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''first variation''

A generalization of the concept of the [[Differential|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 [[#References|[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|du Bois-Reymond lemma]]) is the [[Euler equation|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|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|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|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 [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Gâteaux,   "Fonctions d'une infinités des variables indépendantes"  ''Bull. Soc. Math. France'' , '''47'''  (1919)  pp. 70–96</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Lavrent'ev,   L.A. Lyusternik,   "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Gel'fand,   S.V. Fomin,   "Calculus of variations" , Prentice-Hall  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.G. Luenberger,   "Optimization by vectorspace methods" , Wiley  (1969)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.I. Akhiezer,   "The calculus of variations" , Blaisdell  (1962)  (Translated from Russian)</TD></TR></table>

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[[Category:Mathematics]]