A geometrical approach to the dynamics of spinor condensates I: Hydrodynamics

In this work, we derive the equations of motion governing the hydrodynamics of spin-F spinor condensates. We pursue a description based on standard physical variables (total density and superfluid velocity), alongside 2F `spin-nodes': unit vectors that describe the spin F state, and also exhibit the point-group symmetry of a spinor condensate's mean-field ground state. The hydrodynamic equations of motion consist of a mass continuity equation, 2F Landau-Lifshitz equations for the spin-nodes, and a modified Euler equation. In particular, we provide a generalization of the Mermin-Ho relation to spin one, and find an analytic solution for the skyrmion texture in the incompressible regime of a spin-half condensate. These results exhibit a beautiful geometrical structure that underlies the dynamics of spinor condensates.