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The calculus of variations (or Variational Calculus) is a field of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s. Functionals are often expressed as definite integrals involving functions and their
derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
s. Functions that maximize or minimize functionals may be found using the
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action (physics), action functional. The equations ...
of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...
s''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in
mechanics Mechanics (from Ancient Greek: wikt:μηχανική#Ancient_Greek, μηχανική, ''mēkhanikḗ'', "of machine, machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among Ph ...
is the principle of least/stationary action. Many important problems involve functions of several variables. Solutions of boundary value problems for the
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...
satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
.


History

The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the
brachistochrone curve In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...
problem raised by
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating ...
(1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
first elaborated the subject, beginning in 1733.
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLegendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
and
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
also gave some early attention to the subject. To this discrimination Vincenzo Brunacci (1810),
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by
Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
(1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development. In the 20th century
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
, Oskar Bolza, Gilbert Ames Bliss,
Emmy Noether Amalie Emmy NoetherEmmy (given name), Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promot ...
, Leonida Tonelli,
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are con ...
and
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...
among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory. Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. The
dynamic programming Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. I ...
of Richard Bellman is an alternative to the calculus of variations.


Extrema

The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements y of a given
function space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
defined over a given domain. A functional J /math> is said to have an extremum at the function f if \Delta J = J - J /math> has the same
sign A sign is an Physical object, object, quality (philosophy), quality, event, or Non-physical entity, entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to ...
for all y in an arbitrarily small neighborhood of f. The function f is called an extremal function or extremal. The extremum J /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 that is used for finding weak extrema is the
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action (physics), action functional. The equations ...
.


Euler–Lagrange equation

Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to zero. This leads to solving the associated
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action (physics), action functional. The equations ...
. Consider the functional J = \int_^ L\left(x,y(x),y'(x)\right)\, dx \, . where *x_1, x_2 are constants, *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 Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, ...
at f, and \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 (mathematics), integral of a product (mathematics), product of Function (mathematics), functions in terms o ...
on the second term. The second term on the second line vanishes because \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, 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 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 (mathematics), variable and involves the derivatives of those functions. The term ''ordinary ...
which can be solved to obtain the extremal function f(x). The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum J 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 \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 In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involution (mathematics), involutive List of transforms, transformation on real number, real-valued convex functions of one real variable ...
of L with respect to f'(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 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 Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
. However Weierstrass gave an example of a variational problem with no solution: minimize W 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 equation Second-order linear differential equation, linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic partial differential equation, hyperbolic, or parabolic partial differential equation, parabolic. Any second-or ...
s; see Jost and Li–Jost (1998).


Generalization to other boundary value problems

A more general expression for the potential energy of a membrane is V 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 and eigenvectors, eigenvalues, originated in the context of solving physical Boundary value problem, boundary value problems and named after Lord Rayleigh and Walth ...
: choose an approximating u as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate. The next smallest eigenvalue and eigenfunction can be obtained by minimizing Q under the additional constraint \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).


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 = \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 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 Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a Mathematical formula, formula used to describe the relationship between the angle of incidence (optics), angles of incidence and refraction, when ...
for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.


Fermat's principle in three dimensions

It is expedient to use vector notation: let X = (x_1,x_2,x_3), let t be a parameter, let X(t) be the parametric representation of a curve C, and let \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 The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
for an inhomogeneous medium is u_ = c^2 \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 equations, 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.


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 the
kinetic energy In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical sci ...
of a mechanical system and U its
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
. Hamilton's principle (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral S = \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 Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
results if the conjugate momenta are introduced in place of \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 In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechan ...
: \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 In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or wire rope, cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, ...
shape * Solution to Newton's minimal resistance problem * Solution to the brachistochrone problem * Solution to the tautochrone problem * Solution to isoperimetric problems * Calculating
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...
s * Finding minimal surfaces and solving Plateau's problem *
Optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a Control (optimal control theory), control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applica ...
*
Analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...
, or reformulations of Newton's laws of motion, most notably Lagrangian and
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
; * Geometric optics, especially Lagrangian and Hamiltonian optics; * Variational method (quantum mechanics), one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states; * Variational Bayesian methods, a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning; * Variational methods in general relativity, a family of techniques using calculus of variations to solve problems in Einstein's general theory of relativity; *
Finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...
is a variational method for finding numerical solutions to boundary-value problems in differential equations; * Total variation denoising, an
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
method for filtering high variance or noisy signals.


Variations and sufficient condition for a minimum

Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation is defined as the linear part of the change in the functional, and the second variation is defined as the quadratic part. For example, if J /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 * Isoperimetric inequality * Variational principle * Variational bicomplex * Fermat's principle * Principle of least action * Infinite-dimensional optimization *
Finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...
*
Functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, ...
* Ekeland's variational principle * Inverse problem for Lagrangian mechanics * Obstacle problem * Perturbation methods * Young measure *
Optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a Control (optimal control theory), control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applica ...
* Direct method in calculus of variations * Noether's theorem * De Donder–Weyl theory * Variational Bayesian methods * Chaplygin problem * Nehari manifold * Hu–Washizu principle * Luke's variational principle * Mountain pass theorem * * Measures of central tendency as solutions to variational problems * Stampacchia Medal * Fermat Prize * Convenient vector space


Notes


References


Further reading

* Benesova, B. and Kruzik, M.
"Weak Lower Semicontinuity of Integral Functionals and Applications"
''SIAM Review'' 59(4) (2017), 703–766. * Bolza, O.
Lectures on the Calculus of Variations
Chelsea Publishing Company, 1904, available on Digital Mathematics library. 2nd edition republished in 1961, paperback in 2005, . * Cassel, Kevin W.
Variational Methods with Applications in Science and Engineering
Cambridge University Press, 2013. * Clegg, J.C.

Interscience Publishers Inc., 1968. * Courant, R.
Dirichlet's principle, conformal mapping and minimal surfaces
Interscience, 1950. * Dacorogna, Bernard:
Introduction

Introduction to the Calculus of Variations
', 3rd edition. 2014, World Scientific Publishing, . * Elsgolc, L.E.
Calculus of Variations
Pergamon Press Ltd., 1962. * Forsyth, A.R.
Calculus of Variations
Dover, 1960. * Fox, Charles

Dover Publ., 1987. * Giaquinta, Mariano; Hildebrandt, Stefan: Calculus of Variations I and II, Springer-Verlag, and * Jost, J. and X. Li-Jost
Calculus of Variations
Cambridge University Press, 1998. * Lebedev, L.P. and Cloud, M.J.
The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics
World Scientific, 2003, pages 1–98. * Logan, J. David
Applied Mathematics
3rd edition. Wiley-Interscience, 2006 * * Roubicek, T.:
Calculus of variations
. Chap.17 in:
Mathematical Tools for Physicists
'. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, , pp. 551–588. * Sagan, Hans

Dover, 1992. * Weinstock, Robert
Calculus of Variations with Applications to Physics and Engineering
Dover, 1974 (reprint of 1952 ed.).


External links


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