# What is the symplectic structure of phase space?

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

• 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. – user108787 Oct 25 '16 at 18:25

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 $\{\cdot,\cdot\}: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$.