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Nelin. Dinam., 2006, Volume 2, Number 2, Pages 181–192 (Mi nd161)  

Dynamics of two vortex rings on a sphere

A. V. Borisovab, I. S. Mamaevab

a Udmurt State University
b Institute of Computer Science

Abstract: The motion of two vortex rings on a sphere is considered. This motion generalizes the well-known centrally symmetrical solution of the equations of point vortex dynamics on a plane derived by D. N. Goryachev and H. Aref. The equations of motion in this case are shown to be Liouville integrable, and an explicit reduction to a Hamiltonian system with one degree of freedom is described. Two particular cases in which the solutions are periodical are presented. Explicit quadratures are given for these solutions. Phase portraits are described and bifurcation diagrams are shown for centrally symmetrical motion of four vortices on a sphere.

Keywords: vortex, Hamiltonian, motion on a sphere, phase portrait.

Full text: PDF file (321 kB)

Document Type: Article
UDC: 532.517
MSC: 76M23

Citation: A. V. Borisov, I. S. Mamaev, “Dynamics of two vortex rings on a sphere”, Nelin. Dinam., 2:2 (2006), 181–192

Citation in format AMSBIB
\Bibitem{BorMam06}
\by A.~V.~Borisov, I.~S.~Mamaev
\paper Dynamics of two vortex rings on a sphere
\jour Nelin. Dinam.
\yr 2006
\vol 2
\issue 2
\pages 181--192
\mathnet{http://mi.mathnet.ru/nd161}


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