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

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

2005-02-10

arxiv.org/abs/nlin/0502019v3

Regular and Chaotic Dynamics, 2004 Volume 9 Number 2

Nonlinear Sciences

Chaotic Dynamics

11 pages, 6 figures

Scientific paper

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.

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