Magnetization Dynamics and Geometry: Coupled Möbius Transformations

We present a geometric description of the collective dynamics in the system of spins with the Hamiltonian consisting of exchange interactions and an external magnetic field. This Hamiltonian yields the classical Landau-Lifshitz equation for each individual spin. However, magnetic fields of all spins evolve in a complicated way, due to interactions between particles. Passing to stereographic coordinates, evolution of spins and their magnetic fields can be represented by actions of one-parameter families of Möbius transformations. On the other side, the corresponding evolution on the group of Möbius transformations is determined by the configuration of all local fields in the system. On the whole, local magnetic fields are identified with Möbius transformations acting on them.