/ALL/s/p(x)\_u\text{\'}(x)v\text{\'}(x)\_+\_q(x)u(x)v(x)\_-\backslash lambda\_r(x)\_u(x)\_v(x)\_\backslash right.html"\; ;"title="p(x)\; u\text{\'}(x)v\text{\'}(x)\; +\; q(x)u(x)v(x)\; -\backslash lambda\; r(x)\; u(x)\; v(x)\; \backslash right">p(x)\; u\text{\'}(x)v\text{\'}(x)\; +\; q(x)u(x)v(x)\; -\backslash lambda\; r(x)\; u(x)\; v(x)\; \backslash right$$
where λ is given by the ratio $Q;\; href="/html/ALL/s/.html"\; ;"title="">$Eigenvalue problems in several dimensions

Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain $D$ with boundary $B$ in three dimensions we may define $$Q;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$$ and $$R;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$$ Let $u$ be the function that minimizes the quotient $Q;\; href="/html/ALL/s/varphi.html"\; ;"title="varphi">varphi$ with no condition prescribed on the boundary $B.$ The Euler–Lagrange equation satisfied by $u$ is $$-\backslash nabla\; \backslash cdot\; (p(X)\; \backslash nabla\; u)\; +\; q(x)\; u\; -\; \backslash lambda\; r(x)\; u=0,$$ where $$\backslash lambda\; =\; \backslash frac.$$ The minimizing $u$ must also satisfy the natural boundary condition $$p(S)\; \backslash frac\; +\; \backslash sigma(S)\; u\; =\; 0,$$ on the boundary $B.$ This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilbert (1953).Applications

Optics

Fermat's principle states that light takes a path that (locally) 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 $$A;\; href="/html/ALL/s/.html"\; ;"title="">$$ where the refractive index $n(x,y)$ depends upon the material. If we try $f(x)\; =\; f\_0\; (x)\; +\; \backslash varepsilon\; f\_1\; (x)$ then the first variation of $A$ (the derivative of $A$ with respect to ε) is $$\backslash delta\; A;\; href="/html/ALL/s/\_0,f\_1.html"\; ;"title="\_0,f\_1">\_0,f\_1$$ After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation $$-\backslash frac\; \backslash left;\; href="/html/ALL/s/frac\_\backslash right.html"\; ;"title="frac\; \backslash right">frac\; \backslash right$$ 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 $$n(x,y)\; =\; \backslash begin\; n\_\; \&\; \backslash text\; \backslash quad\; x<0,\; \backslash \backslash \; n\_\; \&\; \backslash text\; \backslash quad\; x>0,\; \backslash end$$ 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 $f\text{'}$ may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form $$\backslash delta\; A;\; href="/html/ALL/s/\_0,f\_1.html"\; ;"title="\_0,f\_1">\_0,f\_1$$ 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.Snell's law
Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a Mathematical formula, formula used to describe the relationship between the angle of incidence (optics), angles of incidence and refraction, when ...

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 $X\; =\; (x\_1,x\_2,x\_3),$ let $t$ be a parameter, let $X(t)$ be the parametric representation of a curve $C,$ and let $\backslash dot\; X(t)$ be its tangent vector. The optical length of the curve is given by $$A;\; href="/html/ALL/s/.html"\; ;"title="">$$ 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 $$\backslash frac\; P\; =\; \backslash sqrt\; \backslash ,\; \backslash nabla\; n,$$ where $$P\; =\; \backslash frac.$$ It follows from the definition that $P$ satisfies $$P\; \backslash cdot\; P\; =\; n(X)^2.$$ Therefore, the integral may also be written as $$A;\; href="/html/ALL/s/.html"\; ;"title="">$$ This form suggests that if we can find a function $\backslash psi$ whose gradient is given by $P,$ then the integral $A$ is given by the difference of $\backslash psi$ 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 $\backslash psi.$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

= Thewave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...

for an inhomogeneous medium is
$$u\_\; =\; c^2\; \backslash nabla\; \backslash cdot\; \backslash nabla\; u,$$
where $c$ is the velocity, which generally depends upon $X.$ Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy
$$\backslash varphi\_t^2\; =\; c(X)^2\; \backslash ,\; \backslash nabla\; \backslash varphi\; \backslash cdot\; \backslash nabla\; \backslash varphi.$$
We may look for solutions in the form
$$\backslash varphi(t,X)\; =\; t\; -\; \backslash psi(X).$$
In that case, $\backslash psi$ satisfies
$$\backslash nabla\; \backslash psi\; \backslash cdot\; \backslash nabla\; \backslash psi\; =\; n^2,$$
where $n=1/c.$ According to the theory of first-order partial differential equations, if $P\; =\; \backslash nabla\; \backslash psi,$ then $P$ satisfies
$$\backslash frac\; =\; n\; \backslash ,\; \backslash nabla\; n,$$
along a system of curves (the light rays) that are given by
$$\backslash frac\; =\; P.$$
These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification
$$\backslash frac\; =\; \backslash frac.$$
We conclude that the function $\backslash psi$ 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 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, $S,$ is defined as the time integral of the Lagrangian, $L.$ The Lagrangian is the difference of energies, $$L\; =\; T\; -\; U,$$ where $T$ is thekinetic energy
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical sci ...

of a mechanical system and $U$ its potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...

