Abstract mechanical connection and abelian reconstruction for almost Kähler manifolds

Abstract

When the phase space $P$ of a Hamiltonian $G$-system $(P,\omega,G,J,H)$ has an almost Kähler structure a preferred connection, called abstract mechanical connection, can be defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for the corresponding connection one-form $\mathcal{A}$ are derived in terms of the momentum map, symplectic and complex structures. Such connection can play the role of the reconstruction connection (due to the work of A. Blaom), thus significantly simplifying computations of the corresponding dynamic and geometric phases for an Abelian group $G$. These ideas are illustrated using the example of the resonant three-wave interaction. Explicit formulas for the connection one-form and the phases are given together with some new results on the symmetry reduction of the Poisson structure.