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, 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.

Keywords

Relative equilibria n-vortex problem Hamiltonian systems Symmetry 

Mathematics Subject Classification

76B47 70F10 13P10 13P15 70H12 

Notes

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.

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

Personalised recommendations