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 ${\displaystyle V}$. Let the hyper-surfaces of constant energy in the 2s-dimensional phase space of the system be metrically indecomposable and let ${\displaystyle \left\langle \cdot \right\rangle _{t}}$ denote time average. Define the quantities ${\displaystyle E}$, ${\displaystyle P}$, ${\displaystyle T}$, ${\displaystyle S}$, as follows:

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