/sci/ - Science & Math

Could someone please explain to me how the symplectic matrix in Hamilton’s equations gives rise to a differential 2-form? I’ve heard that this is apparently a very natural thing, but I cannot quite wrap my head around what makes this so remarkable. I’d really favor a geometric explanation, if possible.

Furthermore, could somebody explain how the differential geometry in Hamiltonian mechanics is at all useful and relevant to actually *solving* Hamilton’s equations? To me, all the topological theorems involving the Hamiltonian seem interesting, but totally useless.