  We discuss the mean-field approximation for a trapped weakly-interacting
Bose-Einstein condensate (BEC) and its connection with the exact many-body
problem by deriving the Gross-Pitaevskii action of the condensate. The
mechanics of the BEC in a harmonic potential is studied by using a variational
approach with time-dependent Gaussian trial wave-functions. In particular, we
analyze the static configurations, the stability and the collective
oscillations for both ground-state and vortices.
