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)
How to Cite This Entry:
Hilbert invariant integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_invariant_integral&oldid=52511
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article