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On the relative equilibrium configurations in the planar five-body problem

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Abstrakty
EN
The number of central configurations in the Grebenicov-Elmabsout model of the planar five-body problem is estimated. An appropriate rational parameterization is used to reduce the equations defining such configurations to some polynomial ones. For the restricted five-body problem a sharp estimation is given by using the Sturm separation theorem.
Rocznik
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495--506
Opis fizyczny
Bibliogr. 24 poz., rys.
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autor
Bibliografia
  • [1] K. Meyer, G.R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, New York, 1992.
  • [2] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton University Press, 1941.
  • [3] F.R. Moulton, The straight line solutions of the problem of n bodies, Ann. Math. 12 (1910), 1–17.
  • [4] S. Smale, Mathematical problems for the next century, Math. Intell. 20 (1998), 7–15.
  • [5] W.D. Macmilian, W. Bartky, Configurations in the problem of four bodies, Trans. Amer. Math. Soc. 34 (1932), 838–875.
  • [6] W.L. Wiliams, Configurations in the problem of five bodies, Trans. Amer. Math. Soc. 44 (1938), 562–579.
  • [7] A. Albouy, The symmetric central configurations of four equal masses, Contemp. Math. 5 (1996) 198, 131–135.
  • [8] M. Hampton, R. Moeckel, Finiteness of relative equilibria of the four-body problem, Invent. Math. 163 (2006), 289–312.
  • [9] B. Elmabsout, Sur l’éxistence de certaines positions d’équilibre relatif dans le probléme des n corps, Celest. Mech. 41 (1988), 131–151.
  • [10] E.A. Grebenicov, Two new dynamical models in celestial mechanics, Rom. Astron. J. 10 (1998) 1, 13–19.
  • [11] J. Llibre, L.F. Mello, New central configurations for the planar 5-body problem, Celestial Mech. Dynam. Astronom. 100 (2008), 141–149.
  • [12] M. Hampton, Stacked central configurations: new examples in the five-body problem, Nonlinearity 18 (2005), 2299–2304.
  • [13] J. Llibre, L.F. Mello, New central configurations for the planar 7-body problem, Nonlinear Anal. Real World Appl. 10 (2009), 2246–2255.
  • [14] E.A. Grebenicov, The existence of the exact symmetric solutions in the plane Newton problem of many bodies, Matem. Model. 10 (1988) 8, 74–80.
  • [15] E.A. Grebenicov, D. Kozak-Skoworodkina, M. Jakubiak, Computer Algebra Methods in the n-Body Problem, Moskwa, 2002 [in Russian].
  • [16] B. Elmabsout, Nouvelles configurations d’equilibre relatif pour le probléme des N corps, C.R. Acad. Sci., Serie II 312 (1991), 467–472.
  • [17] A. Siluszyk, On the linear stability of relative equilibria in the restricted eight-body problem with partial symmetry, Vesnik Brestcaga Universiteta 2 (2004), 20–26.
  • [18] A. Siluszyk, On the Lyapunov stability of relative equilibria in the restricted eight-body problem with partial symmetry, Vesnik Grodzenskaga Universiteta im. J.Kupaly, Ser. 2 (2005) 2, 77–85.
  • [19] E.S.G. Leandro, Finitness and bifurcations of some symmetrical classes of central configurations, Arch. Rational Mech. Anal. 167 (2003), 147–177.
  • [20] W. Barwicz, H. Zoladek, The restricted three body problem revisited, J. Math. Anal. Appl. 366 (2010) 2, 663–672.
  • [21] H. Li, F. van Oystaeyen, A Primer of Algebraic Geometry, New York, 2000.
  • [22] S. Wolfram Mathematica-Book, Cambridge, University Press, 1996.
  • [23] D. Bang, B. Elmabsout, Configurations polygonales d’equilibre relatif, C.R. Acad. Sci. 329, Série IIb, (2001), 243–248.
  • [24] Z. Fortuna, B. Macukov, J. Wasowski, Numerical Methods, WNT, Warsaw, 1993 [in Polish].
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHT-0003-0021
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