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.
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Luca Salasnich (2000) studied this question.
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