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I often think about the phase space with quite deep interpretations. For example, contraction of phase space means losing energy. But, some of the energy is easily restored (free energy?) while some other is hard to restore (enthalpy, internal energy). This is like dividing the phase space to flexible and rough?

Are my insights correct or maybe they can be easily extended more?

Are there any books with such interpretation-like approach to phase space and dynamical systems?

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V.I. Arnold - Mathematical Methods of Classical Mechanics. –  J Verma Apr 16 '11 at 21:26
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Agree with J. Verma that V.I. Arnold's book is a very good start. Another well-written review (which is considerably shorter) is Saunders Mac Lane's Amer. Math. Monthly article "Hamiltonian mechanics and geometry" (@article{Author = {Mac Lane, Saunders}, Journal = {Amer. Math. Monthly}, Pages = {570--586}, Title = {Hamiltonian mechanics and geometry}, Volume = {77}, Year = {1970}}). It's good to keep in mind too that symplectic structures can arise other than via tautological one-forms on cotangent bundles; for example, the Bloch/Riemann sphere has a natural symplectic structure. –  John Sidles Apr 16 '11 at 21:59

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For your kind of reflections I think a useful reference could be the fantastic "The structure of dynamical systems. A symplectic view of physics" of Jean Marie Souriau. In particular take a look at his fourth chapter "Mecanique statistique".

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