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

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.

11 pages, 6 figures

References in corpus (1)

• Lie algebras in vortex dynamics and celestial mechanics - IV

Cited by in corpus (11)

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• A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects
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• Relative periodic solutions of the n-vortex problem on the sphere
• Choreographies in the Discrete Nonlinear Schrödinger Equations
• On one unstable bifurcation in the dynamics of vortex structure
• Phase Topology of Two Vortices of the Identical Intensities in Bose-Einstein Condensate
• Dynamics of a few vortices in an ideal fluid and in a Bose-Einstein condensate
• Choreographic Holomorphic Spheres with Application to Hamiltonian Systems of $N$-Vortex Type