The Helmholtz theorem of classical mechanics reads as follows:

Let

$${\displaystyle H(x,p;V)=K(p)+\varphi (x;V)}$$

$${\displaystyle K={\frac {p^{2)){2m))}$$

$${\displaystyle \varphi (x;V)}$$

${\displaystyle V}$

${\displaystyle \left\langle \cdot \right\rangle _{t))$

• ${\displaystyle E=K+\varphi ,}$
• ${\displaystyle T=2\left\langle K\right\rangle _{t},}$
• ${\displaystyle P=\left\langle -{\frac {\partial \varphi }{\partial V))\right\rangle _{t},}$
• ${\displaystyle S(E,V)=\log \oint {\sqrt {2m\left(E-\varphi \left(x,V\right)\right)))\,dx.}$

Then

$${\displaystyle dS={\frac {dE+PdV}{T)).}$$

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 ${\displaystyle T}$ is given by time average of the kinetic energy, and the entropy ${\displaystyle S}$ by the logarithm of the action (i.e., ${\textstyle \oint dx{\sqrt {2m\left(E-\varphi \left(x,V\right)\right)))}$).
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.

Generalized version

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

Let

${\displaystyle \mathbf {p} =(p_{1},p_{2},...,p_{s}),}$

${\displaystyle \mathbf {q} =(q_{1},q_{2},...,q_{s}),}$

be the canonical coordinates of a s-dimensional Hamiltonian system, and let

${\displaystyle H(\mathbf {p} ,\mathbf {q} ;V)=K(\mathbf {p} )+\varphi (\mathbf {q} ;V)}$

be the Hamiltonian function, where

${\displaystyle K=\sum _{i=1}^{s}{\frac {p_{i}^{2)){2m))}$,

is the kinetic energy and

${\displaystyle \varphi (\mathbf {q} ;V)}$

is the potential energy which depends on a parameter ${\displaystyle V}$. Let the hyper-surfaces of constant energy in the 2s-dimensional phase space of the system be metrically indecomposable and let ${\displaystyle \left\langle \cdot \right\rangle _{t))$ denote time average. Define the quantities ${\displaystyle E}$, ${\displaystyle P}$, ${\displaystyle T}$, ${\displaystyle S}$, as follows:

${\displaystyle E=K+\varphi }$,

${\displaystyle T={\frac {2}{s))\left\langle K\right\rangle _{t))$,

${\displaystyle P=\left\langle -{\frac {\partial \varphi }{\partial V))\right\rangle _{t))$,

${\displaystyle S(E,V)=\log \int _{H(\mathbf {p} ,\mathbf {q} ;V)\leq E}d^{s}\mathbf {p} d^{s}\mathbf {q} .}$

Then:

${\displaystyle dS={\frac {dE+PdV}{T)).}$