The generalized Helmholtz theorem 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:
- ,
- ,
- ,
Then:
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