  We present a simple derivation of the matrix Riccati equations governing the
reduced dynamics as one descends from the group U(N) describing the
Schroedinger evolution of an N-level quantum system to the various coset
spaces, Grassmanian manifolds, associated with it. The special case pertaining
to the geometric phase in N-level systems is described in detail. Further, we
show how the matrix Riccati equation thus obtained can be reformulated as an
equation describing Hamiltonian evolution in a classical phase space and
establish correspondences between the two descriptions.
