The calculus of variations (or Variational Calculus) is a field of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
that uses variations, which are small changes in
functions
and
functionals, to find maxima and minima of functionals:
mappings from a set of
functions to the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. Functionals are often expressed as
definite integrals involving functions and their
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s. Functions that maximize or minimize functionals may be found using the
Euler–Lagrange equation of the calculus of variations.
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a
straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''
geodesics''. A related problem is posed by
Fermat's principle: light follows the path of shortest
optical length connecting two points, which depends upon the material of the medium. One corresponding concept in
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...
is the
principle of least/stationary action.
Many important problems involve functions of several variables. Solutions of
boundary value problems for the
Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \n ...
satisfy the
Dirichlet's principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Formal statement
Dirichlet's principle states that, if the funct ...
.
Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem ...
requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
.
History
The calculus of variations may be said to begin with
Newton's minimal resistance problem in 1687, followed by the
brachistochrone curve problem raised by
Johann Bernoulli (1696).
It immediately occupied the attention of
Jakob Bernoulli and the
Marquis de l'Hôpital, but
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
first elaborated the subject, beginning in 1733.
Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the ''calculus of variations'' in his 1756 lecture ''Elementa Calculi Variationum''.
Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima.
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
and
Gottfried Leibniz also gave some early attention to the subject.
To this discrimination
(1810),
Carl Friedrich Gauss (1829),
Siméon Poisson (1831),
Mikhail Ostrogradsky
Mikhail Vasilyevich Ostrogradsky (transcribed also ''Ostrogradskiy'', Ostrogradskiĭ) (russian: Михаи́л Васи́льевич Острогра́дский, ua, Миха́йло Васи́льович Острогра́дський; 24 Sep ...
(1834), and
Carl Jacobi (1837) have been among the contributors. An important general work is that of
Sarrus (1842) which was condensed and improved by
Cauchy (1844). Other valuable treatises and memoirs have been written by
Strauch
Strauch, a German word meaning ''bush'' or '' shrub'', is a surname. Notable people with it include:
* Adolfo Strauch, (b. 1948), survivor of the Uruguayan Air Force Flight 571 crash
* Adolph Strauch (1822–1883), landscape architect
* Aegidius ...
(1849),
Jellett (1850),
Otto Hesse (1857),
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...
(1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of
Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The
20th
20 (twenty; Roman numeral XX) is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.
In mathematics
*20 is a pronic number.
*20 is a tetrahedral number as 1, 4, 10, 20.
*20 is the ba ...
and the
23rd Hilbert problem published in 1900 encouraged further development.
In the 20th century
David Hilbert,
Oskar Bolza
Oskar Bolza (12 May 1857 – 5 July 1942) was a German mathematician, and student of Felix Klein. He was born in Bad Bergzabern, Palatinate (region), Palatinate, then a district of Bavaria, known for his research in the calculus of variations, p ...
,
Gilbert Ames Bliss,
Emmy Noether,
Leonida Tonelli
Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian people, Italian mathematician, noted for creating Fubini's theorem#Tonelli's theorem for non-negative measurable functions, Tonelli's theorem, a variation of Fubini's theorem, and f ...
,
Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
and
Jacques Hadamard among others made significant contributions.
Marston Morse applied calculus of variations in what is now called
Morse theory.
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely d ...
,
Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in
optimal control theory.
The
dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
...
of
Richard Bellman
Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founde ...
is an alternative to the calculus of variations.
Extrema
The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps
functions to
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements
of a given
function space defined over a given
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
. A functional