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

"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

'' Encyclopedia of Mathematics''.

calculus of variations

'' PlanetMath''.

Calculus of Variations

'' MathWorld''.

Calculus of variations

Example problems.

Mathematics - Calculus of Variations and Integral Equations

Lectures on

Part I

Part II

{{Authority control Optimization in vector spaces

mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...

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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s. Functionals are often expressed as definite integrals involving functions and their derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

s. 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
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...

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 = \n ...

satisfy the Dirichlet's principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Formal statement
Dirichlet's principle states that, if the funct ...

. Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The 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 words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

.
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). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, butLeonhard 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 ...

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''.
Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. 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 also gave some early attention to the subject. To this discrimination Vincenzo Brunacci
Vincenzo Brunacci (3 March 1768 – 16 June 1818) was an Italian mathematician born in Florence.An It ...

(1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky
Mikhail Vasilyevich Ostrogradsky (transcribed also ''Ostrogradskiy'', Ostrogradskiĭ) (russian: Михаи́л Васи́льевич Острогра́дский, ua, Миха́йло Васи́льович Острогра́дський; 24 Sep ...

(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
Strauch, a German word meaning ''bush'' or '' shrub'', is a surname. Notable people with it include:
* Adolfo Strauch, (b. 1948), survivor of the Uruguayan Air Force Flight 571 crash
* Adolph Strauch (1822–1883), landscape architect
* Aegidius ...

(1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...

(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
20 (twenty; Roman numeral XX) is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.
In mathematics
*20 is a pronic number.
*20 is a tetrahedral number as 1, 4, 10, 20.
*20 is the ba ...

and the 23rd Hilbert problem published in 1900 encouraged further development.
In the 20th century David Hilbert, Oskar Bolza
Oskar Bolza (12 May 1857 – 5 July 1942) was a German mathematician, and student of Felix Klein. He was born in Bad Bergzabern, Palatinate (region), Palatinate, then a district of Bavaria, known for his research in the calculus of variations, p ...

, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli
Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian people, Italian mathematician, noted for creating Fubini's theorem#Tonelli's theorem for non-negative measurable functions, Tonelli's theorem, a variation of Fubini's theorem, and f ...

, Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...

and Jacques Hadamard among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory. Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely d ...

, 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.
...

of Richard Bellman
Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founde ...

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 toscalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...

, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements $y$ of a given function space defined over a given domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...

. A functional $J;\; href="/html/ALL/l/.html"\; ;"title="">$necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ...

that is used for finding weak extrema is the Euler–Lagrange equation.
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 thefunctional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on w ...

is equal to zero. This leads to solving the associated Euler–Lagrange equation.
Consider the functional
$$J;\; href="/html/ALL/l/.html"\; ;"title="">$$
where
*$x\_1,\; x\_2$ are constants
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific const ...

,
*$y(x)$ 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 $J;\; href="/html/ALL/l/.html"\; ;"title="">$local minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...

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,
$$J;\; href="/html/ALL/l/.html"\; ;"title="">$$
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 $J;\; href="/html/ALL/l/.html"\; ;"title="">$ the result is a function of $\backslash varepsilon,$
$$\backslash Phi(\backslash varepsilon)\; =\; J;\; href="/html/ALL/l/+\backslash varepsilon\backslash eta.html"\; ;"title="+\backslash varepsilon\backslash eta">+\backslash varepsilon\backslash eta$$
Since the functional $J;\; href="/html/ALL/l/.html"\; ;"title="">$integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...

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
In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point.
Accordingly, the necessary condition of extremum (functional derivative equal ze ...

