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 ${\displaystyle V}$. Let ${\displaystyle \left\langle \cdot \right\rangle _{t}}$ denote the time average. Let

Then

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 ${\displaystyle T}$ is given by time average of the kinetic energy, and the entropy ${\displaystyle S}$ by the logarithm of the action (i.e.${\displaystyle \oint dx{\sqrt {2m\left(E-\varphi \left(x,V\right)\right)}}}$).
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.

...
Wikipedia