The dynamical equation of the magnetization has been reconsidered with enlarging the phase space of the ferromagnetic degrees of freedom to the angular momentum. The generalized LandauLifshitz-Gilbert equation that includes inertial terms, and the corresponding Fokker-Planck equation, are then derived in the framework of mesoscopic non-equilibrium thermodynamics theory. A typical relaxation time τ is introduced describing the relaxation of the magnetization acceleration from the inertial regime towards the precession regime defined by a constant Larmor frequency. For time scales larger than τ , the usual Gilbert equation is recovered. For time scales below τ , nutation and related inertial effects are predicted. The inertial regime offers new opportunities for the implementation of ultrafast magnetization switching in magnetic devices. The range of validity of the Landau-Lifshitz-Gilbert (LLG) equation was established one decade later by W. F. Brown, with a description of a magnetic moment coupled to a heat bath ("thermal fluctuations of a singledomain particle ", 1963 [3]). The magnetic moment is treated as a Brownian particle described by the slow degrees of freedom (10 −9 s), the angles {θ, φ}. The remaining degrees of freedom of the system relax in a much shorter time scale (< 10 −12 s). The time scale separation between the rapidly relaxing environmental degrees of freedom and the slow magnetic degrees of freedom allows the coupling between the magnetization and the environment to be reduced to a single phenomenological damping parameter η, whatever the complexity of the microscopic relaxation involved [4,5].However, important experimental advances towards very short time-resolved response of the magnetization (sub-picoseconds resolution, i.e. below the limit proposed by Brown) have been reported in the last decade [6]. In parallel, industrial needs for very fast memory storage technologies are approaching the limits imposed by the precessional switching [7]. In these experiments and in the corresponding applications, time scale separation between the conserved degrees of freedom {θ, φ} and the other degrees of freedom, assumed by Brown [3], finds its limit.The purpose of this Letter is to investigate the dynamics of the magnetization beyond this limit by extending the phase space to additional degrees of freedom expected to be also out-of-equilibrium at short time scales [5,9]. According to the well-known gyromagnetic relation [10], the next relevant degree of freedom of the ferromagnetic system (beyond the coordinates of position; i.e. the angles {θ, φ}) is the angular momentum L. As will be shown below, the consequence of considering also the conservation of the angular momentum is that inertial terms, i.e. acceleration terms proportional to d 2 M/dt 2 , appear in the equation of motion. The existence of inertial terms in the dynamics of the magnetization opens the way to deterministic ultrafast magnetization switching strategies, beyond the limitations of the precessional regime [11]. We assume however ...

The gyromagnetic relation -i.e. the proportionality between the angular momentum L (defined by an inertial tensor) 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 T. H. Gilbert) is revisited keeping an arbitrary nonzero inertial tensor. 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 at the kinetic limit, i.e. for time scales above the relaxation time τ of the angular momentum.PACS numbers: