Hilbert invariant integral

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)