Helmholtz theorem (classical mechanics)

The Helmholtz theorem of classical mechanics reads as follows:
Let
be the Hamiltonian of a one-dimensional system, where 
is the kinetic energy and 
is a "U-shaped" potential energy profile which depends on a parameter formula_4.
Let formula_5 denote the time average. Let 
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. This in turn allows to define the "thermodynamic state" of a one-dimensional mechanical system. In particular the temperature formula_11 is given by time average of the kinetic energy, and the entropy formula_12 by the logarithm of the action (i.e.formula_13). 
The importance of this theorem has been recognized by Ludwig Boltzmann who saw how to apply it to macroscopic systems (i.e. multidimensional systems), in order to provide a mechanical foundation of equilibrium thermodynamics. This research activity was strictly related to his formulation of the ergodic hypothesis.
A multidimensional version of the Helmholtz theorem, based on the ergodic theorem of George David Birkhoff is known as generalized Helmholtz theorem.