Absolute and relative choreographies in the problem of point vortices moving on a plane

TL;DR

This paper presents new periodic solutions for three and four point vortices on a plane, highlighting differences from celestial mechanics choreographies and analyzing their integrability and motion characteristics.

Contribution

It introduces novel periodic vortex solutions, analyzing their integrability and unique features compared to celestial choreographies.

Findings

New periodic solutions for three and four vortices

Three-vortex system is integrable with one degree of freedom

Four-vortex system is non-integrable with two degrees of freedom

Abstract

We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the n-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.