A Mathematical Revision of the Landau-Lifshitz Equation

For the Landau-Lifshitz equation a Lie type linear system representation in the Minkowski space M(superscript 3+1) has been derived previously [25]. The internal symmetry group is a proper orthochronous Lorentz group SO0(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 iso-spectral 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.

並列關鍵字

magnetization ； Landau-Lifshitz equation ； Lie-Poisson system ； two-generator ； double-bracket flow ； isospectral flow ； coadjoint orbit ； rotation group ； Lorentz group

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