Generalized Helmholtz theorem

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 formula_6.
Let the hyper-surfaces of constant energy in the 2"s"-dimensional phase space of the system be metrically indecomposable and let formula_7 denote time average. Define the quantities formula_8, formula_9, formula_10, formula_11, as follows:
Then:
Remarks.
The thesis of this theorem of classical mechanics reads exactly as the heat theorem of thermodynamics. This fact shows that thermodynamic-like relations exist between certain mechanical quantities in multidimensional ergodic systems. This in turn allows to define the "thermodynamic state" of a multi-dimensional ergodic mechanical system, without the requirement that the system be composed of a large number of degrees of freedom. In particular the temperature formula_10 is given by twice the time average of the kinetic energy per degree of freedom, and the entropy formula_11 by the logarithm of the phase space volume enclosed by the constant energy surface (i.e. the so-called volume entropy).