My training is in mathematics. But during my free time I also try to understand fundamental physics. Recently I came across a material which said that the geometry of classical mechanics is symplectic. I'm not sure of the meaning. It was relating to the Hamiltonian which I'm also not familier. All this while I though that the geometry for Newtonian mechanics is Euclidean, the geometry for quantum mechanic is Hilbert space, the geometry for special relativity is Minkowski and for general relativity .... (don't know which metric space). To study mechanic via Hamitonian do we really need to understand symplectic geometry? By the way can't we just study mechanics as Issac Newton did without going through the Lagragian or Hamitonian? What good are Lagragian and Hamitonian?