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We prove the action-angle theorem in the general, and most natural, contextof integrable systems on Poisson manifolds, thereby generalizing the classicalproof, which is given in the context of symplectic manifolds. The topologicalpart of the proof parallels the proof of the symplectic case, but the rest ofthe proof is quite different, since we are naturally led to using the calculusof polyvector fields, rather than differential forms; in particular, we use inthe end a Poisson version of the classical Caratheodory-Jacobi-Lie theorem,which we also prove. At the end of the article, we generalize the action-angletheorem to the setting of non-commutative integrable systems on Poissonmanifolds.

Action-angle coordinates for integrable systems on Poisson manifolds