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 . 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:

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 is given by twice the time average of the kinetic energy per degree of freedom, and the entropy by the logarithm of the phase space volume enclosed by the constant energy surface (i.e. the so-called volume entropy).