(In classic mechanics, particularly, there is an unusual symmetry to position and velocity, such that the laws of mechanics look roughly a rotation x -> v, v -> -x, which is why this works so well.)
In fact, every symplectic manifold locally looks like this (Darboux's theorem).
There is a barrier of communication that (most) mathematicians don't want to take any time to overcome. Often, the opening is enticing, then it's straight into lemmas, formal language, and citing of famous results by name. Their slight inclination to explain it to others dissipates in the first paragraph, then it's off to the races to impress their peers.
This is easy to see on Wikipedia - most mathematics articles are utterly useless to any non-mathematician who wants to get a general appreciation of an approach to see if it could shed some light on their problem.
In almost all cases, it is better to read domain generalists coming the other way into the higher mathematics.
Of course, one shining exception is John Baez:
For certain systems (e.g., particles in euclidean space interacting via conservative forces dependent only on position), the momenta coincide with velocity, but mathematically they are different objects: velocities are vectors, and momenta are covectors. They transform differently under coordinate transformations.
I wish I knew a concise self-contained exposition off the top of my head, but I don't. You can probably find something on-line. I think there's likely a discussion in Structure and Interpretation of Classical Mechanics (Sussman & Wisdom):
Try looking up "cotangent bundles" (mathematical name for the type of spaces appropriate for Hamiltonian mechanics, formulated in terms of generlized coordinates and momenta) and "tangent bundles" (more appropriate for Lagrangian mechanics, formulated in terms of generalized coordinates and velocities).
That's the way in which terms like "____ geometry" are defined and understood.
And I thought category theory was "math for its own sake"!
Being algebraic symplectic is a much stronger condition than analytic symplectic, but is still interesting enough (and, for geometry related to linear algebra problems, as is often relevant in CS, is not a very strong restriction at all.)
Even as a hydrodynamics software guy, I found the computer graphics research community to be the easiest entry-point for, especially, the topology and modern differential geometry. It's especially nice when they do a simulation paper with a high end geometric/analytic approach.
This might be a good place to go in order to have a start at, say, Arnold's ``topological methods in hydrodynamics'' or anything TQFT-esque.