Hilbert's invariant integral

Summary

Hilbert's invariant integral is a line integral attached to a field of extremals of a variational problem. Its value depends only on the end points of the path, not on the path inside the field. Along curves of the field it equals the functional itself. Comparing it with the functional on an arbitrary curve gives the Weierstrass formula, which expresses $J(y) - J(y_0)$ as an integral of the Weierstrass excess function.