View source for Helmholtz theorem (classical mechanics)

{{Short description|Thermodynamics-like result in classical mechanics}}
{{other uses|Helmholtz theorem (disambiguation)}}

{{No footnotes|date=December 2023}}
The '''Helmholtz theorem of classical mechanics''' reads as follows:

Let <math display="block">H(x,p;V) = K(p) + \varphi(x;V)</math> be the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of a one-dimensional system, where <math display="block">K = \frac{p^2}{2m}</math> is the [[kinetic energy]] and  <math display="block">\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 display="block">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 display="inline">\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 the [[generalized Helmholtz theorem]].

== Generalized version ==
The '''generalized Helmholtz theorem''' is the multi-dimensional generalization of the Helmholtz theorem, and reads as follows.

Let
:<math>\mathbf{p}=(p_1,p_2,...,p_s),</math>
:<math>\mathbf{q}=(q_1,q_2,...,q_s),</math>
be the [[canonical coordinates]] of a ''s''-dimensional [[Hamiltonian system]], and let

:<math> H(\mathbf{p},\mathbf{q};V)=K(\mathbf{p})+\varphi(\mathbf{q};V) </math>

be the [[Hamiltonian mechanics#Mathematical formalism|Hamiltonian]] function, where

:<math>K=\sum_{i=1}^{s}\frac{p_i^2}{2m}</math>,

is the [[kinetic energy]] and

:<math>\varphi(\mathbf{q};V)</math>

is the [[potential energy]] which depends on a parameter <math>V</math>.
Let the hyper-surfaces of constant energy in the 2''s''-dimensional phase space of the system be '''metrically indecomposable''' and let <math>\left\langle \cdot \right\rangle_t </math> denote time average. Define the quantities <math>E</math>, <math>P</math>, <math>T</math>, <math>S</math>, as follows:

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

:<math>T = \frac{2}{s}\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 \int_{H(\mathbf{p},\mathbf{q};V) \leq E} d^s\mathbf{p}d^s \mathbf{q}. </math>

Then:

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

== 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.&nbsp;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.&nbsp;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.&nbsp;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]]
[[Category:Hermann von Helmholtz]]


{{classicalmechanics-stub}}
{{statisticalmechanics-stub}}