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, N. S. Vasiliev 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.
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© 2008 Springer Science + Business Media B.V
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Borisov, A.V., Mamaev, I.S. (2008). Dynamics of Two Rings of Vortices on a Sphere. In: Borisov, A.V., Kozlov, V.V., Mamaev, I.S., Sokolovskiy, M.A. (eds) IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence. IUTAM Bookseries, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6744-0_40
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DOI: https://doi.org/10.1007/978-1-4020-6744-0_40
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-6743-3
Online ISBN: 978-1-4020-6744-0
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