TheInfoList

The calculus of variations is a field of
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
that uses variations, which are small changes in
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
and functionals, to find maxima and minima of functionals: mappings from a set of
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
to the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. Functionals are often expressed as
definite integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s involving functions and their
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s. Functions that maximize or minimize functionals may be found using the
Euler–Lagrange equationIn the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the function (mathematics), functions for which a given functional (mathematics), functional is stationary point, stationary. It wa ...
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
geodesic In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

s. A related problem is posed by
Fermat's principle Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path ...
: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in
mechanics Mechanics (Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximat ...

is the . Many important problems involve functions of several variables. Solutions of
boundary value problem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s for the
Laplace equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
satisfy the
Dirichlet principle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.
Plateau's problem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

.

# History

The calculus of variations may be said to begin with
Newton's minimal resistance problemNewton's Minimal Resistance Problem is a problem of finding a solid of revolution which experiences a minimum resistance when it moves through a homogeneous fluid with constant velocity in the direction of the axis of revolution, named after Isaac N ...
in 1687, followed by the
brachistochrone curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
problem raised by
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss Swiss may refer to: * the adjectival form of Switzerland , french: Suisse(sse), it, svizzero/svizzera or , rm, Svizzer/Svizra , government_type = ...

(1696). It immediately occupied the attention of
Jakob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includ ...

and the , but
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

first elaborated the subject, beginning in 1733.
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

and
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...
(1810),
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

(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 was ...

(1844). Other valuable treatises and memoirs have been written by (1849), Jellett (1850),
Otto Hesse Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg Königsberg (, , ) was the name for the historic Prussia Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a histo ...
(1857),
Alfred Clebsch Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...
(1858), and (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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

. 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 (number), 19 and preceding 21 (number), 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, ...
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 This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
,
Emmy Noether Amalie Emmy Noether 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, ''Promotionsakt Emmy Noeth ...

,
Leonida Tonelli Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quant ...

,
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 Mathematics (from Greek: ) includes the study of such topics as numbers (a ...
and
Jacques Hadamard Jacques Salomon Hadamard ForMemRS (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis of the function . Hue represents the argument, brightness the magnitud ...
Marston Morse Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known as ...

applied calculus of variations in what is now called
Morse theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.
Lev Pontryagin Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet The Soviet Union,. officially the Union of Soviet Socialist Republi ...
, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in
optimal control theory File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt= Mathematical optimization (alter ...
. The
dynamic programming Dynamic programming is both a mathematical optimization File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lo ...
of
Richard Bellman Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics. Biography Bellman was born in 1 ...
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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements of a given
function space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
defined over a given
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
. A functional is said to have an extremum at the function if has the same
sign A sign is an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at ...
for all in an arbitrarily small neighborhood of The function is called an extremal function or extremal. The extremum is called a local maximum if everywhere in an arbitrarily small neighborhood of and a local minimum if there. For a function space of continuous functions, extrema of corresponding functionals are called weak extrema or strong 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 Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a categorical or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical ...
may not hold. Finding strong extrema is more difficult than finding weak extrema. An example of a
necessary condition In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stateme ...
that is used for finding weak extrema is the
Euler–Lagrange equationIn the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the function (mathematics), functions for which a given functional (mathematics), functional is stationary point, stationary. It wa ...
.

