Definition:Hilbert's Invariant Integral

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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 contour integral is known as Hilbert's invariant integral:

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

Source of Name

This entry was named for David Hilbert.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 6.33$: Hilbert's Invariant Integral