Helmholtz theorem (classical mechanics)

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The Helmholtz theorem of classical mechanics reads as follows:

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

• ${\displaystyle E=K+\varphi ,}$
• ${\displaystyle T=2\left\langle K\right\rangle _{t},}$
• ${\displaystyle P=\left\langle -{\frac {\partial \varphi }{\partial V}}\right\rangle _{t},}$
• ${\displaystyle S(E,V)=\log \oint {\sqrt {2m\left(E-\varphi \left(x,V\right)\right)}}\,dx.}$

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