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

Abstract

We examine in detail the relative equilibria in the planar four-vortex problem where two pairs of vortices have equal strength, that is, Γ 1=Γ 2=1 and Γ 3=Γ 4=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.

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Acknowledgements

Part of this work was carried out when the authors were visiting the American Institute of Mathematics in May of 2011. We gratefully acknowledge their hospitality and support. We would also like to thank the two referees for many helpful suggestions and comments. GR was supported by a grant from the National Science Foundation (DMS-1211675), and MS was supported by a NSERC Discovery Grant.

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Correspondence to Gareth E. Roberts.

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Communicated by P. Newton.

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Hampton, M., Roberts, G.E. & Santoprete, M. Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities. J Nonlinear Sci 24, 39–92 (2014). https://doi.org/10.1007/s00332-013-9184-3

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Keywords

  • Relative equilibria
  • n-vortex problem
  • Hamiltonian systems
  • Symmetry

Mathematics Subject Classification

  • 76B47
  • 70F10
  • 13P10
  • 13P15
  • 70H12