Variation of a functional

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first variation

A generalization of the concept of the differential of a function of one variable. It is the principal linear part of the increment of the functional in a certain direction; it is employed in the theory of extremal problems to obtain necessary and sufficient conditions for an extremum. This was the meaning of the term "variation of a functional" imparted to it as early as 1760 by J.L. Lagrange [1]. He considered, in particular, the functionals of the classical calculus of variations of the form


If a given function is replaced by and the latter is substituted in the expression for , one obtains, assuming that the integrand is continuously differentiable, the following equation:


where as . The function is often referred to as the variation of the function , and is sometimes denoted by . The expression , which is a functional with respect to the variation , is said to be the first variation of the functional and is denoted by . As applied to the functional (1), the expression for the first variation has the form



A necessary condition for an extremum of the functional is that the first variation vanishes for all . In the case of the functional (1), a consequence of this necessary condition and the fundamental lemma of variational calculus (cf. du Bois-Reymond lemma) is the Euler equation:

A method similar to (2) is also used to determine variations of higher orders (see, for example, Second variation of a functional).

The general definition of the first variation in infinite-dimensional analysis was given by R. Gâteaux in 1913 (see Gâteaux variation). It is essentially identical with the definition of Lagrange. The first variation of a functional is a homogeneous, but not necessarily linear functional. The usual name under the additional assumption that the expression is linear and continuous with respect to is Gâteaux derivative. Terms such as "Gâteaux variation" , "Gâteaux derivative" , "Gâteaux differential" are more frequently employed than the term "variation of a functional" , which is reserved for the functionals of the classical variational calculus [3].


[1] J.L. Lagrange, "Essai d'une nouvelle methode pour déterminer les maximas et les minimas des formules intégrales indéfinies" , Oevres , 1 , G. Olms (1973) pp. 333–362
[2] R. Gâteaux, "Fonctions d'une infinités des variables indépendantes" Bull. Soc. Math. France , 47 (1919) pp. 70–96
[3] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)



[a1] I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian)
[a2] D.G. Luenberger, "Optimization by vectorspace methods" , Wiley (1969)
[a3] N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian)
How to Cite This Entry:
Variation of a functional. 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