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$\def \cG {\cal G}$ $\def \RR {\mathbf R}$ The moment map is one of the most important constructions in symplectic mechanics. There are at least two main reasons for that: The moment map $\mu : M \to …

See Every Symplectic Manifold Is A (Linear) Coadjoint Orbit.

Actually if you allow infinite dimension, every symplectic manifold is a coadjoint orbit of its group of symplectomorphisms. That is even more... how to say? Symplectic :-) If you want a reference the …

The first attempt to "quantize" a dynamical variable $u$ on a symplectic manifold $(M,\omega)$, that is, to associate a linear operator $\hat u$ on the space of square summable smooth function $\psi …

It is the group of periods of a closed 2-form $\omega$ which plays a role on the different quantizations. Every closed 2-form $\omega$ on a manifold $M$ (more generally on a diffeological space) is th …

I don't know if this question : "when a symplectic manifold is isomorphic to a cotangent bundle" has a complete and simple answer in the literature, in the way you want, but this is some comments that …

Actually the first case in history of a symplectic manifold wasn't a cotangent space. It was the space of Keplerian motions of a planet, represented locally by its Keplerian elements. The Lagrange sym …

The word "sum-plectic" as a greek translation of "com-plexus" was needed also to differentiate the study of "complex geometry" (complex numbers etc) from the study of "complexes de droites" (the geome …

The symplectic area contained in a closed curved, that is the boundary of map of a disc, is the "action along the curve". $$ \int_\sigma \omega = \int_\sigma d\lambda = \int_{\partial \sigma} \lambda …

Just a reformulation of Dick's question: How to describe the orbits of the identity component of the group of diffeomorphisms of a compact manifold, acting naturally on the subspace of cohomology clas …