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Could I ask a conceptual question?

If you have a symplectic manifold ($M$, $\omega$) and a real valued function $f : M \to \mathbb{R}$, you can define a hamiltonian vector field $X$ corresponding to $f$ by the following equation;

$$ i_{X}(\omega) = -df $$

Then could you give me a topological intuition about the hamiltonian vector field?

Thanks.

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    This isn't about topology but about classical mechanics. Think of $f$ as the Hamitonian of a physical system. Then the corresponding Hamiltonian vector field describes the time evolution of the physical system. (That's what the word "Hamiltonian" is doing here.) – Qiaochu Yuan Jan 10 '16 at 20:31

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