The Hamiltonian Tube Of A Cotangent-Lifted Action

TL;DR

This paper derives an explicit Hamiltonian tube model for cotangent-lifted group actions on symplectic manifolds, extending previous work and providing practical coordinates for specific groups like SO(3) and SL(2).

Contribution

It introduces an explicit Hamiltonian tube construction for cotangent-lifted actions that respects fibered structures, generalizing prior results and enabling practical computations.

Findings

Provides explicit symplectic coordinates for SO(3) and SL(2) actions.

Generalizes previous results to a broader class of group actions.

Offers an explicit solution method via differential equations.

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.