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I'm currently attending lectures on Hamiltonian dynamics from a very mathematical viewpoint and I'm having trouble understanding two facts:

1. An inner product defined in every tangent space and a symplectic form both establish a natural isomorphism between tangent and contanget spaces. My question is: what is the nature of this isomorphism?

2. The relationship between the invariance of the symplectic form under a hamiltonian flux and the Liouville theorem.

Can someone help me out? Thanks a lot.

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# Differential geometry and hamiltonian dynamics

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