Monopole and quadrupole shape oscillations of a Bose-Einstein condensate

We use a variational ansatz to derive a closed dynamics involving only the monopole and quadrupole shape degrees of freedom of the condensate, which is a generalization of the work of Perez-Garcia and collaborators. Taking the limit of small oscillations around the equilibrium configuration, we calculate the excitation energies of the low-lying even-parity states of the condensate. In this limit, the diagonal and non-diagonal quadrupole degrees of freedom are decoupled. For the diagonal elements, our results agree exactly with those of Perez-Garcia and collaborators. For the non-diagonal elements we obtain analytical expressions for the excitation energies which are valid for any trap geometry and any particle number and that agree, in the Thomas-Fermi limit and for axially symmetric traps, with Stringari's results. For negative scattering lengths we show that the collapsing state is the breathing mode, independent of the trap geometry.