Spinor formulation of the Landau-Lifshitz-Gilbert equation with geometric algebra

TL;DR

This paper reformulates the Landau-Lifshitz-Gilbert equation using geometric algebra, providing explicit solutions and generalizations, which enhances understanding of magnetization dynamics through a spinor and geometric perspective.

Contribution

It introduces a novel spinor formulation of the Landau-Lifshitz-Gilbert equation using geometric algebra, enabling explicit solutions and extensions to include damping effects.

Findings

Explicit solutions for the undamped case with geometric interpretation

Generalized approach to include damping effects

Discussion on the axial property of magnetization vector

Abstract

The Landau-Lifshitz-Gilbert equation for magnetization dynamics is recast into spinor form using the real-valued Clifford algebra (geometric algebra) of three-space. We show how the undamped case can be explicitly solved to obtain component-wise solutions, with clear geometrical meaning. Generalizations of the approach to include damping are formulated. The implications of the axial property of the magnetization vector are briefly discussed.