Anholonomy of a moving space curve and applications to classical magnetic chains

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Abstract
The subject of space curves finds many applications in physics such as optical fibers, magnetic spin chains, and vortex filaments in a fluid. We show that the time evolution of a space curve is associated with a geometric phase. Using the concept of Fermi-Walker parallel transport, we show that this phase arises because of the path dependence of the rotation of the natural Frenet-Serret triad as one moves along the curve. We employ Lamb’s formalism for space-curve dynamics to derive an expression for the anholonomy density and the geometric phase for a general time evolution. This anholonomy manifests itself as a gauge potential with a monopolelike structure in the space of the tangent vector to the space curve. Our classical approach is amenable to a quantum generalization, which can prove useful in applications. We study the application of our constructive formalism to ferromagnetic and antiferromagnetic (classical) spin chains by first presenting certain classes of exact, physically interesting solutions to these nonlinear dynamical systems and then computing the corresponding geometric phases.

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