  When the phase space P of a Hamiltonian G-system (P, \omega, G, J, H) has an
almost Kahler 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 A are derived in terms of the momentum map,
symplectic and complex structures. Such connection can play the role of the
reconstruction connection (see A. Blaom, Reconstruction phases via Poisson
reduction.
  Diff. Geom. Appl., 12:231--252, 2000.), 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.
