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


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, 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 ...
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. ...
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.


The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to
scalars 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 /math> is said to have an extremum at the function f if \Delta J = J - J /math> has the same sign for all y in an arbitrarily small neighborhood of f. The function f is called an extremal function or extremal. The extremum J /math> is called a local maximum if \Delta J \leq 0 everywhere in an arbitrarily small neighborhood of f, and a local minimum if \Delta J \geq 0 there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not. Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the converse may not hold. Finding strong extrema is more difficult than finding weak extrema. An example of a
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 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 ...
is equal to zero. This leads to solving the associated Euler–Lagrange equation. Consider the functional J = \int_^ L\left(x,y(x),y'(x)\right)\, dx \, . 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'(x) = \frac, *L\left(x, y(x), y'(x)\right) is twice continuously differentiable with respect to its arguments x, y, and y'. If the functional J /math> attains a
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 \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 \varepsilon close to 0, J \le J + \varepsilon \eta\, . The term \varepsilon \eta is called the variation of the function f and is denoted by \delta f. Substituting f + \varepsilon \eta for y in the functional J the result is a function of \varepsilon, \Phi(\varepsilon) = J +\varepsilon\eta\, . Since the functional J /math> has a minimum for y = f the function \Phi(\varepsilon) has a minimum at \varepsilon = 0 and thus, \Phi'(0) \equiv \left.\frac\_ = \int_^ \left.\frac\_ dx = 0 \, . Taking the total derivative of L\left , y, y'\right where y = f + \varepsilon \eta and y' = f' + \varepsilon \eta' are considered as functions of \varepsilon rather than x, yields \frac=\frac\frac + \frac\frac and because \frac = \eta and \frac = \eta', \frac=\frac\eta + \frac\eta'. Therefore, \begin \int_^ \left.\frac\_ dx & = \int_^ \left(\frac \eta + \frac \eta'\right)\, dx \\ & = \int_^ \frac \eta \, dx + \left.\frac \eta \_^ - \int_^ \eta \frac\frac \, dx \\ & = \int_^ \left(\frac \eta - \eta \frac\frac \right)\, dx\\ \end where L\left , y, y'\right\to L\left , f, f'\right/math> when \varepsilon = 0 and we have used
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 \eta = 0 at x_1 and x_2 by definition. Also, as previously mentioned the left side of the equation is zero so that \int_^ \eta (x) \left(\frac - \frac\frac \right) \, dx = 0 \, . 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. \frac -\frac \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 /math> and is denoted \delta J/\delta f(x). In general this gives a second-order
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 A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.


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 \left(x_1, y_1\right) and \left(x_2, y_2\right). The
arc 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 = \int_^ \sqrt \, dx \, , with y'(x) = \frac \, , \ \ y_1=f(x_1) \, , \ \ y_2=f(x_2) \, . 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 \frac -\frac \frac=0 with L = \sqrt \, . Since f does not appear explicitly in L, the first term in the Euler–Lagrange equation vanishes for all f(x) and thus, \frac \frac = 0 \, . Substituting for L and taking the derivative, \frac \ \frac \ = 0 \, . Thus \frac = c \, , for some constant c. Then \frac = c^2 \, , where 0 \le c^2<1. Solving, we get '(x)2=\frac which implies that f'(x)=m is a constant and therefore that the shortest curve that connects two points \left(x_1, y_1\right) and \left(x_2, y_2\right) is f(x) = m x + b \qquad \text \ \ m = \frac \quad \text \quad b = \frac and we have thus found the extremal function f(x) that minimizes the functional A /math> so that A /math> is a minimum. The equation for a straight line is y = f(x). In other words, the shortest distance between two points is a straight line.

Beltrami's identity

In physics problems it may be the case that \frac = 0, meaning the integrand is a function of f(x) and f'(x) but x does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity L - f' \frac = C \, , where C is a constant. The left hand side is the Legendre transformation of L with respect to f'(x). The intuition behind this result is that, if the variable x is actually time, then the statement \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 = \int_^ f(x, y(x), y'(x), \dots, y^(x)) dx, then y must satisfy the Euler– Poisson equation, \frac - \frac \left( \frac \right) + \dots + (-1)^ \frac \left \frac \right 0.

Du Bois-Reymond's theorem

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral 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 \frac \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 = \int_0^1 (x^3-t)^2 x'^6, = \. Clearly, x(t) = t^minimizes the functional, but we find any function x \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 \leq p < q < \infty. There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals. Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.