. 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
$$S\; =\; \backslash int\_^\; L(x,\; \backslash dot\; x,\; t)\; \backslash ,\; dt$$
is stationary with respect to variations in the path $x(t).$
The Euler–Lagrange equations for this system are known as Lagrange's equations:
$$\backslash frac\; \backslash frac\; =\; \backslash frac,$$
and they are equivalent to Newton's equations of motion (for such systems).
The conjugate momenta $P$ are defined by
$$p\; =\; \backslash frac.$$
For example, if
$$T\; =\; \backslash frac\; m\; \backslash dot\; x^2,$$
then $$p\; =\; m\; \backslash dot\; x.$$
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...

results if the conjugate momenta are introduced in place of $\backslash dot\; x$ by a Legendre transformation of the Lagrangian $L$ into the Hamiltonian $H$ defined by
$$H(x,\; p,\; t)\; =\; p\; \backslash ,\backslash dot\; x\; -\; L(x,\backslash dot\; x,\; t).$$
The Hamiltonian is the total energy of the system: $H\; =\; T\; +\; U.$
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 $X.$ This function is a solution of the Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechan ...

:
$$\backslash frac\; +\; H\backslash left(x,\backslash frac,t\backslash right)\; =\; 0.$$
Further applications

Further applications of the calculus of variations include the following: * The derivation of thecatenary
In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or wire rope, cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
The catenary curve has a U-like shape, ...

shape
* Solution to Newton's minimal resistance problem
* Solution to the brachistochrone problem
* Solution to the tautochrone problem
* Solution to isoperimetric problems
* Calculating geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...

s
* Finding minimal surfaces and solving Plateau's problem
* Optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a Control (optimal control theory), control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applica ...

* Analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...

, or reformulations of Newton's laws of motion, most notably Lagrangian and Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...

;
* 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
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...

is a variational method for finding numerical solutions to boundary-value problems in differential equations;
* Total variation denoising, an image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

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 is defined as the linear part of the change in the functional, and the second variation is defined as the quadratic part. For example, if $J;\; href="/html/ALL/s/.html"\; ;"title="">$See also

* First variation * Isoperimetric inequality * Variational principle * Variational bicomplex * Fermat's principle * Principle of least action * Infinite-dimensional optimization *Finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...

* Functional analysis
Functional analysis is a branch of mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, ...

* Ekeland's variational principle
* Inverse problem for Lagrangian mechanics
* Obstacle problem
* Perturbation methods
* Young measure
* Optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a Control (optimal control theory), control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applica ...

* Direct method in calculus of variations
* Noether's theorem
* De Donder–Weyl theory
* Variational Bayesian methods
* Chaplygin problem
* Nehari manifold
* Hu–Washizu principle
* Luke's variational principle
* Mountain pass theorem
*
* Measures of central tendency as solutions to variational problems
* Stampacchia Medal
* Fermat Prize
* Convenient vector space
Notes

References

Further reading

* Benesova, B. and Kruzik, M."Weak Lower Semicontinuity of Integral Functionals and Applications"

''SIAM Review'' 59(4) (2017), 703–766. * Bolza, O.

Lectures on the Calculus of Variations

Chelsea Publishing Company, 1904, available on Digital Mathematics library. 2nd edition republished in 1961, paperback in 2005, . * Cassel, Kevin W.

Variational Methods with Applications in Science and Engineering

Cambridge University Press, 2013. * Clegg, J.C.

Interscience Publishers Inc., 1968. * Courant, R.

Dirichlet's principle, conformal mapping and minimal surfaces

Interscience, 1950. * Dacorogna, Bernard:

Introduction

Introduction to the Calculus of Variations

', 3rd edition. 2014, World Scientific Publishing, . * Elsgolc, L.E.

Calculus of Variations

Pergamon Press Ltd., 1962. * Forsyth, A.R.

Calculus of Variations

Dover, 1960. * Fox, Charles

Dover Publ., 1987. * Giaquinta, Mariano; Hildebrandt, Stefan: Calculus of Variations I and II, Springer-Verlag, and * Jost, J. and X. Li-Jost

Calculus of Variations

Cambridge University Press, 1998. * 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. * Logan, J. David

Applied Mathematics

3rd edition. Wiley-Interscience, 2006 * * Roubicek, T.:

Calculus of variations

. Chap.17 in:

Mathematical Tools for Physicists

'. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, , pp. 551–588. * Sagan, Hans

Dover, 1992. * Weinstock, Robert

Calculus of Variations with Applications to Physics and Engineering

Dover, 1974 (reprint of 1952 ed.).

External links

Variational calculus

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'' PlanetMath''.

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Calculus of variations

Example problems.

Mathematics - Calculus of Variations and Integral Equations

Lectures on

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* Selected papers on Geodesic FieldsPart I

Part II

{{Authority control Optimization in vector spaces