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2008-11-25T09:57:53Z

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:''For other uses, see [[Helmholtz theorem]]''.

The '''Helmholtz theorem of classical mechanics''' reads as follows:

Let

:<math>H(x,p;V)=K(p)+\varphi(x;V)</math>

be the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of a one-dimensional system, where

:<math>K=\frac{p^2}{2m}</math>

is the [[kinetic energy]] and

:<math>\varphi(x;V)</math>

is a "U-shaped" [[potential energy]] profile which depends on a parameter <math>V</math>.
Let <math>\left\langle \cdot \right\rangle _{t}</math> denote the time average. Let

:<math>E = K + \varphi, </math>

:<math>T = 2\left\langle K\right\rangle _{t},</math>

:<math>P = \left\langle -\frac{\partial \varphi }{\partial V}\right\rangle _{t},</math>

:<math>S(E,V)=\log \oint \sqrt{2m\left( E-\varphi \left( x,V\right) \right) }\,dx.</math>

Then

:<math>dS = \frac{dE+PdV}{T}.</math>

== 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]] <math>T</math> is given by time average of the kinetic energy, and the [[entropy]] <math>S</math> by the logarithm of the [[Action (physics)|action]] (i.e.<math>\oint
dx\sqrt{2m\left( E-\varphi \left( x,V\right) \right) }</math>). <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 [[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

[[Category:Classical mechanics]]
[[Category:Statistical mechanics theorems]]

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