A geometrical approach to the dynamics of spinor condensates II: Collective modes

In this paper we study the linearized dynamics of spinor condensates using the spin-node formalism developed in arXiv:0812.3403. We provide a general method to linearize the equations of motion based on the symmetry of the mean-field ground state using the local stereographic projection of the spin nodes. We also provide a simple construction to extract the collective modes from symmetry considerations alone akin to the analysis of vibrational excitations of polyatomic molecules. Finally, we will present a mapping between the spin-wave modes, and the wave functions of electrons in atoms, where the spherical symmetry is degraded by a crystal field.