Magnetization dynamics, gyromagnetic relation, and inertial effects

The gyromagnetic relation-that is, the proportionality between the angular momentum L→ and the magnetization M→-is evidence of the intimate connections between the magnetic properties and the inertial properties of ferromagnetic bodies. However, inertia is absent from the dynamics of a magnetic dipole: The Landau-Lifshitz equation, the Gilbert equation, and the Bloch equation contain only the first derivative of the magnetization with respect to time. In order to investigate this paradoxical situation, the Lagrangian approach, proposed originally by Gilbert, is revisited keeping an arbitrary nonzero inertia tensor. The corresponding physical picture is a generalization to three dimensions of Ampère's hypothesis of molecular currents. A dynamic equation generalized to the inertial regime is obtained. It is shown how both the usual gyromagnetic relation and the well-known Landau-Lifshitz-Gilbert equation are recovered in the kinetic limit, that is, for time scales longer than the relaxation time of the angular momentum.