A non-dissipative model for vortex motion in thin superconductors is considered. The Lagrangian is a Galilean invariant version of the Ginzburg Landau model for time-dependent fields, with kinetic terms linear in the first time derivatives of the fields. It is shown how, for certain values of the coupling constants, the field dynamics can be reduced to first order differential equations for the vortex positions. Two vortices circle around one another at constant speed and separation in this model. 1997 Academic Press 1. INTRODUCTION Magnetic flux penetrates a Type II superconductor in the form of vortices [1], and recently it has become possible to produce images of vortices sufficiently rapidly that their motion can be observed directly [2]. In the Ginzburg Landau theory of superconductivity, a charged scalar field representing the electron-pair condensate is coupled to the electromagnetic field. The basic vortex solution, discovered by Abrikosov [3], is a localised magnetic flux tube surrounded by a circulating supercurrent.
The Ginzburg Landau potential energy functional contains only one dimensionless coupling constant *. The value *=1 (in our units) is mathematically particularly interesting, because in this case there are no forces between static vortices, and there is a continuous family of static multivortex solutions. A Type II superconductor is modelled by *>1. In this case, the potential energy of a two-vortex configuration decreases as the separation increases, in other words, vortices repel [4]. However, there are several possibilities for how the vortices might move, depending on the nature of the dynamical equations for the fields. Let us ignore pinning, which tends to prevent vortex motion at all. The first possibility is that the vortex acceleration is proportional to the force acting. This is what occurs in the relativistic generalisation of the Ginzburg Landau model, known as the Abelian Higgs model. Relativistic vortices may be interpreted as a solitonic version of fundamental strings [5], or as strings joining confined quarks, or as cosmic strings produced at a phase transition early in the universe's history [6]. The second article no. PH975672 114 0003-4916Γ97 25.00