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Abstract

For the Landau-Lifshitz equation a Lie type linear system representation in the Minkowski space M3 1+ has been derived previously [25]. The internal symmetry group is a proper orthochronous Lorentz group SOo(3, 1), and the numerical method based on the internal symmetry was developed in [29]. This paper derives another four new representations of the Landau-Lifshitz equation. We prove that this equation admits two generators: one conservative and one dissipative, as well as two brackets: Poisson bracket and dissipative bracket. Upon embedding the Landau-Lifshitz equation into a skew-symmetric matrix space, we can develop a double-bracket flow representation. The conserved magnetization magnitude is just the result of the isospectrality for an isospectral flow equation. Finally, on the cotangent bundle of an invariant manifold of the constant magnetization magnitude, we introduce the Lie-Poisson bracket to construct an evolutional differential equations system. The magnetization trajectory traces a coadjoint orbit in the Poisson manifold under a coadjoint action of the rotation group SO(3). The six different representations including the one by Bloch et al. [3] are compared.

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