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(Comment $\to$ answer as requested.) Let $G=\mathbf R$ act on the 2-torus $Z=\mathrm U(1)\times\mathrm U(1)$ by $g(z_1,z_2)=(e^{ig}z_1, e^{i\pi g}z_2)$. Lift the action to $T^*Z$ and use any $G$-invar …

The direct connection between Poisson bracket and non-commutativity is pretty clear, at least if you agree that the (later introduced) Lie bracket $[X,Y]$ of vector fields measures the non-commutativi …

For the desired symplectic emphasis, I’d warmly recommend these (of which Panagiotis’ ref. cites #2–5): Atiyah, Michael F.; Bott, Raoul, The moment map and equivariant cohomology, Topology 23, 1-28 …

There are: Bertrand’s theorem, which says that the isotropic oscillator and Kepler potentials are the only analytic radial ones all of whose nonrectilinear bounded orbits are closed. (Recommendation …

Flag manifolds $G/C(S)$ even exhaust homogeneous symplectic manifolds of $G$: Borel-Weil (1954, Thm 1). Also restated with fewer details in (1954, Thm 1).

The reduction you require is a (very) special case of Marsden-Weinstein (1974). Your one constant of the motion, say $\psi$, is the moment map of an action of the additive group $G=\mathbf R$ on the s …

Let your manifold be $X=G/H$. First of all, since it is simply connected, we can write it as $K/U$ where $K$ and $U=K\cap H$ are compact in $G$ (Montgomery’s theorem, 1950). Next, since $K/U$ is homog …

There is always an equivariant local symplectomorphism with $T_pM$ with its 2-form and linear isotropy action, by the Moser-Weinstein proof. But that constant 2-form then has more possible “equivarian …

For $G$ semisimple as you assume, the Killing form gives a $G$-equivariant map $\mathfrak g\to\mathfrak g^*$, which identifies adjoint to coadjoint orbits. That is all you need.

Nice examples are worked out in Audin (2004, §VI.3.d), Arvanitoyeorgos (1999)(pdf), McDuff-Salamon (1998, §5.6). It’s not true that $\omega$ is rarely explicit: e.g. on all coadjoint orbits (including …

For question 1 (references): this is covered in Besse, Einstein Manifolds, pp. 230-233 for even and then odd $n$. For question 2 (nice formula), I think you want to do something like what works for a …

No. In $\mathbf C^2$ with standard 2-form and complex structure, the real span of $U=\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)$ and $V=\left(\begin{smallmatrix}i\\1\end{smallmatrix}\right) …

I don’t think your characterization of the vector field is quite correct; I would recommend Cartan (1922, §§184-185), Godbillon (1969, Proposition XI.3.7), or Souriau (1970, Theorem 7.29). Or this ans …

The preimage of a coadjoint orbit under a moment map is, under a mild transversality assumption, a coisotropic submanifold; so its null foliation $\smash{\ker(\omega_{|\Phi^{-1}(\mathcal O)})}$ is not …

1) is correct: by saying “hamiltonian diffeomorphism of” you imply that the torus has a symplectic structure $\omega$ and the diffeo is (something like) time 1 flow of a hamiltonian vector field $X$: …