Regular and Chaotic Dynamics

, Volume 12, Issue 2, pp 117–126 | Cite as

Relative equilibrium and collapse configurations of four point vortices

  • K. A. O’neil
Research Articles


Relative equilibrium configurations of point vortices in the plane can be related to a system of polynomial equations in the vortex positions and circulations. For systems of four vortices the solution set to this system is proved to be finite, so long as a number of polynomial expressions in the vortex circulations are nonzero, and the number of relative equilibrium configurations is thereby shown to have an upper bound of 56. A sharper upper bound is found for the special case of vanishing total circulation. The polynomial system is simple enough to allow the complete set of relative equilibrium configurations to be found numerically when the circulations are chosen appropriately. Collapse configurations of four vortices are also considered; while finiteness is not proved, the approach provides an effective computational method that yields all configurations with a given ratio of velocity to position.

Key words

point vortices relative equilibrium 

MSC2000 numbers

76B47 70F10 70H12 


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  1. 1.
    Aref, H., Newton, P.K., Stremler, M., Tokieda, T., and Vainchtein, D., Vortex Crystals, Adv. Appl. Mech., 2003, vol. 39, pp. 1–79.CrossRefGoogle Scholar
  2. 2.
    Aref, H. and van Buren, M., Vortex Triple Rings, Phys. Fluids, 2005, vol. 17, 057104, 21 pp.Google Scholar
  3. 3.
    Campbell, L.J. and Ziff, R., Vortex Patterns and Energies in a Rotating Superfluid, Phys. Rev. B, 1979, vol.20, pp. 1886–1902.CrossRefGoogle Scholar
  4. 4.
    Hampton, M. and Moeckel, R., Finiteness of Relative Equilibria of the Four-Body Problem, Invent. Math., 2006, vol. 163, pp. 289–312.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hampton, M. and Moeckel, R., Finiteness of Stationary Configurations of the Four Vortex Problem, 2006, (preprint).Google Scholar
  6. 6.
    Novikov, E.A. and Sedov, B., Vortex Collapse, Soviet Phys. JETP, 1979, vol. 22, pp. 297–301.Google Scholar
  7. 7.
    O’Neil, K., Stationary Configurations of Point Vortices, Trans. Amer. Math. Soc., 1987, vol. 302, pp. 383–425.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    O’Neil, K., Minimal Polynomial Systems for Point Vortex Equilibria, Phys. D, 2006, vol. 219, pp. 69–79.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Palmore, J.I., Relative Equilibria of Vortices in Two Dimensions, Proc. Nat. Acad. Sci. U.S.A., 1982, vol. 79, pp. 716–718.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Roberts, G., A Continuum of Relative Equilibria in the Five-Body Problem, Phys. D, 1999, vol. 127, pp. 141–145.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Shafarevich, I.R., Basic Algebraic Geometry, Revised printing of vol. 213 of Grundlehren der mathematischen Wissenschaften, 1974. Springer Study Edition. Berlin: Springer-Verlag, 1977.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  • K. A. O’neil
    • 1
  1. 1.Department of MathematicsThe University of TulsaTulsaUSA

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