When the phase space
P of a Hamiltonian
G-system
(P,, 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
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.