Time-dependent variational approach to Bose-Einstein condensation

We discuss the mean-held approximation far a trapped weakly-interacting Bos e-Einstein condensate (BEC) and its connection with the exact many-body pro blem by deriving the Gross-Pitaevskii action of the condensate. The mechani cs of the BEC in a harmonic potential is studied by using a variational app roach with time-dependent Gaussian trial wave-functions. In particular, we analyze the static configurations, the stability and the collective oscilla tions for both ground-state and vortices.