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What is called the symplectic structure of the phase space? How is Liuville theorem connected with smoothness of symplectic structure?

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  • $\begingroup$ I am sure you have read this en.wikipedia.org/wiki/Symplectic_vector_space but I wonder, rather than a math treatment, are you looking (as am I), for an intuitive picture, if one exists. I think you should clarify this. $\endgroup$ – user108787 Oct 25 '16 at 18:25
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  1. Here we will for simplicity only consider manifolds within the category of $C^{\infty}$-smooth manifolds.

  2. Assuming that the phase space is a symplectic manifold $(M,\omega)$, the symplectic structure is provided by a closed non-degenerate 2-form $\omega$.

  3. The symplectic 2-form $\omega$ gives rise to a non-degenerate Poisson bracket $\{\cdot,\cdot\}: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M) $.

  4. Liouville's theorem can be viewed as the fact that Hamiltonian vector fields are divergencefree.

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