\begin{definition}[Definition:Hilbert's Invariant Integral]
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''':
:$\ds \map g {x, \mathbf y} = \int_\Gamma \paren {-H \rd x + \mathbf p \rd \mathbf y}$
{{NamedforDef|David Hilbert|cat = Hilbert}}
\end{definition}