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:

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