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.
Best Answer
The symplectic framework paves the way to classical mechanics on a Poisson manifold (i.e., with a Poisson bracket). See the book Mechanics and Symmetry by Marsden & Ratiu. https://cdn.preterhuman.net/texts/science_and_technology/physics/Introduction%20to%20Mechanics%20and%20Symmetry.pdf
The Poisson point of view unifies many classical systems that cannot be seen otherwise in a uniform way. For example, many fluid flow equations have a Hamiltonian description only when viewed on an infinite-dimensional Poisson manifold. See the survey article
PJ Morrison, Hamiltonian description of the ideal fluid, Reviews of Modern Physics 70 (1998), 467. http://www.ph.utexas.edu/~morrison/98RMP_morrison.pdf
and the book
Beris and Edwards, Thermodynamics of flowing systems.