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:


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