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Finiteness of stationary configurations of the four-vortex problem


Authors: Marshall Hampton and Richard Moeckel
Journal: Trans. Amer. Math. Soc. 361 (2009), 1317-1332
MSC (2000): Primary 70F10, 70F15, 37N05, 76Bxx
Published electronically: October 16, 2008
MathSciNet review: 2457400
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that the number of relative equilibria, equilibria, and rigidly translating configurations in the problem of four point vortices is finite. The proof is based on symbolic and exact integer computations which are carried out by computer. We also provide upper bounds for these classes of stationary configurations.


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  • 1. Alain Albouy and Alain Chenciner, Le problème des 𝑛 corps et les distances mutuelles, Invent. Math. 131 (1998), no. 1, 151–184 (French). MR 1489897, 10.1007/s002220050200
  • 2. A. Albouy, On a paper of Moeckel on central configurations, Regul. Chaotic Dyn. 8 (2003), no. 2, 133–142. MR 1988854, 10.1070/RD2003v008n02ABEH000232
  • 3. Alain Albouy and Richard Moeckel, The inverse problem for collinear central configurations, Celestial Mech. Dynam. Astronom. 77 (2000), no. 2, 77–91 (2001). MR 1820352, 10.1023/A:1008345830461
  • 4. Hassan Aref and Mark A. Stremler, Four-vortex motion with zero total circulation and impulse, Phys. Fluids 11 (1999), no. 12, 3704–3715. MR 1723520, 10.1063/1.870233
  • 5. H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda, and D. L. Vainchtein, Vortex Crystals, Advances In Applied Mechanics 39 (2003) 1-79.
  • 6. D. N. Bernstein, The number of roots of a system of equations, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 1–4 (Russian). MR 0435072
  • 7. M. Celli, 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, Thèse, Université Paris 7 (2005).
  • 8. T. Christof and A. Loebel. PORTA: Polyhedron Representation Transformation Algorithm, Version 1.3.2. http://www.iwr.uni-heidelberg.de/ iwr/comopt/soft/PORTA/readme.html
  • 9. David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. An introduction to computational algebraic geometry and commutative algebra. MR 1417938
  • 10. O. Dziobek, Über einen merkwürdigen Fall des Vielkörperproblems, Astron. Nach. 152 (1900) 33-46.
  • 11. D.R. Grayson and M. E. Sullivan, Macaulay 2, a software system for research in algebraic geometry and commutative algebra, http://www.math.uic.edu/Macaulay2/.
  • 12. W. Gröbli, Specielle Probleme über die Bewegung geradliniger paralleler Wirfbelfäden, Vierteljahrschrift der naturforschenden Gesellschaft in Zürich 22 (1877) 37-81, 129-165.
  • 13. Marshall Hampton and Richard Moeckel, Finiteness of relative equilibria of the four-body problem, Invent. Math. 163 (2006), no. 2, 289–312. MR 2207019, 10.1007/s00222-005-0461-0
  • 14. M. Hampton and R. Moeckel, VortexCalculations.nb, Mathematica notebook available from http://www.math.umn.edu/ rick
  • 15. H. Helmholtz, Uber Integrale der hydrodynamischen Gleichungen, Welche den Wirbelbewegungen entsprechen, Crelle's Journal für Mathematik, 55 (1858) 25-55. English translation by P. G. Tait, P.G., On the integrals of the hydrodynamical equations which express vortex-motion, Philosophical Magazine, (1867) 485-512.
  • 16. Heron of Alexandria, Metrica, (circa 60).
  • 17. A. G. Khovansky, Newton polyhedra and toric varieties, Fun. Anal. Appl. 11 (1977) 289-296.
  • 18. G. R. Kirchhoff, Vorlesungen über Mathenatische Physik I, Teubner, Leipzig (1876).
  • 19. A. G. Kushnirenko, Newton polytopes and the Bézout theorem, Fun. Anal. Appl. 10 (1976) 233-235.
  • 20. C. C. MacDuffee, Theory of Matrices, Chelsea Publishing Co., New York (1946).
  • 21. Computational Algebra Group, MAGMA, version 2.11.11, University of Sydney.
  • 22. Richard Moeckel, Chaotic dynamics near triple collision, Arch. Rational Mech. Anal. 107 (1989), no. 1, 37–69. MR 1000223, 10.1007/BF00251426
  • 23. Richard Moeckel, A computer-assisted proof of Saari’s conjecture for the planar three-body problem, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3105–3117 (electronic). MR 2135737, 10.1090/S0002-9947-04-03527-5
  • 24. T. S. Motzkin, H. Raiffa, G. L. Thompson, and R. M. Thrall, The double description method, Contributions to the theory of games, vol. 2, Annals of Mathematics Studies, no. 28, Princeton University Press, Princeton, N. J., 1953, pp. 51–73. MR 0060202
  • 25. E. A. Novikov, Dynamics and statistics of a system of vortices, Soviet Phys. JETP 41 (1975) 937-943.
  • 26. Kevin Anthony O’Neil, Stationary configurations of point vortices, Trans. Amer. Math. Soc. 302 (1987), no. 2, 383–425. MR 891628, 10.1090/S0002-9947-1987-0891628-1
  • 27. K. A. O'Neil, Notes on relative equilibria of point vortices, personal communication, (2006).
  • 28. Gareth E. Roberts, A continuum of relative equilibria in the five-body problem, Phys. D 127 (1999), no. 3-4, 141–145. MR 1669486, 10.1016/S0167-2789(98)00315-7
  • 29. A. Seidenberg, Elements of the theory of algebraic curves, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0248139
  • 30. Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
  • 31. J. L. Synge, On the motion of three vortices, Canadian J. Math. 1 (1949), 257–270. MR 0030841
  • 32. Sir W. Thomson (Lord Kelvin), On vortex atoms, Proc. R. Soc. Edinburgh 6 (1867) 94-105.
  • 33. J. Verschelde, 1999, Algorithm 795: PHCpack: A General-Purpose Solver for Polynomial Systems by Homotopy Continuation, ACM Trans. Math. Softw. 25, 251-276.
  • 34. S. Wolfram, Mathematica, version 5.0.1.0, Wolfram Research, Inc.

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Additional Information

Marshall Hampton
Affiliation: School of Mathematics, University of Minnesota, Duluth, Minnesota 55812
Email: mhampton@d.umn.edu

Richard Moeckel
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: rick@math.umn.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04685-0
Keywords: Relative equilibria, vortices
Received by editor(s): December 15, 2006
Published electronically: October 16, 2008
Additional Notes: The first author was partially supported by NSF grant DMS-0202268. The second author was partially supported by NSF grant DMS-0500443.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.