# 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 where the
functional derivative Functional may refer to: * Movements in architecture: ** Functionalism (architecture) In architecture File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Pl ...
is equal to zero. This leads to solving the associated
Euler–Lagrange equationIn the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the function (mathematics), functions for which a given functional (mathematics), functional is stationary point, stationary. It wa ...
. Consider the functional : where : are constants, : is twice continuously differentiable, :, : is twice continuously differentiable with respect to its arguments . If the functional 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 ra ...
at and is an arbitrary function that has at least one derivative and vanishes at the endpoints and then for any number close to 0, : The term is called the variation of the function and is denoted by Substituting   for   in the functional the result is a function of , : Since the functional has a minimum for the function has a minimum at and thus, :$\Phi\text{'}\left(0\right) \equiv \left.\frac\_ = \int_^ \left.\frac\_ dx = 0 \, .$ Taking the
total derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of where and are considered as functions of rather than , yields :$\frac=\frac\frac + \frac\frac$ and since    and  , :$\frac=\frac\eta + \frac\eta\text{'}.$ Therefore, :$\begin \int_^ \left.\frac\_ dx & = \int_^ \left\left(\frac \eta + \frac \eta\text{'}\right\right)\, dx \\ & = \int_^ \frac \eta \, dx + \left.\frac \eta \_^ - \int_^ \eta \frac\frac \, dx \\ & = \int_^ \left\left(\frac \eta - \eta \frac\frac \right\right)\, dx\\ \end$ where when ''ε'' = 0 and we have used
integration by parts In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...

on the second term. The second term on the second line vanishes because at and by definition. Also, as previously mentioned the left side of the equation is zero so that :$\int_^ \eta \left(x\right) \left\left(\frac - \frac\frac \right\right) \, dx = 0 \, .$ According to the
fundamental lemma of calculus of variations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 Functional may refer to: * Movements in architecture: ** Functionalism (architecture) In architecture File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Pl ...
of and is denoted In general this gives a second-order
ordinary differential equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
which can be solved to obtain the extremal function The Euler–Lagrange equation is a
necessary Necessary or necessity may refer to: * Need A need is something that is necessary Necessary or necessity may refer to: * Need ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by ...
, but not sufficient, condition for an extremum . 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 which is the shortest curve that connects two points and The
arc length Arc length is the distance between two points along a section of a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. In ...

of the curve is given by : with :$y\,\text{'}\left(x\right) = \frac \, , \ \ y_1=f\left(x_1\right) \, , \ \ y_2=f\left(x_2\right) \, .$ The Euler–Lagrange equation will now be used to find the extremal function that minimizes the functional :$\frac -\frac \frac=0$ with :$L = \sqrt \, .$ Since does not appear explicitly in the first term in the Euler–Lagrange equation vanishes for all and thus, :$\frac \frac = 0 \, .$ Substituting for 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 : which implies that :$f\text{'}\left(x\right)=m$ is a constant and therefore that the shortest curve that connects two points and is :$f\left(x\right) = m x + b \qquad \text \ \ m = \frac \quad \text \quad b = \frac$ and we have thus found the extremal function that minimizes the functional so that is a minimum. The equation for a straight line is . 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 $f\left(x\right)$ and $f\text{'}\left(x\right)$ but $x$ does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity :$L-f\text{'}\frac=C \, ,$ where $C$ is a constant. The left hand side is the
Legendre transformation In mathematics and physics, the Legendre transformation, named after Adrien-Marie Legendre, is an involution (mathematics), involutive List of transforms, transformation on the real number, real-valued convex functions of one real variable. In phys ...
of $L$ with respect to $f\text{'}\left(x\right)$. The intuition behind this result is that, if the variable is actually time, then the statement $\frac = 0$ implies that the Lagrangian is time-independent. By
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...
, 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 $y\left(x\right)$, that is, if
$S = \int\limits_^ f\left(x, y\left(x\right), y\text{'}\left(x\right), ..., y^\left(x\right)\right) dx,$
then $y$ must satisfy the Euler– Poisson equation,

# 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: : :$= \.$ Clearly, $x\left(t\right) = t^$ 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 φ(''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 Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
= \iint_D \sqrt \,dx\,dy.\,
Plateau's problem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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: :$\varphi_\left(1 + \varphi_y^2\right) + \varphi_\left(1 + \varphi_x^2\right) - 2\varphi_x \varphi_y \varphi_ = 0.\,$ 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 Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
= \frac\iint_D \nabla \varphi \cdot \nabla \varphi \, dx\, dy.\, The functional ''V'' is to be minimized among all trial functions φ 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
Dirichlet principle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
in honor of his teacher
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For c ...

