What is the symplectic structure of phase space?

Question

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

Answer 1

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.