Open Problems in Integrable Systems — Poisson geometry and action-angle variables

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(M^N,\Pi)$ be a regular Poisson manifold of rank $2r$. Put $s=N-r$. A regular Liouville integrable system is a map $F=(f_1,\ldots,f_s):M\to\mathbb R^s$ with independent differentials and $\{f_i,f_j\}=0$, whose Hamiltonian leaves are compact $r$-tori. Locally the action-angle theorem gives coordinates $(\theta_1,\ldots,\theta_r,\sigma_1,\ldots,\sigma_s)$ with $\theta_i\in\mathbb R/\mathbb Z$ and \[\Pi=\sum_{i=1}^r\frac{\partial}{\partial\theta_i}\wedge\frac{\partial}{\partial\sigma_i};\] the remaining $\sigma_{r+1},\ldots,\sigma_s$ are Casimirs. Global action-angle coordinates mean a global product-coordinate representation $M\simeq\mathbb T^r\times B$, $B\subseteq\mathbb R^s$ open, with the same Poisson formula and with the level-set foliation of $F$ given by projection to $B$.

Statement recorded in the supplied source.

Determine the obstructions to patching local action-angle coordinates into global action-angle coordinates for regular Poisson integrable systems. Characterize when those obstructions vanish, taking account of the foliation by affine leaves on the moment-map image and the global transverse Casimir coordinates. [S2]

Short English statement

What prevents local Poisson action-angle charts from combining into one global system of angle, action and transverse coordinates?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100603 (AMR-020-0603). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexey Bolsinov, Vladimir S. Matveev, Eva Miranda and Serge Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 6.5 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1804.03737.