From Encyclopedia of Mathematics
Jump to: navigation, search

A mathematical term introduced by J.L. Lagrange [1] to denote a small displacement of an independent variable or of a functional. The method of variations is a method of studying an extremal problem in which the variations of the functional caused by small displacements of the argument are studied. This is one of the principal methods for studying extremal problems (hence the name variational calculus).

Let a functional $f$ be given on a space $X$, and let $V$ be a parameter space. A variation of the argument $x_0\in X$ is an ordinary curve $x(t,v)$, $\alpha\leq t\leq\beta$, $\alpha\leq0$, $\beta\geq0$, $v\in V$, in $X$ which passes through $x_0$ in a certain neighbourhood defined by the restrictions that are in force. Let the value $t=0$ correspond to $x_0$. As $v$ runs through the set of all parameters, the variations run through a certain family of curves issuing from $x_0$. In finite-dimensional and infinite-dimensional analysis, beginning with Lagrange, it is usual to employ the directional variation where $V=X$ and $x(t,v)=x_0+tv$. In this case the vector $v$ is referred to as the variation. However, other classes of variations are employed in geometry, in variational calculus and, in particular, in the theory of optimal control; these include polygonal variations, needle-shaped or spiked variations and variations connected with sliding regimes [2], [3]. The choice of the space of variations and the construction of the variations themselves are a very important element in obtaining necessary conditions for an extremum. See also Variation of a functional; Gâteaux derivative; Fréchet derivative; Functional derivative.


[1] J.L. Lagrange, "Essai d'une nouvelle methode pour déterminer les maximas et les minimas des formules intégrales indéfinies" , Oevres , 1 , G. Olms , New York (1973) pp. 333–362
[2] G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947)
[3] L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Wiley (1962) (Translated from Russian)



[a1] I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian)
[a2] L. Cesari, "Optimization - Theory and applications" , Springer (1983)
How to Cite This Entry:
Variation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article