Spinor representation of curves and complex forces on filaments

Abstract

We present a theoretical framework to study equilibrium configurations of filaments within a spinor representation of curves. The curve representing the filament is described by a unit two-component spinor field and its charge conjugate satisfying two-dimensional equations coupled by the curvature and torsion. The spinor field replaces the Frenet-Serret frame, whereas its structure equations replace the Frenet-Serret equations. Employing this spinorial description of curves, we derive the Euler-Lagrange equations of curves whose energies depend on their curvature and torsion. We analyze the conservation laws of the spinors representing the balance of the forces and torques along the filament. We illustrate this framework by applying these results to the Euler Elastica, whose bending energy is quadratic in the curvature.