# Definition:Hilbert's Invariant Integral

## Definition

Let $\mathbf y$ be an $n$-dimensional vector.

Let $H$ be Hamiltonian and $\mathbf p$ momenta.

Let $\Gamma$ be a curve connecting points $\paren{x_0,\map{\mathbf y} {x_0} }$ and $\paren{x,\mathbf y}$.

Then the following line integral is known as **Hilbert's Invariant Integral**:

- $\displaystyle \map g {x,\mathbf y}=\int_\Gamma\paren{-H\rd x+\mathbf p\rd\mathbf y}$

## Source of Name

This entry was named for David Hilbert.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 6.33$: Hilbert's Invariant Integral