# Generalized Helmholtz theorem

The generalized Helmholtz theorem is the multi-dimensional generalization of the Helmholtz theorem which is valid only in one dimension. The generalized Helmholtz theorem 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:

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

## 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 in multidimensional ergodic systems. This in turn allows to define the "thermodynamic state" of a multi-dimensional ergodic mechanical system, without the requirement that the system be composed of a large number of degrees of freedom. In particular the temperature ${\displaystyle T}$ is given by twice the time average of the kinetic energy per degree of freedom, and the entropy ${\displaystyle S}$ by the logarithm of the phase space volume enclosed by the constant energy surface (i.e. the so-called volume entropy).