, 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
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on w ...

of $J;\; href="/html/ALL/l/.html"\; ;"title="">$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 and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

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 $J;\; href="/html/ALL/l/.html"\; ;"title="">$ 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
$$A;\; href="/html/ALL/l/.html"\; ;"title="">$$
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 $A;\; href="/html/ALL/l/.html"\; ;"title="">$
$$\backslash frac\; -\backslash frac\; \backslash frac=0$$
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
$$;\; href="/html/ALL/l/\text{\'}(x).html"\; ;"title="\text{\'}(x)">\text{\'}(x)$$
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 $A;\; href="/html/ALL/l/.html"\; ;"title="">$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 the Legendre transformation 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, $$\backslash frac\; -\; \backslash frac\; \backslash left(\; \backslash frac\; \backslash right)\; +\; \backslash dots\; +\; (-1)^\; \backslash frac\; \backslash left;\; href="/html/ALL/l/\backslash frac\_\backslash right.html"\; ;"title="\backslash frac\; \backslash right">\backslash frac\; \backslash right$$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: $$L;\; href="/html/ALL/l/.html"\; ;"title="">$$ $$=\; \backslash .$$ 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: $$U;\; href="/html/ALL/l/varphi.html"\; ;"title="varphi">varphi$$Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The 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 $$V;\; href="/html/ALL/l/varphi.html"\; ;"title="varphi">varphi$$ 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 $V;\; href="/html/ALL/l/\_+\_\backslash varepsilon\_v.html"\; ;"title="\; +\; \backslash varepsilon\; v">\; +\; \backslash varepsilon\; v$Generalization to other boundary value problems

A more general expression for the potential energy of a membrane is $$V;\; href="/html/ALL/l/varphi.html"\; ;"title="varphi">varphi$$ 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 $V;\; href="/html/ALL/l/\_+\_\backslash varepsilon\_v.html"\; ;"title="\; +\; \backslash varepsilon\; v">\; +\; \backslash varepsilon\; v$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 $$Q;\; href="/html/ALL/l/varphi.html"\; ;"title="varphi">varphi$$ 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 $$R;\; href="/html/ALL/l/varphi.html"\; ;"title="varphi">varphi$$ 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, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz.
The name Rayleigh–Ritz is being debate ...

: 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
$$Q;\; href="/html/ALL/l/varphi.html"\; ;"title="varphi">varphi$$
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
$$V\_1\; =\; \backslash frac\; \backslash left(\; \backslash int\_^\; \backslash left;\; href="/html/ALL/l/p(x)\_u\text{\'}(x)v\text{\'}(x)\_+\_q(x)u(x)v(x)\_-\backslash lambda\_r(x)\_u(x)\_v(x)\_\backslash right.html"\; ;"title="p(x)\; u\text{\'}(x)v\text{\'}(x)\; +\; q(x)u(x)v(x)\; -\backslash lambda\; r(x)\; u(x)\; v(x)\; \backslash right">p(x)\; u\text{\'}(x)v\text{\'}(x)\; +\; q(x)u(x)v(x)\; -\backslash lambda\; r(x)\; u(x)\; v(x)\; \backslash right$$
where λ is given by the ratio $Q;\; href="/html/ALL/l/.html"\; ;"title="">$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 $$Q;\; href="/html/ALL/l/varphi.html"\; ;"title="varphi">varphi$$ and $$R;\; href="/html/ALL/l/varphi.html"\; ;"title="varphi">varphi$$ Let $u$ be the function that minimizes the quotient $Q;\; href="/html/ALL/l/varphi.html"\; ;"title="varphi">varphi$ 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 $$A;\; href="/html/ALL/l/.html"\; ;"title="">$$ 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 thefirst variation In applied mathematics and the calculus of variations, the first variation of a functional ''J''(''y'') is defined as the linear functional \delta J(y) mapping the function ''h'' to
:\delta J(y,h) = \lim_ \frac = \left.\frac J(y + \varepsilon h ...

of $A$ (the derivative of $A$ with respect to ε) is
$$\backslash delta\; A;\; href="/html/ALL/l/\_0,f\_1.html"\; ;"title="\_0,f\_1">\_0,f\_1$$
After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation
$$-\backslash frac\; \backslash left;\; href="/html/ALL/l/frac\_\backslash right.html"\; ;"title="frac\; \backslash right">frac\; \backslash right$$
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 $$\backslash delta\; A;\; href="/html/ALL/l/\_0,f\_1.html"\; ;"title="\_0,f\_1">\_0,f\_1$$ 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 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 $$A;\; href="/html/ALL/l/.html"\; ;"title="">$$ 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 $$A;\; href="/html/ALL/l/.html"\; ;"title="">$$ 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

