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In looking at various notes on Symplectic Geometry (SG) as well as the book by McDuff/Salamon, it is either said or implied that the interested student should have seen the Hamiltonian before. It goes without saying that smooth manifold theory is needed but, I have an entire undergraduate degree in Math and never saw the Hamiltonian or heard it mentioned at all. Also, I've seen many students working in areas in or around SG and I know for a fact that they took no physics at all as undergraduates so, was there some course that I should have taken as a math undergrad that maybe I missed in order to have had this knowledge? What courses/books would have covered this topic aside from those typically given for physics programs?

NoodleNami
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    Hamiltonians are not part of the standard coursework for any pure math curricula here in the US ( and I believe in most of the world). A mathematicians first introduction to them would be in an analytical mechanics course or in a symplectic geometry course. At the beginning of a good course in SG one will cover the case of a symplectic vector space and Hamilton's equations on this space which is generalized in SG and symplectic mathifolds. – J.V.Gaiter Jun 10 '21 at 00:33
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    I think it should be also noted that Hamiltonians in math are not so intimidating!, there are things in symplectic geometry that are considered "intimidating", for example the hard analysis used in the study of J-holomorphic curves. But to define Hamiltonians on symplectic manifold only needs differential forms on smooth manifolds and the definition is a couple of lines. I have been working on Hamiltonian circle actions for years and I know very little about Hamiltonian mechanics, The two subjects are rather disjoint on a technical level (although one motivates the other in some way). – Nick L Jun 17 '21 at 18:44

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