Generalized Landau–Lifshitz–Gilbert equation for uniformly magnetized bodies

We consider generalized Landau–Lifshitz–Gilbert (LLG) deterministic dynamics in uniformly magnetized bodies. The dynamics take place on the unit sphere ΣΣ, and are characterized by a vector field vv tangential to ΣΣ. By using Helmholtz decomposition on ΣΣ, it is proven that vv is uniquely defined by two potentials χχ and ψψ. Potential χχ can be identified with the free energy of the system, while ψψ describes non-conservative interactions of the system with the environment. The presence of ψψ modifies the usual energy balance of LLG dynamics. Instead of purely relaxation dynamics we may have steady injection of energy through non-conservative interactions. The implications of the new form of the energy balance are discussed in detail.