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)