Recall
We left off with:
- momentum maps;
- coadjoint orbits.
Recall - Momentum map
Let $\liealg{g}$ be a Lie algebra, and $\liealg{h}$ a Lie algebra of smooth hamiltonians on $(M, \pb{}{})$. An homomorphism \begin{align*} \Psi &: \liealg{g} \to \liealg{h} && \xi \mapsto h_{\xi} \\ \text{induces}\\ \psi &: M \to \liealg{g}^* && \ip{\psi(x)}{\xi} = h_{\xi}(x). \end{align*} We call $\psi$ a momentum map.
If $\psi$ is a momentum map, then it is a Poisson map.
Convexity
Assumption: $(M^{2n}, \omega)$ is a compact symplectic manifold.
(Atiyah-Guillemin-Sternberg) Let $T=\Torus{n}$ have an effective Hamiltonian action on $M$ with momentum map $\psi : M \to \liealg{t}^*$. Then $\psi(M)$ is a compact, convex subset of $\liealg{t}^*$.
(Kirwan) If $G$ is a compact Lie group with an effective Hamiltonian action on $M$ with momentum map $\psi$, and $T \lhd G$ is a maximal torus, then $\psi(M) \cap \liealg{t}^*_+$ is a compact, convex polytope.
(Delzant) There is a distinguished family $D$ of convex polytopes which are the image of a torus momentum map. Given such a polytope, one can reconstruct $(M, \omega)$, the $T$-action and $\psi$.
Let $x \in M$ have the stabilizer subgroup $T_x \lhd T$ of dimension $k \leq n$. There are symplectic coordinates $(x_1, y_1, \ldots, x_k, y_k, \ldots, x_n, y_n)$ on a neighbourhood of $x$ and a basis $t_1, \ldots, t_n$ of $\liealg{t}^*$ such that \begin{align*} \psi(p)-\psi(x) &= \sum_{a=1}^k \frac{1}{2} (x_a^2 + y_a^2) t_a + \sum_{b=k+1}^n x_b t_b. \end{align*}
Example
Let $(M, \omega)$ be $\sphere{2}$ with its unit area form. Let $T = \Torus{1}$ act by rotation about the vertical $z$-axis. The momentum map is $\psi = z$.
 |
| The momentum map. |
Let $M = \sphere{2} \times \sphere{2}$ with $\omega$ the sum of the unit area forms, and $T = \Torus{2}$ acting by rotating about each vertical axis.
 |
| The image of the momentum map. |
Liouville-Arnold Theorem
Let $(M^{2n}, \omega)$ be a symplectic manifold, $f_1, \ldots, f_n$ be Poisson commuting, independent functions.
Let $T$ be a connected, compact component of a regular level set $F_c = \set{f_1=c_1, \ldots, f_n=c_n}$. Then $T \cong \Torus{n}$ and there is a neighbourhood $U \supset T$ and coordinates $(\theta, I) : U \to \Torus{n} \times \R^n$ such that \begin{align*} 1. &&& \omega|U = \sum_{i=1}^n \D{\theta_i} \wedge \D{I_i}, \\ 2. &&& f_i|U = F_i(I), \\ 3. &&& X_{f_i} = \sum_{j=1}^n \didi{F_i}{I_j} \didi{}{\theta_j}. \end{align*}
Simple Pendulum
Take \[ H = \frac{1}{2} p^2 + \cos(x). \]
Pendulum.
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