Groebli solution for three magnetic vortices

TL;DR

This paper extends the classical Groebli solution for three point vortices to the context of magnetic vortices, providing analytical insights into their dynamics within a generalized Hamiltonian framework.

Contribution

It introduces a generalized solution for three magnetic vortices based on Groebli's classical approach, bridging fluid vortex dynamics and magnetic vortex behavior.

Findings

Derived a generalized analytical solution for three magnetic vortices.

Analyzed the dynamics and stability of magnetic vortex configurations.

Discussed implications for ferromagnetic element behavior.

Abstract

The dynamics of N point vortices in a fluid is described by the Helmholtz-Kirchhoff (HK) equations which lead to a completely integrable Hamiltonian system for N=2 or 3 but chaotic dynamics for N>3. Here we consider a generalization of the HK equations to describe the dynamics of magnetic vortices within a collective-coordinate approximation. In particular, we analyze in detail the dynamics of a system of three magnetic vortices by a suitable generalization of the solution for three point vortices in an ordinary fluid obtained by Groebli more than a century ago. The significance of our results for the dynamics of ferromagnetic elements is briefly discussed.