Portal:Gravitation/Mass

For other uses, see Helmholtz theorem.

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 = \frac{p^{2}}{2 m}

φ (x; V)

is a "U-shaped" potential energy profile which depends on a parameter $V$ . Let ${⟨ \cdot ⟩}_{t}$ denote the time average. Let

E = K + φ,

T = 2 {⟨ K ⟩}_{t},

P = {⟨ - \frac{\partial φ}{\partial V} ⟩}_{t},

S (E, V) = \log \oint \sqrt{2 m (E - φ (x, V))} d x .

Then

d S = \frac{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. $\oint d x \sqrt{2 m (E - φ (x, V))}$ ).
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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Campisi, M. (2005) On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem Studies in History and Philosophy of Modern Physics 36: 275–290