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

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

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

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

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

be the Hamiltonian function, where

K=\sum _{i=1}^{s}{\frac {p_{i}^{2}}{2m}}

is the kinetic energy and

\varphi (\mathbf {q} ;V)

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

E=K+\varphi

T={\frac {2}{s}}\left\langle K\right\rangle _{t}

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

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

Then:

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 $T$ is given by twice the time average of the kinetic energy per degree of freedom, and the entropy $S$ by the logarithm of the phase space volume enclosed by the constant energy surface (i.e. the so-called volume entropy).

Generalized Helmholtz theorem

Remarks

Further reading