Coupling symmetries with Poisson structures
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COUPLING SYMMETRIES WITH POISSON STRUCTURES Camille Laurent-Gengoux · Eva Miranda
Received: 30 October 2012 / Published online: 20 February 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein’s splitting theorem for the integrable system is also studied giving some examples in which such a splitting does not exist, i.e. when the integrable system is not, locally, a product of an integrable system on the symplectic leaf and an integrable system on a transversal. The problem of splitting for integrable systems with additional symmetries is also considered. Keywords Group actions · Poisson manifolds · Integrable systems · Splitting theorem · Equivariant Carathéodory–Jacobi–Lie theorem · Coupling Mathematics Subject Classification 53D17 · 37J35 1 Introduction Integrable Hamiltonian systems have been widely studied in the context of symplectic manifolds. The existence of action-angle coordinates (semilocal and global) under some additional conditions have become a main goal in this area. For an extensive study of the existence of action-angle coordinates in symplectic manifolds, we refer to Arnold [1] and Duistermaat [7]. Several extensions of the concept of complete integrability have been done in the symplectic context. We refer to Dazord and Delzant [6] and Nekhoroshev [18] for more details. However, the most natural framework for several dynamical systems is the framework of Poisson manifolds. For instance, the Gelfand–Cetlin system, whose underlying Poisson structure is the dual of a Lie algebra [9], arises from a non-symplectic Poisson structure. C. Laurent-Gengoux Institut Elie Cartan de Lorraine, Université de Lorraine, UMR 7122, site de Metz, Metz, France
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E. Miranda ( ) Departament de Matemàtica Aplicada I, EPSEB, Universitat Politècnica de Catalunya, Barcelona, Spain e-mail: [email protected]
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C. LAURENT-GENGOUX, E. MIRANDA
In [11] an action-angle theorem for completely integrable systems is proved within the Poisson context. Indeed, [11] proves we proved an action-angle theorem for noncommutative integrable systems, non-commutative integrable systems being systems for which there are more constants of motion than required in order to prove Liouville integrability. An important point is that a non-commutative integrable systems may be regular even at singular points of the Poisson structure. The construction of these action-angle coordinates go through the construction of a natural Hamiltonian Tn action tangent to the fibers of the moment map. So, as happens in the symplectic case, we can naturally let a compact Abelian group act in the integrable system. In [11], however, we did not address what happens when there are additional symmetries encoded in actions of compact Lie groups. As
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