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
Article

Abstract

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.

Keywords

Relative equilibria n-vortex problem Hamiltonian systems Symmetry 

Mathematics Subject Classification

76B47 70F10 13P10 13P15 70H12 

References

  1. Albouy, A.: The symmetric central configurations of four equal masses. In: Hamiltonian Dynamics and Celestial Mechanics, Seattle, WA, 1995. Contemp. Math., vol. 198, pp. 131–135. Amer. Math. Soc., Providence (1996) CrossRefGoogle Scholar
  2. Albouy, A., Chenciner, A.: Le problème des n corps et les distances mutuelles. Invent. Math. 131(1), 151–184 (1997) CrossRefMathSciNetGoogle Scholar
  3. Albouy, A., Fu, Y., Sun, S.: Symmetry of planar four-body convex central configurations. Proc. R. Soc. A, Math. Phys. Eng. Sci. 464(2093), 1355–1365 (2008) CrossRefMATHMathSciNetGoogle Scholar
  4. Albouy, A., Cabral, H.E., Santos, A.A.: Some problems on the classical n-body problem. Celest. Mech. Dyn. Astron. 113, 369–375 (2012) CrossRefMathSciNetGoogle Scholar
  5. Aref, H.: Point vortex dynamics: a classical mathematics playground. J. Math. Phys. 48(6), 065401 (2007a), 23 pp. CrossRefMathSciNetGoogle Scholar
  6. Aref, H.: Vortices and polynomials. Fluid Dyn. Res. 39, 5–23 (2007b) CrossRefMATHMathSciNetGoogle Scholar
  7. Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., Vainchtein, D.L.: Vortex crystals. Adv. Appl. Mech. 39, 1–79 (2003) CrossRefGoogle Scholar
  8. Barros, J.F., Leandro, E.S.G.: The set of degenerate central configurations in the planar restricted four-body problem. SIAM J. Math. Anal. 43(2), 634 (2011) CrossRefMATHMathSciNetGoogle Scholar
  9. Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998) CrossRefMATHGoogle Scholar
  10. Cardano, G.: Artis magnae. Johann Petreius, Nuremberg (1545) Google Scholar
  11. Celli, M.: Sur les mouvements homographiques de n corps associés à des masses de signe quelconque, le cas particulier où la somme des masses est nulle, et une application à la recherche de choréographies perverse. Ph.D. thesis, Université Paris 7, France (2005) Google Scholar
  12. Chen, C., Davenport, J.H., May, J.P., Maza, M.M., Xia, B., Xiao, R.: Triangular decomposition of semi-algebraic systems. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC’10, pp. 187–194. ACM Press, New York (2010) CrossRefGoogle Scholar
  13. Chen, Y., Kolokolnikov, T., Zhirov, D.: Collective behavior of large number of vortices in the plane. Proc. R. Soc. A 469(2156), 1–6 (2013) CrossRefGoogle Scholar
  14. Cors, J.M., Roberts, G.E.: Four-body co-circular central configurations. Nonlinearity 25(2), 343–370 (2012) CrossRefMATHMathSciNetGoogle Scholar
  15. Cox, D.A., Little, J.B., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, Berlin (2007) CrossRefGoogle Scholar
  16. Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-3—a computer algebra system for polynomial computations (2011). http://www.singular.uni-kl.de
  17. Dziobek, O.: Über einen merkwürdigen fall des vielkörperproblems. Astron. Nachr. 152, 33 (1900) CrossRefGoogle Scholar
  18. Hampton, M.: Concave central configurations in the four-body problem. Ph.D. thesis, University of Washington, Seattle (2002) Google Scholar
  19. Hampton, M., Moeckel, R.: Finiteness of relative equilibria of the four-body problem. Invent. Math. 163(2), 289–312 (2005) CrossRefMathSciNetGoogle Scholar
  20. Hampton, M., Moeckel, R.: Finiteness of stationary configurations of the four-vortex problem. Trans. Am. Math. Soc. 361(3), 1317–1332 (2009) CrossRefMATHMathSciNetGoogle Scholar
  21. Kirchhoff, G.: Vorlesungen über mathematische physik. B.G. Teubner, Leipzig (1883) Google Scholar
  22. Lord Kelvin: On vortex atoms. Proc. R. Soc. Edinb. 6, 94–105 (1867) Google Scholar
  23. Meyer, K., Schmidt, D.: Bifurcations of relative equilibria in the n-body and Kirchhoff problems. SIAM J. Math. Anal. 19(6), 1295–1313 (1988) CrossRefMATHMathSciNetGoogle Scholar
  24. Meyer, K.R., Hall, G.R., Offin, D.: Introduction to Hamiltonian Dynamical Systems and the n-Body Problem, 2nd edn. Springer, Berlin (2009) MATHGoogle Scholar
  25. Moulton, F.R.: The straight line solutions of the problem of n bodies. Ann. Math. 12(1), 1–17 (1910) CrossRefMATHMathSciNetGoogle Scholar
  26. Newton, P.K.: The n-Vortex Problem: Analytical Techniques. Springer, Berlin (2001) CrossRefGoogle Scholar
  27. Newton, P.K., Chamoun, G.: Construction of point vortex equilibria via Brownian ratchets. Proc. R. Soc. A 463, 1525–1540 (2007) CrossRefMATHMathSciNetGoogle Scholar
  28. O’Neil, K.A.: Stationary configurations of point vortices. Trans. Am. Math. Soc. 302(2), 383–425 (1987) MATHMathSciNetGoogle Scholar
  29. Palmore, J.: Relative equilibria of vortices in two dimensions. Proc. Natl. Acad. Sci. USA 79(2), 716–718 (1982) CrossRefMATHMathSciNetGoogle Scholar
  30. Perez-Chavela, E., Santoprete, M.: Convex four-body central configurations with some equal masses. Arch. Ration. Mech. Anal. 185(3), 481–494 (2007) CrossRefMATHMathSciNetGoogle Scholar
  31. Roberts, G.E.: Stability of relative equilibria in the planar n-vortex problem. SIAM J. Appl. Dyn. Syst. 12(2), 1114–1134 (2013) CrossRefMATHMathSciNetGoogle Scholar
  32. Schmidt, D.: Central configurations and relative equilibria for the n-body problem. In: Classical and Celestial Mechanics (Recife, 1993/1999), pp. 1–33. Princeton Univ. Press, Princeton (2002) Google Scholar
  33. Spang, S.: A zero-dimensional approach to compute real radicals. Comput. Sci. J. Mold. 16(1), 64–92 (2008) MATHMathSciNetGoogle Scholar
  34. Spang, S.: realrad.lib. A singular 3-1-3 library for computing real radicals (2011) Google Scholar
  35. Stein, W.A., et al.: Sage mathematics software (Version 4.6.2) (2011). http://www.sagemath.org
  36. Uspensky, J.V.: Theory of Equations. McGraw-Hill Book Co., New York (1948) Google Scholar

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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