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A curvilinear integral over a closed [[Differential form|differential form]] which is the derivative of the action of a functional of variational calculus. For the functional
 
A curvilinear integral over a closed [[Differential form|differential form]] which is the derivative of the action of a functional of variational calculus. For the functional
  
<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/h/h047/h047280/h0472801.png" /></td> </tr></table>
+
$$
 +
J ( x)  = \int\limits L ( t, x  ^ {i} , {\dot{x} } {}  ^ {i} )  dt
 +
$$
 +
 
 +
it is necessary to find a vector function  $  U  ^ {i} ( t, x  ^ {i} ) $,
 +
known as a field, such that the integral
 +
 
 +
$$
 +
J  ^ {*}  = \int\limits _  \gamma
 +
\left ( L ( t, x  ^ {i} , U  ^ {i} ( t, x  ^ {i} )) -
 +
\right .$$
  
it is necessary to find a vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h0472802.png" />, known as a field, such that the integral
+
$$
 +
- \left .
 +
\sum _ {k = 1 } ^ { n }  U  ^ {k} ( t, x  ^ {i} )
  
<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/h/h047/h047280/h0472803.png" /></td> </tr></table>
+
\frac{\partial  L ( t, x  ^ {i} , U  ^ {i} ( t, x
 +
^ {i} )) }{\partial  x  ^ {k} }
 +
\right )  dt +
 +
$$
  
<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/h/h047/h047280/h0472804.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ {k = 1 } ^ { n } 
 +
\frac{\partial  L ( t, x  ^ {i} , U  ^ {i} ( t, x  ^ {i} )) }{\partial  \dot{x}  ^ {k} }
 +
  dx  ^ {k}
 +
$$
  
<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/h/h047/h047280/h0472805.png" /></td> </tr></table>
+
is independent of the path of integration. If such a function exists,  $  J  ^ {*} $
 +
is said to be a Hilbert invariant integral. The condition of closure of the differential form in the integrand generates a system of partial differential equations of the first order.
  
is independent of the path of integration. If such a function exists, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h0472806.png" /> is said to be a Hilbert invariant integral. The condition of closure of the differential form in the integrand generates a system of partial differential equations of the first order.
+
The Hilbert invariant integral is the most natural connection between the theory of Weierstrass and the theory of Hamilton–Jacobi. Since  $  J  ^ {*} $
 +
is invariant, the value of the Hilbert invariant integral on the curves joining the points  $  P _ {0} = ( t _ {0} , x _ {0}  ^ {i} ) $
 +
and  $  P _ {1} = ( t _ {1} , x _ {1}  ^ {i} ) $
 +
becomes a function  $  S ( P _ {1} , P _ {2} ) $
 +
of this pair of points, called the action. A level line  $  S = \textrm{ const } $
 +
is said to be a transversal of  $  U  ^ {i} ( t, x  ^ {i} ) $.  
 +
The solutions of  $  \dot{x}  ^ {i} = U  ^ {i} ( t, x  ^ {i} ) $
 +
are the extremals of $  J( x) $.
 +
Conversely, if a domain is covered by a field of extremals, the integral  $  J  ^ {*} $
 +
constructed from the function  $  U  ^ {i} ( t, x  ^ {i} ) $,
 +
which is equal to the derivative of the extremal passing through  $  ( t, x  ^ {i} ) $,
 +
is a Hilbert invariant integral. The possibility of an appropriate contour i.e. of constructing the Hilbert invariant integral, is usually formulated as the [[Jacobi condition|Jacobi condition]].
  
The Hilbert invariant integral is the most natural connection between the theory of Weierstrass and the theory of Hamilton–Jacobi. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h0472807.png" /> is invariant, the value of the Hilbert invariant integral on the curves joining the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h0472808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h0472809.png" /> becomes a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728010.png" /> of this pair of points, called the action. A level line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728011.png" /> is said to be a transversal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728012.png" />. The solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728013.png" /> are the extremals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728014.png" />. Conversely, if a domain is covered by a field of extremals, the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728015.png" /> constructed from the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728016.png" />, which is equal to the derivative of the extremal passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728017.png" />, is a Hilbert invariant integral. The possibility of an appropriate contour i.e. of constructing the Hilbert invariant integral, is usually formulated as the [[Jacobi condition|Jacobi condition]].
+
If the curve  $  x  ^ {i} ( t) $
 +
passes in a domain covered by a field through the points  $  P _ {0} $
 +
and $  P _ {1} $,
 +
which are also connected by an extremal  $  x _ {0}  ^ {i} ( t) $,  
 +
then the invariance of Hilbert's invariant integral and the equality  $  dx _ {0}  ^ {i} /dt = U  ^ {i} ( t, x _ {0}  ^ {i} ( t)) $
 +
yield the [[Weierstrass formula|Weierstrass formula]] for the increment of the functional, and hence also a sufficient Weierstrass condition for an extremum (cf. [[Weierstrass conditions (for a variational extremum)|Weierstrass conditions (for a variational extremum)]]).
  
