Update: it may be Spivak's new book Physics for Mathematicians: Mechanics I covers most of the material that this answer had in mind. I've just ordered a copy, and will report on it when it arrives.
Neither Milnor's book nor Guillemin and Pollack's book contains the word "symplectic" ... which is a great pity!
Since the manifolds under study are smooth, they have a cotangent bundle; this bundle is associated to a tautological one-form whose exterior derivative is a (canonical) symplectic form.
If in addition the base manifold has a metric, then a canonical (quadratic) Hamiltonian function too is defined on the tangent bundle.
Hmmm ... what might be the integral curves of this Hamiltonian function? It is instructive for students to discover for themselves that the curves are simply the geodesics of the base manifold.
In this way, students gain an appreciation that all of dynamics (both classical and quantum) is intimately linked to the geometry and topology of smooth manifolds ... this appreciation is good preparation for many careers in math, science, and engineering.