Advertisement

Regular and Chaotic Dynamics

, Volume 23, Issue 1, pp 127–134 | Cite as

Dynamics of Three Vortices on a Sphere

  • Alexey V. Borisov
  • Ivan S. Mamaev
  • Alexander A. Kilin
Article
  • 70 Downloads

Abstract

This paper is concerned with the dynamics of vortices on a sphere. It is shown that, as a result of reduction, the problem reduces to investigating a system with a nonlinear Poisson bracket. The topology of a symplectic leaf in the case of three vortices is studied.

Keywords

point vortices space of constant curvature nonlinear Poisson bracket reduction 

MSC2010 numbers

76M23 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bogomolov, V.A., Two-Dimensional Hydrodynamics on a Sphere, Izv. Atmos. Oc. Phys., 1979, vol. 15, no. 1, pp. 18–22.Google Scholar
  2. 2.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Reduction and Chaotic Behavior of Point Vortices on a Plane and a Sphere, Rus. J. Nonlin. Dyn., 2005, vol. 1, no. 2, pp. 233–246.zbMATHGoogle Scholar
  3. 3.
    Borisov, A.V. and Mamaev, I. S., Poisson Structures and Lie Algebras in Hamiltonian Mechanics, Izhevsk: Izd. UdSU, 1999, 464 pp.zbMATHGoogle Scholar
  4. 4.
    Volterra V. Leçons sur la théorie mathématique de la lutte pous la vie. Paris: Gauthier-Villars, 1931.zbMATHGoogle Scholar
  5. 5.
    Borisov, A. V. and Lebedev, V.G., Dynamics of Three Vortices on a Plane and a Sphere. II. General compact case, Regul. Chaotic Dyn., 1998, vol. 3, no. 2, pp. 99–114.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Reduction and Chaotic Behavior of Point Vortices on a Plane and a Sphere, Discrete and Continuous Dynamical Systems. Ser. B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, pp. 100–109.Google Scholar
  7. 7.
    Kidambi, R. and Newton, P.K., Collision of Three Vortices on a Sphere, Il Nuovo Cimento, 1999, vol. 22, no. C(6), pp. 779–791.Google Scholar
  8. 8.
    Kidambi, R. and Newton, P.K., Motion of Three Point Vortices on a Sphere, Physica D, 1998, vol. 116, pp. 143–175.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Newton, P.K., The N-Vortex problem. Analytical Techniques. Springer, 2001.CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
  • Ivan S. Mamaev
    • 2
    • 3
  • Alexander A. Kilin
    • 1
    • 2
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  3. 3.Izhevsk State Technical UniversityIzhevskRussia

Personalised recommendations