This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities are a variable and its derivative.
(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 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.)