# 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 $\tuple {x_0, \map{\mathbf y} {x_0} }$ and $\tuple {x, \mathbf y}$.

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.