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My question essentially breaks down to How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...

I'm trying to learn about moment maps in symplectic topology (suppose our Lie group is $G$ with Lie algebra $\mathfrak g$, acting on the symplectic manifold $(M,\omega)$ by symplectomorphisms). I'm ...