Action-angle variables and a KAM theorem for b-Poisson manifolds

Abstract

Abstract. In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds.

Herman's Approach to Quasi-Periodic Perturbations in the Reversible KAM Context 2

“…In KAM theory, there are also results for some "exotic" classes of dynamical systems, for instance, for weakly reversible systems (where the reversing diffeomorphism of the phase space is not assumed to be an involution) [4,30], for locally Hamiltonian vector fields V (defined by the condition that the 1-form i V ω 2 is closed but not necessarily exact, so that the Hamilton function can be multi-valued) [25,26,39], for conformally Hamiltonian vector fields V (defined by the identity d(i V ω 2 ) ≡ ηω 2 with constant η = 0) [12], for generalized Hamiltonian (or Poisson-Hamilton) systems defined on Poisson manifolds [23,24] (see [11,39] for more references), for presymplectic systems (defined in another way on Poisson manifolds where the role of the symplectic form ω 2 is played by a closed degenerate 2-form with constant rank) [1], for b-Hamiltonian vector fields on the so-called b-Poisson (or log-symplectic) manifolds [18], or for equivariant vector fields [45]. Here i V ω 2 is the interior product, or the contraction, of ω 2 with V .…”

Noncommutative integrable systems on b-symplectic manifolds

“…In Theorem 8 we recalled the action-angle coordinate theorem for noncommutative integrable systems on Poisson manifolds, which was proved in [LMV11]. For b-symplectic manifolds and the commutative b-integrable systems defined there, we have proved an action-angle coordinate theorem [KMS15], which is similar to the symplectic case in the sense that even on the hypersurface Z where the Poisson structure drops rank there is a foliation by Liouville tori (with dimension equal to the rank of the system) and a semilocal neighborhood with "action-angle coordinates" around them. The main goal of this paper is to establish a similar result in the non-commutative case, proving the existence of r-dimensional invariant tori on Z and action-angle coordinates around them.…”

“…Proof. (of Theorem 21) In the first step we perform "uniformization of periods" similar to [LMV11] and [KMS15]. The joint flow of the vector fields X f 1 , .…”

“…A non-commutative integrable system on a symplectic manifold with boundary yields a non-commutative system on a class of Poisson manifolds called b-Poisson manifolds, as long as the asymptotics of the system satisfy certain conditions near the boundary. b-Poisson manifolds constitute a class of Poisson manifolds which recently has been studied extensively(see for instance [GMP11], [GMP12], [GMPS13] and [GLPR14]) and integrable systems on such manifolds have been the object of study in [KMS15], [KM16] and [DKM15].…”

“…a torus action attached to a non-commutative integrable system and extend the Darboux-Carathéodory coordinates in a neighbourhood of the invariant subset. The upshot is the use of b-symplectic techniques and toric actions on these manifolds [GMPS13], [GMPS2] as we did in [KMS15] and [KM16] for commutative systems on b-manifolds. The proof is a combination of the theory of torus actions with a refinement of the commutative proof by considering Cas-basic forms and working with them as a subcomplex of the b-De Rham complex.…”