. However Weierstrass gave an example of a variational problem with no solution: minimize :$W$
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
= \int_^ (x\varphi')^2 \, dx\, among all functions ''φ'' that satisfy $\varphi\left(-1\right)=-1$ and $\varphi\left(1\right)=1.$ $W$ 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 Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be pr ...
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 Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
= \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 $f\left(x,y\right)$ in ''D'', an external force $g\left(s\right)$ on the boundary ''C'', and elastic forces with modulus $\sigma\left(s\right)$ 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

# 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 Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
= \int_^ \left p(x) \varphi'(x)^2 + q(x) \varphi(x)^2 \right\, dx, \, where $\varphi$ is restricted to functions that satisfy the boundary conditions :$\varphi\left(x_1\right)=0, \quad \varphi\left(x_2\right)=0. \,$ Let ''R'' be a normalization integral :$R$
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
=\int_^ r(x)\varphi(x)^2 \, dx.\, The functions $p\left(x\right)$ and $r\left(x\right)$ are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio ''Q''/''R'' among all φ satisfying the endpoint conditions. It is shown below that the Euler–Lagrange equation for the minimizing ''u'' is :$-\left(pu\text{'}\right)\text{'} +q u -\lambda r u =0, \,$ where λ 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 λ 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 $u_1\left(x\right)$. 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, eigenvalue, 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 :$\int_^ r\left(x\right) u_1\left(x\right) \varphi\left(x\right) \, dx=0. \,$ 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 φ vanish at the endpoints, we may not impose any condition at the endpoints, and set :$Q$
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
= \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 $\varphi = u + \varepsilon v$ the first variation for the ratio $Q/R$ is : where λ is given by the ratio

## 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 Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
= \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 Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
= \iiint_D r(X) \varphi(X)^2 \, dx \, dy \, dz.\, Let ''u'' be the function that minimizes the quotient $Q$
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
/ R
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term ...
with no condition prescribed on the boundary ''B''. The Euler–Lagrange equation satisfied by ''u'' is :$-\nabla \cdot \left(p\left(X\right) \nabla u\right) + q\left(x\right) u - \lambda r\left(x\right) u=0,\,$ where :$\lambda = \frac.\,$ The minimizing ''u'' must also satisfy the natural boundary condition :$p\left(S\right) \frac + \sigma\left(S\right) u =0,$ 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 Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path ...
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 $y=f\left(x\right)$ along the path, then the optical length is given by : where the refractive index $n\left(x,y\right)$ depends upon the material. If we try $f\left(x\right) = f_0 \left(x\right) + \varepsilon f_1 \left(x\right)$ then the first variation of ''A'' (the derivative of ''A'' with respect to ε) is : After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation : The light rays may be determined by integrating this equation. This formalism is used in the context of
Lagrangian opticsHamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
and
Hamiltonian opticsHamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
.

### Snell's law

There is a discontinuity of the refractive index when light enters or leaves a lens. Let :$n\left(x,y\right) = n_ \quad \hbox \quad x<0, \,$ :$n\left(x,y\right) = n_ \quad \hbox \quad x>0,\,$ 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 : 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 of light at the interface between two media of different refractive index, refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of in ...

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 $X=\left(x_1,x_2,x_3\right),$ let ''t'' be a parameter, let $X\left(t\right)$ be the parametric representation of a curve ''C'', and let $\dot X\left(t\right)$ be its tangent vector. The optical length of the curve is given by : 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 :$P \cdot P = n\left(X\right)^2. \,$ Therefore, the integral may also be written as : This form suggests that if we can find a function ψ whose gradient is given by ''P'', then the integral ''A'' is given by the difference of ψ 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 ψ. 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 opticsHamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
and
Hamiltonian opticsHamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
.

