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''first variation''
 
''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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v0961201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
J( x)  = \int\limits _ { t _ {0} } ^ { {t _ 1 } }
 +
L( t, x ( t), \dot{x} ( t)) dt.
 +
$$
  
If a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v0961202.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v0961203.png" /> and the latter is substituted in the expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v0961204.png" />, one obtains, assuming that the integrand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v0961205.png" /> is continuously differentiable, the following equation:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v0961206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
J( x _ {0} + \alpha h)  = J( x _ {0} )+
 +
\alpha J _ {1} ( x _ {0} )( h)+ r( \alpha ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v0961207.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v0961208.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v0961209.png" /> is often referred to as the variation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612010.png" />, and is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612011.png" />. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612012.png" />, which is a functional with respect to the variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612013.png" />, is said to be the first variation of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612014.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612015.png" />. As applied to the functional (1), the expression for the first variation has the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\delta J( x _ {0} , h)  = \
 +
\int\limits _ { t _ {0} } ^ { {t _ 1 } }
 +
( p( t) \dot{h} ( t) + q( t) h( t)) dt ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612017.png" /></td> </tr></table>
+
$$
 +
p( t)  = L _ {\dot{x} }  ( t, x _ {0} ( t), {\dot{x} } _ {0} ( t)),\ \
 +
q( t)  = L _ {x} ( t, x _ {0} ( t), {\dot{x} } _ {0} ( t)).
 +
$$
  
A necessary condition for an extremum of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612018.png" /> is that the first variation vanishes for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612019.png" />. 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]]:
+
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]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612020.png" /></td> </tr></table>
+
$$
 +
-  
 +
\frac{d}{dt}
 +
L _ {\dot{x} }  ( t, x _ {0} ( t), {\dot{x} } _ {0} ( t)) +
 +
L _ {x} ( t, x _ {0} ( t), {\dot{x} } _ {0} ( t))  = 0.
 +
$$
  
 
A method similar to (2) is also used to determine variations of higher orders (see, for example, [[Second variation|Second variation]] of a functional).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612021.png" /> is linear and continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096120/v09612022.png" /> 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]]].
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:27, 6 June 2020


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)

Comments

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