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.
Let 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 is given by time average of the kinetic energy, and the entropy by the logarithm of the action.

The importance of this theorem has been recognized by Ludwig Boltzmann who saw how to apply it to macroscopic 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.