Generalized Helmholtz theorem - Alchetron, the free social encyclopedia

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

p = ( p 1 , p 2 , . . . , p s ) , q = ( q 1 , q 2 , . . . , q s ) ,

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

H ( p , q ; V ) = K ( p ) + φ ( q ; V )

be the Hamiltonian function, where

K = ∑ i = 1 s p i 2 2 m ,

is the kinetic energy and

φ ( 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 ⟨ ⋅ ⟩ t denote time average. Define the quantities E , P , T , S , as follows:

E = K + φ , T = 2 s ⟨ K ⟩ t , P = ⟨ − ∂ φ ∂ V ⟩ t , S ( E , V ) = log ⁡ ∫ H ( p , q ; V ) ≤ E d s p d s q .

Then:

d S = d E + P d V 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).

References

Generalized Helmholtz theorem Wikipedia

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