Hello everybody! 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.