Variation of a functional

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

$$\tag{1 } J( x) = \int\limits _ { t _ {0} } ^ { {t _ 1 } } L( t, x ( t), \dot{x} ( t)) dt.$$

If a given function ${x _ {0} } ( t)$ is replaced by ${x _ {0} } ( t) + \alpha h( t)$ and the latter is substituted in the expression for $J( x)$, one obtains, assuming that the integrand $L$ is continuously differentiable, the following equation:

$$\tag{2 } J( x _ {0} + \alpha h) = J( x _ {0} )+ \alpha J _ {1} ( x _ {0} )( h)+ r( \alpha ),$$

where $| r( \alpha ) | \rightarrow 0$ as $\alpha \rightarrow 0$. The function $h( t)$ is often referred to as the variation of the function ${x _ {0} } ( t)$, and is sometimes denoted by $\delta x ( t)$. The expression ${J _ {1} } ( {x _ {0} } )( h)$, which is a functional with respect to the variation $h$, is said to be the first variation of the functional $J( x)$ and is denoted by $\delta J( x _ {0} , h)$. As applied to the functional (1), the expression for the first variation has the form

$$\tag{3 } \delta J( x _ {0} , h) = \ \int\limits _ { t _ {0} } ^ { {t _ 1 } } ( p( t) \dot{h} ( t) + q( t) h( t)) dt ,$$

where

$$p( t) = L _ {\dot{x} } ( t, x _ {0} ( t), {\dot{x} } _ {0} ( t)),\ \ q( t) = L _ {x} ( t, x _ {0} ( t), {\dot{x} } _ {0} ( t)).$$

A necessary condition for an extremum of the functional $J( x)$ is that the first variation vanishes for all $h$. 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:

$$- \frac{d}{dt} L _ {\dot{x} } ( t, x _ {0} ( t), {\dot{x} } _ {0} ( t)) + L _ {x} ( t, x _ {0} ( t), {\dot{x} } _ {0} ( t)) = 0.$$

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 $\delta J ( {x _ {0} } , h)$ is linear and continuous with respect to $h$ 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].

References

 [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)

References

 [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: http://encyclopediaofmath.org/index.php?title=Variation_of_a_functional&oldid=49114
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article