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
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 (i.e., ). <br/> 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 the generalized Helmholtz theorem.
The generalized Helmholtz theorem is the multi-dimensional generalization of the Helmholtz theorem, and 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: