Coupling symmetries with Poisson structures

Camille Laurent-Gengoux (1) , Eva Miranda (2)

1 IECL - Institut Élie Cartan de Lorraine
2 CDS - Cadence Design Systems

We study local normal forms for completely integrable sys-tems 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.

Domaines: Mathématiques [math]

Camille Laurent-Gengoux, Eva Miranda. Coupling symmetries with Poisson structures. Acta Mathematica Vietnamica, 2013, 38, pp.21 - 32. ⟨10.1007/s40306-013-0008-1⟩. ⟨hal-01093314⟩