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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 …

There is another way around, starting with Lagrange, but bypassing his variational construction to reach immediately the presymplectic construction. Considering the motion of a point submitted to a fo …

I learned geometric quantization from Jean-Marie Souriau — one of the initiators of the subject with Bertram Kostant — in two texts essentially: "Structure des systèmes dynamiques" (chapter 5). It i …

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 …

For a closed 2-form $\omega$ on a manifold $M$, the integrality of the closed 2-form, that is, $$ \int_\sigma \omega \in a{\bf Z}, \quad \mbox{for all} \quad \sigma \in H_2(M,{\bf Z}), $$ for some rea …

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 …

If you really know nothing about physics I suggest you begin with any text book on physics for undergrad. Easy to read, it will introduce the main usual suspects. After, you'll ask again :) I am not …