If the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728018.png" /> passes in a domain covered by a field through the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728020.png" />, which are also connected by an extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728021.png" />, then the invariance of Hilbert's invariant integral and the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728022.png" /> yield the [[Weierstrass formula|Weierstrass formula]] for the increment of the functional, and hence also a sufficient Weierstrass condition for an extremum (cf. [[Weierstrass conditions (for a variational extremum)|Weierstrass conditions (for a variational extremum)]]).
+
For a fixed point  $  P _ {0} $
 +
the action  $  S( P _ {0} , P) $
 +
is a function  $  S( t, x  ^ {i} ) $
 +
of the point  $  P = ( t, x  ^ {i} ) $,  
 +
and $  J  ^ {*} = \int dS $.  
 +
The transition to the canonical coordinates
  
For a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728023.png" /> the action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728024.png" /> is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728025.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728027.png" />. The transition to the canonical coordinates
+
$$
 +
p _ {k} ( t, x  ^ {i} )  = \
  
<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/h/h047/h047280/h04728028.png" /></td> </tr></table>
+
\frac{\partial  L ( t, x  ^ {i} , U  ^ {i} ( t, x  ^ {i} )) }{\partial  \dot{x}  ^ {k} }
 +
 
 +
$$
  
 
makes it possible to write the Hilbert invariant integral as
 
makes it possible to write the Hilbert invariant integral as
  
<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/h/h047/h047280/h04728029.png" /></td> </tr></table>
+
$$
 +
J  ^ {*}  = \int\limits dS  = \int\limits
 +
- H ( t, x  ^ {i} , p _ {i} ( t, x  ^ {i} ))  dt +
 +
\sum _ {k = 1 } ^ { n }
 +
p _ {k} ( t, x  ^ {i} )  dx  ^ {i} ;
 +
$$
  
 
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/h/h047/h047280/h04728030.png" /></td> </tr></table>
+
$$
 +
= \sum _ {k = 1 } ^ { n }  p _ {k} U  ^ {k} - L,
 +
$$
 +
 
 +
$$
  
<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/h/h047/h047280/h04728031.png" /></td> </tr></table>
+
\frac{\partial  S ( t, x  ^ {i} ) }{\partial  t }
 +
+ H ( t, x
 +
^ {i} , p _ {i} ( t, x  ^ {i} ))  = 0;
 +
\frac{\partial  S ( t,\
 +
x  ^ {i} ) }{\partial  x  ^ {i} }
 +
  = p _ {i} ( t, x  ^ {i} ).
 +
$$
  
 
These relations are equivalent to the Hamilton–Jacobi equation (cf. [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]).
 
These relations are equivalent to the Hamilton–Jacobi equation (cf. [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]).
  
The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047280/h04728032.png" /> for fields of geodesics was introduced by E. Beltrami [[#References|[1]]] in 1868, and, for the general case, by D. Hilbert [[#References|[2]]], [[#References|[3]]], [[#References|[4]]] in 1900.
+
The integral $  J  ^ {*} $
 +
for fields of geodesics was introduced by E. Beltrami [[#References|[1]]] in 1868, and, for the general case, by D. Hilbert [[#References|[2]]], [[#References|[3]]], [[#References|[4]]] in 1900.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Beltrami,  ''Rend. R. Istor. Lombardo Sci. Let.'' , '''1''' :  2  (1868)  pp. 708–718</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Hilbert,  "Mathematische Probleme"  ''Nachr. Ges. Wiss. Göttingen''  (1900)  pp. 253–297</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  "Hilbert problems"  ''Bull. Amer. Math. Soc.'' , '''8'''  (1902)  pp. 437–479  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Hilbert,  "Zur Variationsrechnung"  ''Math. Ann.'' , '''62'''  (1906)  pp. 351–370</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.I. Akhiezer,  "The calculus of variations" , Blaisdell  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</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">[7]</TD> <TD valign="top">  C. Carathéodory,  "Variationsrechnung und partielle Differentialgleichungen erster Ordnung" , Teubner  (1956)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L. Young,  "Lectures on the calculus of variations and optimal control theory" , Saunders  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Beltrami,  ''Rend. R. Istor. Lombardo Sci. Let.'' , '''1''' :  2  (1868)  pp. 708–718</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Hilbert,  "Mathematische Probleme"  ''Nachr. Ges. Wiss. Göttingen''  (1900)  pp. 253–297</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  "Hilbert problems"  ''Bull. Amer. Math. Soc.'' , '''8'''  (1902)  pp. 437–479  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Hilbert,  "Zur Variationsrechnung"  ''Math. Ann.'' , '''62'''  (1906)  pp. 351–370</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.I. Akhiezer,  "The calculus of variations" , Blaisdell  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</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">[7]</TD> <TD valign="top">  C. Carathéodory,  "Variationsrechnung und partielle Differentialgleichungen erster Ordnung" , Teubner  (1956)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L. Young,  "Lectures on the calculus of variations and optimal control theory" , Saunders  (1969)</TD></TR></table>

