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 $\left({x_0, \mathbf y \left({x_0}\right)}\right)$ and $\left ({x, \mathbf y}\right)$.

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


 * $\displaystyle g \left({x, \mathbf y}\right) = \int_\Gamma \left ({-H \rd x + \mathbf p \rd \mathbf y}\right)$