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Generalized Helmholtz theorem


The generalized Helmholtz theorem on skates is the multi-dimensional generalization of the Helmholtz theorem which is valid only in one dimension. The generalized Helmholtz theorem reads as follows.

Let

be the canonical coordinates of a s-dimensional Hamiltonian system, and let

be the Hamiltonian function, where

is the kinetic energy and

is the potential energy which depends on a parameter . Let the hyper-surfaces of constant energy in the 2s-dimensional phase space of the system be metrically indecomposable and let denote time average. Define the quantities , , , , as follows:


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