Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1401–1425 | Cite as

Relative Periodic Solutions of the N-Vortex Problem Via the Variational Method

  • Qun WangEmail author


This article studies the N-vortex problem in the plane with positive vorticities. After an investigation of some properties for normalised relative equilibria of the system, we use symplectic capacity theory to show that there exist infinitely many normalised relative periodic orbits on a dense subset of all energy levels, which are neither fixed points nor relative equilibria.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham R., Marsden J.E., Marsden J.E.: Foundations of Mechanics, vol.36. Benjamin/Cummings Publishing Company Reading, Massachusetts (1978)zbMATHGoogle Scholar
  2. 2.
    Albouy, A., Kaloshin, V.: Finiteness of central configurations of five bodies in the plane. Ann. Math. 176, 535–588 2012MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aref H.: Motion of three vortices. Phys. Fluids 22(3), 393–400 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aref H.: Point vortex motions with a center of symmetry. Phys. Fluids 25(12), 2183–2187 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., Vainchtein, D.L.: Vortex crystals. Tech. rep., Department of Theoretical and Applied Mechanics (UIUC) 2002Google Scholar
  6. 6.
    Bartsch T., Dai Q.: Periodic solutions of the N-vortex Hamiltonian system in planar domains. J. Differ. Equ. 260(3), 2275–2295 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bartsch T., Gebhard B.: Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type. Math. Ann. 369(1–2), 627–651 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Borisov A.V., Mamaev I.S., Kilin A.: Absolute and relative choreographies in the problem of point vortices moving on a plane. Regul. Chaotic Dyn. 9(2), 101–111 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borisov A.V., Pavlov A.: Dynamics and statics of vortices on a plane and a sphere-I. Regul. Chaotic Dyn. 3(1), 28–38 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Carminati C., Sere E., Tanaka K.: The fixed energy problem for a class of nonconvex singular Hamiltonian systems. J. Differ. Equ. 230(1), 362–377 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carvalho A.C., Cabral H.E.: Lyapunov orbits in the N-vortex problem. Regul. Chaotic Dyn. 19(3), 348–362 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chenciner, A.: Action minimizing solutions of the Newtonian N-body problem: from homology to symmetry. arXiv preprint arXiv:math/0304449 2003
  13. 13.
    Chenciner A., Féjoz J.: Unchained polygons and the N-body problem. Regul. Chaotic Dyn. 14(1), 64–115 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chenciner A., Montgomery R.: A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. Math. Second Ser. 152(3), 881–902 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ekeland I.: Convexity Methods in Hamiltonian Mechanics: vol.19. Springer, Berlin (2012)Google Scholar
  16. 16.
    Hampton M., Moeckel R.: Finiteness of stationary configurations of the four-vortex problem. Trans. Am. Math. Soc. 361(3), 1317–1332 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hofer H., Viterbo C.: The Weinstein conjecture in the presence of holomorphic spheres. Commun. Pure Appl. Math. 45(5), 583–622 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hofer H., Zehnder E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, Basel (2012)zbMATHGoogle Scholar
  19. 19.
    Khanin K.: Quasi-periodic motions of vortex systems. Physica D Nonlinear Phenom. 4(2), 261–269 (1982)ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Kirchhoff G.R.: Vorlesungen über Mathematische Physik: Mechanik, vol. 1. Teubner. Teubner, Stuttgart (1876)zbMATHGoogle Scholar
  21. 21.
    Koiller J., Carvalho S.P.: Non-integrability of the 4-vortex system: Analytical proof. Commun. Math. Phys. 120(4), 643–652 (1989)ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Koiller J., De Carvalho S.P., Da Silva R.R., De Oliveira L.C.G.: On Aref’s vortex motions with a symmetry center. Physica D Nonlinear Phenom. 16(1), 27–61 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Laurent-Polz F.: Relative periodic orbits in point vortex systems. Nonlinearity 17(6), 1989 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lim C.C.: On theMotion of Vortices in Two Dimensions. 5. University of Toronto Press. , Toronto (1943)Google Scholar
  25. 25.
    Lim C.C.: Canonical transformations and graph theory. Phys. Lett. A 138(6–7), 258–266 (1989)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Lugt H.J.: Vortex flow in nature and technology. Wiley-Interscience, New York (1983) 305 p. Translation. 1983Google Scholar
  27. 27.
    Marchal C.: How the method of minimization of action avoids singularities. Celest. Mech. Dyn. Astron. 83(1–4), 325–353 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Montaldi J., Souliere A., Tokieda T.: Vortex dynamics on a cylinder. SIAM J. Appl. Dyn. Syst. 2(3), 417–430 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mumford, D.: Algebraic Geometry. I, Complex Projective Varieties. Grundl. Math. Wiss., vol. 221, Springer, New York 1976Google Scholar
  30. 30.
    O’Neil K.A.: Stationary configurations of point vortices. Trans. Am. Math. Soc. 302(2), 383–425 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    O’Neil K.A.: Relative equilibrium and collapse configurations of four point vortices. Regul. Chaotic Dyn. 12(2), 117–126 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Palmore J.I.: Relative equilibria of vortices in two dimensions. Proc. Natl. Acad. Sci. 79(2), 716–718 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Poincaré H.: Lesméthodes Nouvelles de laMécanique Céleste. Gauthier-Villars, Paris (1892)Google Scholar
  34. 34.
    Rabinowitz P.H.: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 31(2), 157–184 (1978)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Roberts G.E.: Stability of relative equilibria in the planar N-vortex problem. SIAM J. Appl. Dyn. Syst. 12(2), 1114–1134 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Roberts G.E.: Morse theory and relative equilibria in the planar N-vortex problem. Arch. Ration. Mech. Anal. 228, 209–236 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Routh E.J.: Some applications of conjugate functions. Proc. Lond. Math. Soc. 1(1), 73–89 (1880)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Soulière A., Tokieda T.: Periodic motions of vortices on surfaces with symmetry. J. Fluid Mech. 460, 83–92 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Synge J.: On the motion of three vortices. Can. J. Math. 1(3), 257–270 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Thomson J.J.: A Treatise on the Motion of Vortex Rings: An Essay to Which the Adams Prize was Adjudged in 1882, in the University of Cambridge. Macmillan, New York (1883)Google Scholar
  41. 41.
    Tokieda T.: Tourbillons dansants. C. R. l’Acad. Sci. Ser. I-Math. 333(10), 943–946 (2001)ADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    von Helmholtz H.: Über integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Math. Bd. LV. Heft 1, 4 (1858)MathSciNetGoogle Scholar
  43. 43.
    Viterbo, C.: Capacités symplectiques et applications. Sémin. Bourbaki 31, 1988–1989Google Scholar
  44. 44.
    Viterbo, C.: A proof of Weinstein’s conjecture in R2n. In: Annales de l’Institut Henri Poincare (C) Non Linear Analysis , vol. 4, pp. 337–356. Elsevier 1987Google Scholar
  45. 45.
    Weinstein A.: Periodic orbits for convex Hamiltonian systems. Ann. Math. 108(3), 507–518 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Yarmchuk E., Gordon M., Packard R.: Observation of stationary vortex arrays in rotating superfluid helium. Phys. Rev. Lett. 43(3), 214 (1979)ADSCrossRefGoogle Scholar
  47. 47.
    Ziglin S.: Nonintegrability of a problem on the motion of four point vortices. Sov. Math. Dokl. 21, 296–299 (1980)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS, UMR 7534, CEREMADE, Université Paris-Dauphine, PSL Research UniversityParisFrance

Personalised recommendations