On the Poisson structure of the time-dependent mean-field equations for systems of bosons out of equilibrium

We analyze the Poisson structure of the time-dependent meanfield equations for condensed bosons and construct the Lie-Poisson bracket associated to these equations. The latter follow from the time-dependent variational principle of Balian and Veneroni when a Gaussian Ansatz is chosen for the density operator. We perform a stability analysis of both the full and the linearized equations. We also search for canonically conjugate variables. In certain cases, the evolution equations can indeed be cast in a Hamiltonian form. (C) 1998 Academic Press.