An Arnold-Type Stability Analysis of Vortex Filaments

We establish nonlinear orbital stability of circular arcs with opening angle $\theta \le \pi$ and closed circles under the Localized Induction Equation (LIE), with prescribed endpoint constraints for arcs. Our Arnold-type variational approach combines two conserved quantities, the bending energy and a component of the fluid impulse, into a functional that we prove is comparable to the squared $H^2$-distance from the corresponding orbit for all admissible curves. For circular arcs with angle $\theta >\pi$, we prove nonlinear orbital instability using the inverse generalized Hasimoto transformation.