Journal of Nonlinear Science

, Volume 24, Issue 1, pp 39–92 | Cite as

Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities

  • Marshall Hampton
  • Gareth E. Roberts
  • Manuele Santoprete


We examine in detail the relative equilibria in the planar four-vortex problem where two pairs of vortices have equal strength, that is, Γ1=Γ2=1 and Γ3=Γ4=m where \(m \in \mathbb{R} - \{0\}\) is a parameter. One main result is that, for m>0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. In fact, there exist asymmetric convex configurations when m<0. In contrast to the Newtonian four-body problem with two equal pairs of masses, where the symmetry of all convex central configurations is unproven, the equations in the vortex case are easier to handle, allowing for a complete classification of all solutions. Precise counts on the number and type of solutions (equivalence classes) for different values of m, as well as a description of some of the bifurcations that occur, are provided. Our techniques involve a combination of analysis, and modern and computational algebraic geometry.


Relative equilibria n-vortex problem Hamiltonian systems Symmetry 

Mathematics Subject Classification

76B47 70F10 13P10 13P15 70H12 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Marshall Hampton
    • 1
  • Gareth E. Roberts
    • 2
  • Manuele Santoprete
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of Minnesota DuluthDuluthUSA
  2. 2.Department of Mathematics and Computer ScienceCollege of the Holy CrossWorcesterUSA
  3. 3.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada

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