Generalized Helmholtz theorem

The generalized Helmholtz theorem on skates 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