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 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
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 functional. The equations were discovered ...
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 objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
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
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 pat ...
: 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 object ...
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.
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 Newton'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 ...
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 un ...
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 Le ...
(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 in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leib ...
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 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, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
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 mathem ...
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
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
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
Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Prussia, and died in Munich, Bavaria. He worked mainly on algebraic invariants, and geometry. The Hessian matrix, the Hesse nor ...
(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 analysis". Despite leaving university without a degree, he studied mathematics ...
. 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,
Oskar Bolza,
Gilbert Ames Bliss
Gilbert Ames Bliss, (9 May 1876 – 8 May 1951), was an American mathematician, known for his work on the calculus of variations.
Life
Bliss grew up in a Chicago family that eventually became affluent; in 1907, his father became president of the ...
,
Emmy Noether
Amalie Emmy NoetherEmmy 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, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calc ...
,
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
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
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 a ...
applied calculus of variations in what is now called
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
.
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 due ...
,
Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in
optimal control theory
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
.
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 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
of a given
function space defined over a given
domain. A functional