Journal of Symplectic Geometry

Volume 16 (2018)

Number 3

Global action-angle variables for non-commutative integrable systems

Pages: 645 – 699



Rui Loja Fernandes (Department of Mathematics, University of Illinois, Urbana-Champaign, Il., U.S.A.)

Camille Laurent-Gengoux (Institut Elie Cartan de Lorraine, Université de Lorraine, Metz, France)

Pol Vanhaecke (Laboratoire de Mathématiques et Applications, Université de Poitiers, Futuroscope Chasseneuil, France)


In this paper we analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (regular NCI systems) on Poisson manifolds. In contrast with local action-angle variables, which always exist in the neighborhood of a compact connected component of the regular fibers of the momentum map, global action-angle variables rarely exist. The fact that there are obstructions to the existence of global action-angle variables was first observed and analyzed by Duistermaat in the case of Liouville integrable systems on symplectic manifolds and later by Dazord–Delzant in the case of non-commutative integrable systems on symplectic manifolds. In our more general case where phase space is an arbitrary Poisson manifold, there are more obstructions. Our approach makes use of a few new features which we introduce: the action bundle and the action lattice bundle of the NCI system (these bundles are canonically defined) and three foliations (the action, angle and transverse foliation), whose existence is also subject to obstructions, often of a cohomological nature.

Received 28 June 2015

Accepted 8 December 2017

Published 26 November 2018