Magnetic vortices

Abstract

In spaces with constant curvature, magnetic trajectories of a Killing magnetic field are showed to be centerlines of Kirchhoff elastic rods. If the space is also simply connected, then a Killing magnetic flow is proved to be equivalent with the Kirchhoff elastic rod variational model. This allows to see Killing magnetic flows as vortex filament flows and so use the Hasimoto transformation to state the solitonic nature of the magnetic trajectories as solutions of the cubic Schrödinger equation.

Magnetic curves in Sasakian manifolds

“…An important class of magnetic fields consists of Killing vector fields. Studying Killing magnetic curves in 3-dimensional space forms, Barros and Romero showed in [10] that these curves are centerlines of Kirchhoff elastic rods. Further on, Barros et al in [9] showed that these curves are solitons of the localized induction equation (LIE).…”

Notes on magnetic curves in 3D semi-Riemannian manifolds

“…If the charged particle moves parallel to magnetic field, the Lorentzian force acting on the particle is zero. When the two vectors (velocity and the magnetic field) are perpendicular to each other, the Lorentz force is maximum (for details, see [4,2,3,6,7,9,8]). …”

Killing magnetic curves in three-dimensional almost paracontact manifolds

“…In fact, given a nontrivial magnetic field on a Riemannian manifold, there exists no linear connection, whose geodesics coincide with the magnetic curves of F [2, Prop. In recent years, magnetic curves have been intensively studied (see for example [1][2][3][4]6,8,10] and the references therein), since they are a natural object of investigation under several points of view. In fact, besides their interpretation as a generalization of geodesics, they also encode further geometrical and physical meanings.…”

“…Observe that on a three-dimensional pseudo-Riemannian manifold (M, g), the Lorentz force is provided via the cross product, and magnetic fields are in a one-to-one correspondence with divergence-free vector fields. In the case of a three-manifold of constant curvature, the Killing magnetic flow permits to clarify the relationships among phenomena of different physical meaning, codified by the Lorentz force equation for Killing magnetic fields, the field equation for Kirchhoff elastic rods (so giving a variational interpretation of magnetic curves), the Betchov-Da Rios equation for vortex filaments and the cubic Schröedinger equation [1].…”

“…In particular, the function x(s) = −( 4 q 2 sinh 2 ( 3qs 4 )) 1 3 is a solution of (5.9) when c 1 = −1 and c 2 = 0. Hence, γ(s) = − 4 q 2 sinh 2 3qs 4 1 3 , q 2 4 q 2 sinh 2 3qs 4 1 3 ds, a 0 s + z 0 is a magnetic curve.…”