On the dynamics of spin systems in the Landau-Lifshitz theory

In the framework of the Landau-Lifshitz equations without any dissipation (an approximation which may also be helpful for finite but weak Gilbert damping), with all interactions included, for general ground states, geometries and domain structures, and many types of effective fields the dynamics of the spin precession around this ground state is considered. At first the precession is treated in the linear approximation. For the eigenmodes of the precession one has a `rule of geometric mean' for the eigenfrequencies. For the eigenmodes pseudo-orthogonality relations are obtained, which reflect the gyrotropic and elliptic character of the spin precession and differ from those known from the Schrodinger equation. Moreover, pseudo-orthogonality relations are valid 'everywhere' (e.g., both in the outer region and in the core region of a magnetic vortex). Then also some aspects of the nonlinear mode coupling with emphasis on `confluence' and `splitting' processes of elementary magnetic spin-wave excitations are considered. At the same time these processes contribute to the Gilbert damping. There are thus essential differences to quantum mechanics, although at a first glance one discovers many similarities. From the results one may also get insights of why these systems are so complex that (although the essential quantities depend only on the local values of the partially long-ranged effective magnetic fields) practically only detailed experiments and computer simulations make sense.