Geometrical phase and inertial regime of the magnetization: Hannay angle and magnetic monopole

It is well known that the Landau-Lifshitz-Gilbert (LLG) equation for a macroscopic magnetic moment find its limit of validity at very short time scales or equivalently at very high frequencies. The reason for this limit of validity is well understood in terms of separation of the characteristic times between slow (the magnetization) and fast (the environment) degrees of freedom, as pointed-out in the stochastic derivation of the LLG equation first proposed by W. F. Brown in 1963. Indeed, the ferromagnetic moment is a slow collective variable, but fast degrees of freedom are also playing a role in the dynamics, and especially the variation of the angular momentum responsible for inertia. In the last couple of years, the generalization of the LLG equation with inertia (ILLG) has been derived by different means (see list of references). The signature of the inertial regime of the magnetization is the nutation that can be measured by resonance experiments (but it has not been observed up to know). We developed an approach in terms of geometrical phase (defining the corresponding Hannay angle, which is the classical analog to the quantum Berry phase : see references), that has recently been used with success to analogous problems. We calculated the Hannay angle for the precession of the magnetization in the case of the inertial effect, and the corresponding magnetic monopole. This analysis allows the slow vs. fast variable expansion to be calculated in the specific case of pure precession.