Dynamics of Two Rings of Vortices on a Sphere

  • Alexey V. Borisov
  • Ivan S. Mamaev
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 6)

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.

Keywords

Vortices Hamiltonian motion on a sphere phase portrait 

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Copyright information

© Springer Science + Business Media B.V 2008

Authors and Affiliations

  • Alexey V. Borisov
    • 1
  • Ivan S. Mamaev
    • 1
  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

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