# Force, momentum and topology of a moving magnetic domain

## Abstract

The following general relations involving force, momentum and topological winding number of a translating magnetic domain are derived from the Landau-Lifshifz equation in a context appropriate to bubbles: The gyrotropic force tending to deflect a steadily moving domain is proportional to a mean winding number linear in Bloch-point coordinates. The time derivative of the canonical momentum for a domain of integer winding number is equal to the total force, which must include gyrotropic and dissipative terms. A new contour integral expresses the momentum in the limit of vanishing wall thickness. Approximate equations of quasi-steady domain motion are cast into a form resembling Hamiltion's equations for a particle. Discussion centers on applications to gradientless propagation, bubble saturation velocity, and the Blochline model of inertial effects, and on general limitations of the theory. © 1979.