Latest revision as of 08:35, 18 August 2022


A curvilinear integral over a closed differential form which is the derivative of the action of a functional of variational calculus. For the functional

$$ J ( x) = \int\limits L ( t, x ^ {i} , {\dot{x} } {} ^ {i} ) dt $$

it is necessary to find a vector function $ U ^ {i} ( t, x ^ {i} ) $, known as a field, such that the integral

$$ J ^ {*} = \int\limits _ \gamma \left ( L ( t, x ^ {i} , U ^ {i} ( t, x ^ {i} )) - \right .$$

$$ - \left . \sum _ {k = 1 } ^ { n } U ^ {k} ( t, x ^ {i} ) \frac{\partial L ( t, x ^ {i} , U ^ {i} ( t, x ^ {i} )) }{\partial x ^ {k} } \right ) dt + $$

$$ + \sum _ {k = 1 } ^ { n } \frac{\partial L ( t, x ^ {i} , U ^ {i} ( t, x ^ {i} )) }{\partial \dot{x} ^ {k} } dx ^ {k} $$

is independent of the path of integration. If such a function exists, $ J ^ {*} $ is said to be a Hilbert invariant integral. The condition of closure of the differential form in the integrand generates a system of partial differential equations of the first order.

The Hilbert invariant integral is the most natural connection between the theory of Weierstrass and the theory of Hamilton–Jacobi. Since $ J ^ {*} $ is invariant, the value of the Hilbert invariant integral on the curves joining the points $ P _ {0} = ( t _ {0} , x _ {0} ^ {i} ) $ and $ P _ {1} = ( t _ {1} , x _ {1} ^ {i} ) $ becomes a function $ S ( P _ {1} , P _ {2} ) $ of this pair of points, called the action. A level line $ S = \textrm{ const } $ is said to be a transversal of $ U ^ {i} ( t, x ^ {i} ) $. The solutions of $ \dot{x} ^ {i} = U ^ {i} ( t, x ^ {i} ) $ are the extremals of $ J( x) $. Conversely, if a domain is covered by a field of extremals, the integral $ J ^ {*} $ constructed from the function $ U ^ {i} ( t, x ^ {i} ) $, which is equal to the derivative of the extremal passing through $ ( t, x ^ {i} ) $, is a Hilbert invariant integral. The possibility of an appropriate contour i.e. of constructing the Hilbert invariant integral, is usually formulated as the Jacobi condition.

If the curve $ x ^ {i} ( t) $ passes in a domain covered by a field through the points $ P _ {0} $ and $ P _ {1} $, which are also connected by an extremal $ x _ {0} ^ {i} ( t) $, then the invariance of Hilbert's invariant integral and the equality $ dx _ {0} ^ {i} /dt = U ^ {i} ( t, x _ {0} ^ {i} ( t)) $ yield the Weierstrass formula for the increment of the functional, and hence also a sufficient Weierstrass condition for an extremum (cf. Weierstrass conditions (for a variational extremum)).

For a fixed point $ P _ {0} $ the action $ S( P _ {0} , P) $ is a function $ S( t, x ^ {i} ) $ of the point $ P = ( t, x ^ {i} ) $, and $ J ^ {*} = \int dS $. The transition to the canonical coordinates

$$ p _ {k} ( t, x ^ {i} ) = \ \frac{\partial L ( t, x ^ {i} , U ^ {i} ( t, x ^ {i} )) }{\partial \dot{x} ^ {k} } $$

makes it possible to write the Hilbert invariant integral as

$$ J ^ {*} = \int\limits dS = \int\limits - H ( t, x ^ {i} , p _ {i} ( t, x ^ {i} )) dt + \sum _ {k = 1 } ^ { n } p _ {k} ( t, x ^ {i} ) dx ^ {i} ; $$

where

$$ H = \sum _ {k = 1 } ^ { n } p _ {k} U ^ {k} - L, $$

$$ \frac{\partial S ( t, x ^ {i} ) }{\partial t } + H ( t, x ^ {i} , p _ {i} ( t, x ^ {i} )) = 0; \ \frac{\partial S ( t,\ x ^ {i} ) }{\partial x ^ {i} } = p _ {i} ( t, x ^ {i} ). $$

These relations are equivalent to the Hamilton–Jacobi equation (cf. Hamilton–Jacobi theory).

The integral $ J ^ {*} $ for fields of geodesics was introduced by E. Beltrami [1] in 1868, and, for the general case, by D. Hilbert [2], [3], [4] in 1900.

References

[1] E. Beltrami, Rend. R. Istor. Lombardo Sci. Let. , 1 : 2 (1868) pp. 708–718
[2] D. Hilbert, "Mathematische Probleme" Nachr. Ges. Wiss. Göttingen (1900) pp. 253–297
[3] "Hilbert problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German)
[4] D. Hilbert, "Zur Variationsrechnung" Math. Ann. , 62 (1906) pp. 351–370
[5] N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian)
[6] I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian)
[7] C. Carathéodory, "Variationsrechnung und partielle Differentialgleichungen erster Ordnung" , Teubner (1956)
[8] L. Young, "Lectures on the calculus of variations and optimal control theory" , Saunders (1969)
How to Cite This Entry:
Hilbert invariant integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_invariant_integral&oldid=13085
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article