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This is Homework 22 in Ana Cannas da Silva's Lectures on Symplectic Geometry (2001). It's also described in her articles Symplectic toric manifolds (2003, p.124) and Symplectic Geometry (2006, p.172). …

(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 answer is yes, due to formal properties of moment maps (that depend on almost no details of your situation). Namely, suppose $K$ acts transitively on any symplectic manifold $(X,\omega_X)$, with e …

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 …

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, …

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: " …

$ \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 …

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 …

This is not so much an answer as a suggestion to change the question. When $(P,\Omega)$ is prequantizable, i.e. there exists over $P$ a hermitian line bundle with connection $(L,\nabla)$ having curva …

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 …

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 …

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 …

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 …