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 i s a generalization of the work of Perez-Garcia and collaborators. Taking th e limit of small oscillations around the equilibrium configuration, we calc ulate the excitation energies of the low-lying even-parity states of the co ndensate. In this limit, the diagonal and non-diagonal quadrupole degrees o f freedom are decoupled. For the diagonal elements, our results agree exact ly with those of Perez-Garcia and collaborators. For the non-diagonal eleme nts 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 result s. For negative scattering lengths we show that the collapsing state is the breathing mode, independent of the trap geometry.