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 is given by twice the time average of the kinetic energy per degree of freedom, and the entropy by the logarithm of the phase space volume enclosed by the constant energy surface (i.e. the so-called volume entropy.
Read more about this topic: Generalized Helmholtz Theorem
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