Reddit reviews Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics Vol. 60)
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A mathematican who wrote fairly extensively about topological/geometric methods in classical mechanics was Vladimir Arnold. His book Mathematical Methods of Classical Mechanics is all about formulating classical mechanics as problems on sympletic manifolds and other similar ideas. It's widely regarded as an absolute classic as well. I've had a read of the first few bits and it does look very good. His books on Dynamical Systems are excellent as well. Indeed, there are a lot of links with Dynmanical Systems within classical mechanics: as an example, a Hamiltonian and Hamilton's equations of motion form a DS in the generalised coordinates.