Books such as Mathematical methods of classical mechanics describe an approach to classical (Newtonian/Galilean) mechanics where Hamiltonian mechanics turn into a theory of symplectic forms on manifolds.
I'm wondering why it's at all interesting to consider such things in the classical case. In more advanced theories of physics, manifolds become relevant, but in classical mechanics, everything is Euclidean and homogeneous, so this isn't really needed. There, I'm wondering what motivated people to develop the theory of Hamiltonian mechanics through the theory of symplectic manifolds.
One idea: There wasn't really and physical motivation, but someone noticed it, and it was mathematically nice. If this is the case, then how did people notice it?
However, maybe this theory was not developed until general relativity or quantum mechanics came about. That is, manifolds became relevant to more advanced physics, and Arnold thought it reasonable to teach students this manifold formalism in classical mechanics in order to better prepare them for more advanced physics. But, if this is true, no one thought of using symplectic manifolds purely for reasons of classical physics.