Journal of Symplectic Geometry

Volume 15 (2017)

Number 3

The Hamiltonian tube of a cotangent-lifted action

Pages: 803 – 852

DOI: http://dx.doi.org/10.4310/JSG.2017.v15.n3.a7

Authors

Miguel Rodríguez-Olmos (Department of Applied Mathematics, Technical University of Catalonia, Barcelona, Spain)

Miguel Teixidó-Román (Department of Applied Mathematics, Technical University of Catalonia, Barcelona, Spain)

Abstract

The Marle–Guillemin–Sternberg (MGS) form is local model for a neighborhood of an orbit of a Hamiltonian Lie group action on a symplectic manifold. One of the main features of the MGS form is that it puts simultaneously in normal form the existing symplectic structure and momentum map. The main drawback of the MGS form is that it does not have an explicit expression. We will obtain a MGS form for cotangent-lifted actions on cotangent bundles that, in addition to its defining features, respects the additional fibered structure present. This model generalizes previous results obtained by T. Schmah for orbits with fully-isotropic momentum. In addition, our construction is explicit up to the integration of a differential equation on $G$. This equation can be easily solved for the groups $SO(3)$ or $SL(2)$, thus giving explicit symplectic coordinates for arbitrary canonical actions of these groups on any cotangent bundle.

2010 Mathematics Subject Classification

37J15, 53D20, 70H33

Full Text (PDF format)

Paper received on 17 November 2014.