### = Connection with the wave equation

= The
wave equation The wave equation is a second-order linear for the description of s—as they occur in —such as (e.g. waves, and ) or waves. It arises in fields like , , and . Historically, the problem of a such as that of a was studied by , , , and ...
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 :$\varphi_t^2 = c\left(X\right)^2 \, \nabla \varphi \cdot \nabla \varphi. \,$ We may look for solutions in the form :$\varphi\left(t,X\right) = t - \psi\left(X\right). \,$ In that case, ψ satisfies :$\nabla \psi \cdot \nabla \psi = n^2, \,$ where $n=1/c.$ According to the theory of
first-order partial differential equationIn 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. \, Su ...
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 ψ 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 that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
of a mechanical system and ''U'' its
potential energy In physics, potential energy is the energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, but not created or destroyed. The unit of measure ...

.
Hamilton's principle In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
(or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral :$S = \int_^ L\left(x, \dot x, t\right) \, dt$ 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 the Italian-French mathematician and astronomer Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaHamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton Sir William Rowan Hamilton MRIA (3 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College, Dublin, Trinit ...
: :$\frac + H\left\left(x,\frac,t\right\right) =0.\,$

## Further applications

Further applications of the calculus of variations include the following: * The derivation of the
catenary In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

shape * Solution to
Newton's minimal resistance problemNewton's Minimal Resistance Problem is a problem of finding a solid of revolution which experiences a minimum resistance when it moves through a homogeneous fluid with constant velocity in the direction of the axis of revolution, named after Isaac N ...
* Solution to the
brachistochrone In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
problem * Solution to
isoperimetric In mathematics, the isoperimetric inequality is a geometry, geometric inequality (mathematics), 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 \opera ...
problems * Calculating
geodesic In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

s * Finding
minimal surface 180px, A helicoid minimal surface formed by a soap film on a helical frame In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

s and solving
Plateau's problem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
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 applicatio ...

# 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 is a functional with the function as its argument, and there is a small change in its argument from to , where is a function in the same function space as , then the corresponding change in the functional is :   The functional is said to be differentiable if : where is a linear functional, is the norm of , and as . The linear functional is the first variation of and is denoted by, : The functional is said to be twice differentiable if : where is a linear functional (the first variation), is a quadratic functional, and as . The quadratic functional is the second variation of and is denoted by, : The second variation is said to be strongly positive if : for all and for some constant . 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.

* First variation * Isoperimetric inequality *
Variational principle In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are s ...
* Variational bicomplex *
Fermat's principle Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path ...
*
Principle of least action :''This article discusses the history of the principle of least action. For the application, please refer to action (physics) In physics, action is an attribute of the dynamics (physics), dynamics of a physical system from which the equations ...

* Infinite-dimensional optimization *
Functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
* 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 applicatio ...
* Direct method in calculus of variations *
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...
*
De Donder–Weyl theoryIn mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this framew ...
*
Variational Bayesian methods Variational Bayesian methods are a family of techniques for approximating intractable integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combin ...
* Chaplygin problem * Nehari manifold * Hu–Washizu principle * Luke's variational principle * Mountain pass theorem * * Central tendency#Solutions to variational problems, Measures of central tendency as solutions to variational problems * Stampacchia Medal * Fermat Prize * Convenient vector space

# References

* Benesova, B. and Kruzik, M.
"Weak Lower Semicontinuity of Integral Functionals and Applications"
''SIAM Review'' 59(4) (2017), 703–766. * Oskar Bolza, 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. * Richard Courant, Courant, R.
Dirichlet's principle, conformal mapping and minimal surfaces
Interscience, 1950. * Bernard Dacorogna, Dacorogna, Bernard:
Introduction

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

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

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

Variational calculus
''Encyclopedia of Mathematics''.
calculus of variations
''PlanetMath''.
Calculus of Variations
''MathWorld''.
Calculus of variations
Example problems.
Mathematics - Calculus of Variations and Integral Equations
Lectures on YouTube. * Selected papers on Geodesic Fields
Part IPart II
{{Authority control Calculus of variations, Optimization in vector spaces