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 \mathrm d x + \mathrm{ \mathbf p d \mathbf y } } \right)$