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: