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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

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

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

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

$$ 
- \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|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)]]).

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|Hamilton–Jacobi theory]]).

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====
<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>