Samiksha Jaiswal (Editor)

Helmholtz theorem (classical mechanics)

Updated on
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

The Helmholtz theorem of classical mechanics reads as follows:


H ( x , p ; V ) = K ( p ) + φ ( x ; V )

be the Hamiltonian of a one-dimensional system, where

K = p 2 2 m

is the kinetic energy and

φ ( x ; V )

is a "U-shaped" potential energy profile which depends on a parameter V . Let t denote the time average. Let

E = K + φ , T = 2 K t , P = φ V t , S ( E , V ) = log 2 m ( E φ ( x , V ) ) d x .


d S = d E + P d V T .


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 T is given by time average of the kinetic energy, and the entropy S by the logarithm of the action (i.e. d x 2 m ( E φ ( x , V ) ) ).
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.


Helmholtz theorem (classical mechanics) Wikipedia

Similar Topics
Kirbyjon Caldwell
Rick Hahn
Soo Kim