 # Helmholtz theorem (classical mechanics)

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The Helmholtz theorem of classical mechanics reads as follows:

Let

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 .

Then

d S = d E + P d V T .

## 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 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.

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