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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 …

You want to read Chapter IV "Statistical Mechanics" in Structure of Dynamical Systems (1970 French original available here) by J.-M. Souriau, one of the pioneers of symplectic mechanics. Given a symp …

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 think you have in mind the integrability (a.k.a. "flatness") problem for $G$-structures. Beyond the cases mentioned at that link (symplectic, Kähler, and complex structures, corresponding to $G=Sp(n …

The difference $\frac12\dim(X)-\dim(T)$ is known as the complexity of the $T$-space (assumed effective), so that's the keyword you want to use. Such results as I've heard of are mainly for complexity …

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.

$ \def\C{{\mathbf C}} \def\d{\delta} \def\e{{\mathbf e}} \def\r{{\mathbf r}} \def\u{{\mathbf u}} \def\x{{\mathbf x}} \def\y{{\mathbf y}} \def\z{{\mathbf z}} \def\<{\langle} \def\>{\rangle} $ If I unde …

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 …

Michael F. Atiyah, Convexity and commuting Hamiltonians (1982), Lemma 2.3. Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology (2nd ed., 1998), Lemmas 5.51 and 5.54. Michèle Audin, …

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 …

That reminds me of a paper that I believe should answer your question: $\mathbf R^{2n}$ is a universal symplectic manifold for reduction (available here): The authors show that if a manifold $Q$ i …

There is a large literature on this. I would recommend the papers MR0379752 Vergne, Michèle La structure de Poisson sur l'algèbre symétrique d'une algèbre de Lie nilpotente. Bull. Soc. Math. France …

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 $X$: …

You already know that the pair $(M,L)$ of a symplectic manifold and a Lagrangian submanifold is locally isomorphic to $(T^*L, L)$. This is the beginning of Corollary 6.2 of Weinstein, who continues: " …

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).