Functions of several variables

For example, if \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 varphi= \iint_D \sqrt \,dx\,dy.
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: \varphi_(1 + \varphi_y^2) + \varphi_(1 + \varphi_x^2) - 2\varphi_x \varphi_y \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 varphi= \frac\iint_D \nabla \varphi \cdot \nabla \varphi \, dx\, dy. The functional V is to be minimized among all trial functions \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 + \varepsilon v/math> must vanish: \left.\frac V + \varepsilon v_ = \iint_D \nabla u \cdot \nabla v \, dx\,dy = 0. Provided that u has two derivatives, we may apply the divergence theorem to obtain \iint_D \nabla \cdot (v \nabla u) \,dx\,dy = \iint_D \nabla u \cdot \nabla v + v \nabla \cdot \nabla u \,dx\,dy = \int_C v \frac \, ds, where C is the boundary of D, s is arclength along C and \partial u / \partial n is the normal derivative of u on C. Since v vanishes on C and the first variation vanishes, the result is \iint_D v\nabla \cdot \nabla u \,dx\,dy =0 for all smooth functions v that vanish on the boundary of D. The proof for the case of one dimensional integrals may be adapted to this case to show that \nabla \cdot \nabla u= 0 in D. The difficulty with this reasoning is the assumption that the minimizing function u must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize W varphi= \int_^ (x\varphi')^2 \, dx among all functions \varphi that satisfy \varphi(-1)=-1 and \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 equations; 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 varphi= \iint_D \left \frac \nabla \varphi \cdot \nabla \varphi + f(x,y) \varphi \right\, dx\,dy \, + \int_C \left \frac \sigma(s) \varphi^2 + g(s) \varphi \right\, ds. 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 \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 + \varepsilon v/math> is given by \iint_D \left \nabla u \cdot \nabla v + f v \right\, dx\, dy + \int_C \left \sigma u v + g v \right\, ds = 0. If we apply the divergence theorem, the result is \iint_D \left -v \nabla \cdot \nabla u + v f \right\, dx \, dy + \int_C v \left \frac + \sigma u + g \right\, ds =0. If we first set v = 0 on C, the boundary integral vanishes, and we conclude as before that - \nabla \cdot \nabla u + f =0 in D. Then if we allow v to assume arbitrary boundary values, this implies that u must satisfy the boundary condition \frac + \sigma u + g =0, on C. This boundary condition is a consequence of the minimizing property of u: it is not imposed beforehand. Such conditions are called natural boundary conditions. The preceding reasoning is not valid if \sigma vanishes identically on C. In such a case, we could allow a trial function \varphi \equiv c, where c is a constant. For such a trial function, V = c\left \iint_D f \, dx\,dy + \int_C g \, ds \right By appropriate choice of c, V can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless \iint_D f \, dx\,dy + \int_C g \, ds =0. This condition implies that net external forces on the system are in equilibrium. If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added. Further details and examples are in Courant and Hilbert (1953).

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 varphi= \int_^ \left p(x) \varphi'(x)^2 + q(x) \varphi(x)^2 \right\, dx, where \varphiis restricted to functions that satisfy the boundary conditions \varphi(x_1)=0, \quad \varphi(x_2)=0. Let R be a normalization integral R varphi=\int_^ r(x)\varphi(x)^2 \, dx. 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 \varphi satisfying the endpoint conditions. It is shown below that the Euler–Lagrange equation for the minimizing u is -(p u')' +q u -\lambda r u = 0, where \lambda is the quotient \lambda = \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 \lambda will be denoted by \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 the
Rayleigh–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 \int_^ r(x) u_1(x) \varphi(x) \, 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 \varphi vanish at the endpoints, we may not impose any condition at the endpoints, and set Q varphi= \int_^ \left p(x) \varphi'(x)^2 + q(x)\varphi(x)^2 \right\, dx + a_1 \varphi(x_1)^2 + a_2 \varphi(x_2)^2, where a_1 and a_2 are arbitrary. If we set \varphi = u + \varepsilon vthe first variation for the ratio Q/R is V_1 = \frac \left( \int_^ \left p(x) u'(x)v'(x) + q(x)u(x)v(x) -\lambda r(x) u(x) v(x) \right\, dx + a_1 u(x_1)v(x_1) + a_2 u(x_2)v(x_2) \right), where λ is given by the ratio Q R /math> as previously. After integration by parts, \frac V_1 = \int_^ v(x) \left -(p u')' + q u -\lambda r u \right\, dx + v(x_1) -p(x_1)u'(x_1) + a_1 u(x_1)+ v(x_2) (x_2) u'(x_2) + a_2 u(x_2) If we first require that v vanish at the endpoints, the first variation will vanish for all such v only if -(p u')' + q u -\lambda r u =0 \quad \hbox \quad x_1 < x < x_2. If u satisfies this condition, then the first variation will vanish for arbitrary v only if -p(x_1)u'(x_1) + a_1 u(x_1)=0, \quad \hbox \quad p(x_2) u'(x_2) + a_2 u(x_2)=0. These latter conditions are the natural boundary conditions for this problem, since they are not imposed on trial functions for the minimization, but are instead a consequence of the minimization.

