Helmholtz theorem (classical mechanics)

The Helmholtz theorem of classical mechanics reads as follows:

Let $$H(x,p;V) = K(p) + \varphi(x;V)$$ be the Hamiltonian of a one-dimensional system, where $$K = \frac{p^2}{2m}$$ is the kinetic energy and $$\varphi(x;V)$$ is a "U-shaped" potential energy profile which depends on a parameter $$V$$. Let $$\left\langle \cdot \right\rangle _{t}$$ denote the time average. Let
 * $$E = K + \varphi, $$
 * $$T = 2\left\langle K\right\rangle _{t},$$
 * $$P = \left\langle -\frac{\partial \varphi }{\partial V}\right\rangle _{t},$$
 * $$S(E,V)=\log \oint \sqrt{2m\left( E-\varphi \left( x,V\right) \right) }\,dx.$$

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. This in turn allows to define the "thermodynamic state" of a one-dimensional mechanical system. In particular the temperature $$T$$ is given by time average of the kinetic energy, and the entropy $$S$$ by the logarithm of the action (i.e., $\oint dx \sqrt{2m\left( E - \varphi \left( x, V\right) \right) }$ ). 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 the generalized Helmholtz theorem.

Generalized version
The generalized Helmholtz theorem is the multi-dimensional generalization of the Helmholtz theorem, and 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:


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