= The wave equation 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 offirst-order partial differential equation In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of ''n'' variables. The equation takes the form
: F(x_1,\ldots,x_n,u,u_,\ldots u_) =0. \,
...

s, 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, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...

of a mechanical system and $U$ 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
$$S\; =\; \backslash int\_^\; L(x,\; \backslash dot\; x,\; t)\; \backslash ,\; dt$$
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 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:
$$\backslash frac\; +\; H\backslash left(x,\backslash frac,t\backslash right)\; =\; 0.$$
Further applications

Further applications of the calculus of variations include the following: * The derivation of the catenary shape * Solution to Newton's minimal resistance problem * Solution to the brachistochrone problem * Solution to thetautochrone problem
A tautochrone or isochrone curve (from Greek prefixes tauto- meaning ''same'' or iso- ''equal'', and chrono ''time'') is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is indepen ...

* Solution to isoperimetric
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by ...

problems
* Calculating geodesics
* Finding minimal surfaces and solving Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The 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
Variational methods in general relativity refers to various mathematical techniques that employ the use of variational calculus in Einstein's theory of general relativity. The most commonly used tools are Lagrangians and Hamiltonians and are used ...

, 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
In signal processing, particularly image processing, total variation denoising, also known as total variation regularization or total variation filtering, is a noise removal process ( filter). It is based on the principle that signals with excessi ...

, 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 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 $J;\; href="/html/ALL/l/.html"\; ;"title="">$See also

*First variation In applied mathematics and the calculus of variations, the first variation of a functional ''J''(''y'') is defined as the linear functional \delta J(y) mapping the function ''h'' to
:\delta J(y,h) = \lim_ \frac = \left.\frac J(y + \varepsilon h ...

* Isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...

* Variational principle
* Variational bicomplex
In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bun ...

* Fermat's principle
* Principle of least action
* Infinite-dimensional optimization In certain optimization problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a function or the shape of a body. Such a problem is an infinite-dimensional optimization problem, becaus ...

* Finite element method
* Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

* Ekeland's variational principle In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist nearly optimal solutions to some optimization problems.
Ekeland's principle can be used when the lower level set of a ...

* Inverse problem for Lagrangian mechanics
* Obstacle problem The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which i ...

* Perturbation methods
* Young measure
In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limi ...

* Optimal control
* Direct method in calculus of variations
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method reli ...

* Noether's theorem
* De Donder–Weyl theory
* Variational Bayesian methods
* Chaplygin problem
In mathematics, particularly in the fields of nonlinear dynamics and the calculus of variations, the Chaplygin problem is an isoperimetric problem with a differential constraint. Specifically, the problem is to determine what flight path an airp ...

* Nehari manifold
* Hu–Washizu principle
* Luke's variational principle
* Mountain pass theorem The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The th ...

*
* Measures of central tendency as solutions to variational problems
* Stampacchia Medal The Stampacchia Gold Medal is an international prize awarded every three years by the Italian Mathematical Union (''Unione Matematica Italiana'' - ''UMI'' ) together with the Ettore Majorana Foundation (Erice), in recognition of outstanding contrib ...

* Fermat Prize
* Convenient vector space In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition.
Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. ...

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

'' Encyclopedia of Mathematics''.

calculus of variations

'' PlanetMath''.

Calculus of Variations

'' MathWorld''.

Calculus of variations

Example problems.

Mathematics - Calculus of Variations and Integral Equations

Lectures on

YouTube
YouTube is a global online video sharing and social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the second mo ...

.
* Selected papers on Geodesic FieldsPart I

Part II

{{Authority control Optimization in vector spaces