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 varphi= \iiint_D p(X) \nabla \varphi \cdot \nabla \varphi + q(X) \varphi^2 \, dx \, dy \, dz + \iint_B \sigma(S) \varphi^2 \, dS, and R varphi= \iiint_D r(X) \varphi(X)^2 \, dx \, dy \, dz. Let u be the function that minimizes the quotient Q varphi/ R varphi with no condition prescribed on the boundary B. The Euler–Lagrange equation satisfied by u is -\nabla \cdot (p(X) \nabla u) + q(x) u - \lambda r(x) u=0, where \lambda = \frac. The minimizing u must also satisfy the natural boundary condition p(S) \frac + \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).



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 = \int_^ n(x,f(x)) \sqrt dx, where the refractive index n(x,y) depends upon the material. If we try f(x) = f_0 (x) + \varepsilon f_1 (x) then the
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 ...
of A (the derivative of A with respect to ε) is \delta A _0,f_1= \int_^ \left \frac + n_y (x,f_0) f_1 \sqrt \rightdx. After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation -\frac \left frac \right+ n_y (x,f_0) \sqrt = 0. The light rays may be determined by integrating this equation. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

Snell's law

There is a discontinuity of the refractive index when light enters or leaves a lens. Let n(x,y) = \begin n_ & \text \quad x<0, \\ n_ & \text \quad x>0, \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' may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form \delta A _0,f_1= f_1(0)\left n_\frac - n_\frac \right The factor multiplying n_ is the sine of angle of the incident ray with the x axis, and the factor multiplying n_ is the sine of angle of the refracted ray with the x axis. 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 \dot X(t) be its tangent vector. The optical length of the curve is given by A = \int_^ n(X) \sqrt \, dt. Note that this integral is invariant with respect to changes in the parametric representation of C. The Euler–Lagrange equations for a minimizing curve have the symmetric form \frac P = \sqrt \, \nabla n, where P = \frac. It follows from the definition that P satisfies P \cdot P = n(X)^2. Therefore, the integral may also be written as A = \int_^ P \cdot \dot X \, dt. This form suggests that if we can find a function \psi whose gradient is given by P, then the integral A is given by the difference of \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 \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 \nabla \cdot \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 \varphi_t^2 = c(X)^2 \, \nabla \varphi \cdot \nabla \varphi. We may look for solutions in the form \varphi(t,X) = t - \psi(X). In that case, \psi satisfies \nabla \psi \cdot \nabla \psi = n^2, where n=1/c. According to the theory of
first-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 = \nabla \psi, then P satisfies \frac = n \, \nabla n, along a system of curves (the light rays) that are given by \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 \frac = \frac. We conclude that the function \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.


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 the
kinetic 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 = \int_^ L(x, \dot x, t) \, 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: \frac \frac = \frac, and they are equivalent to Newton's equations of motion (for such systems). The conjugate momenta P are defined by p = \frac. For example, if T = \frac m \dot x^2, then p = m \dot x. Hamiltonian mechanics results if the conjugate momenta are introduced in place of \dot x by a Legendre transformation of the Lagrangian L into the Hamiltonian H defined by H(x, p, t) = p \,\dot x - L(x,\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: \frac + H\left(x,\frac,t\right) = 0.

Further applications

Further applications of the calculus of variations include the following: * The derivation of the catenary shape * Solution to Newton's minimal resistance problem * Solution to the brachistochrone problem * Solution to the
tautochrone problem 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 /math> is a functional with the function y = y(x) as its argument, and there is a small change in its argument from y to y + h, where h = h(x) is a function in the same function space as y, then the corresponding change in the functional is \Delta J = J +h- J The functional J /math> is said to be differentiable if \Delta J = \varphi + \varepsilon \, h\, , where \varphi /math> is a linear functional, \, h\, is the norm of h, and \varepsilon \to 0 as \, h\, \to 0. The linear functional \varphi /math> is the first variation of J /math> and is denoted by, \delta J = \varphi The functional J /math> is said to be twice differentiable if \Delta J = \varphi_1 + \varphi_2 + \varepsilon \, h\, ^2, where \varphi_1 /math> is a linear functional (the first variation), \varphi_2 /math> is a quadratic functional, and \varepsilon \to 0 as \, h\, \to 0. The quadratic functional \varphi_2 /math> is the second variation of J /math> and is denoted by, \delta^2 J = \varphi_2 The second variation \delta^2 J /math> is said to be strongly positive if \delta^2J \ge k \, h\, ^2, for all h and for some constant k > 0. Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.

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. ...



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 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 Fields
Part IPart II
{{Authority control Optimization in vector spaces