Helmholtz theorem (classical mechanics)

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=p22m 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⁡∮2m(E−φ(x,V))dx.

Then dS=dE+PdVT.

## 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., ∮dx2m(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.

## References

* Helmholtz, H., von (1884a). Principien der Statik monocyklischer Systeme. Borchardt-Crelle’s Journal für die reine und angewandte Mathematik, 97, 111–140 (also in Wiedemann G. (Ed.) (1895) Wissenschafltliche Abhandlungen. Vol. 3 (pp. 142–162, 179–202). Leipzig: Johann Ambrosious Barth).
* Helmholtz, H., von (1884b). Studien zur Statik monocyklischer Systeme. Sitzungsberichte der Kö niglich Preussischen Akademie der Wissenschaften zu Berlin, I, 159–177 (also in Wiedemann G. (Ed.) (1895) Wissenschafltliche Abhandlungen. Vol. 3 (pp. 163–178). Leipzig: Johann Ambrosious Barth).
* Boltzmann, L. (1884). Über die Eigenschaften monocyklischer und anderer damit verwandter Systeme.Crelles Journal, 98: 68–94 (also in Boltzmann, L. (1909). Wissenschaftliche Abhandlungen (Vol. 3, pp. 122–152), F. Hasenöhrl (Ed.). Leipzig. Reissued New York: Chelsea, 1969).
* Gallavotti, G. (1999). Statistical mechanics: A short treatise. Berlin: Springer.
* Campisi, M. (2005) On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem Studies in History and Philosophy of Modern Physics 36: 275–290

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