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Linear Stability of the n-gon Relative Equilibria of the (1 + n)-Body Problem

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Abstract

We consider the linear stability of the regular n-gon relative equilibria of the (1 + n)-body problem. It is shown that there exist at most two kinds of infinitesimal bodies arranged alternatively at the vertices of a regular n-gon when n is even, and only one set of identical infinitesimal bodies when n is odd. In the case of n even, the regular n-gon relative equilibrium is shown to be linearly stable when \({n \geqslant 14}\). In each case of n = 8, 10 and 12, linear stability can also be preserved if the ratio of two kinds of masses belongs to an open interval. When n is odd, the related conclusion on the linear stability is recalled.

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References

  1. Maxwell, J.C.: On the stability of the motion of Saturn’s rings, An essay, which obtained the Adams Prize for the year 1956, in the University of Cambridge, pp. 1–71. Macmillan and CO, Cambridge (1859)

  2. Tisserand, F.: Traité de Mécanique Céleste, Tome II (1889), pp. 171–185. Gauthier-Villars, Paris (1960)

  3. Pendse C.G.: The theory of Saturn’s rings. Philos. Trans. R. Soc. Lond. Ser. A Math.Phys. Sci. 234(735), 145–176 (1935)

    Article  Google Scholar 

  4. Scheeres D.J., Vinh N.X.: Linear stability of a self-gravitating ring. Celest. Mech. Dyn. Astron. 51, 83–103 (1991)

    Article  MATH  Google Scholar 

  5. Roberts, G.E.: Linear stability in the (1 + n)-gon relative equilibrium. In: Delgado, J. (ed.) Hamiltonian Systems and Celestial Mechanics, vol. 303. River Edge, Singapore (2000)

  6. Vanderbei R.J., Kolemen E.: Linear stability of ring systems. Astron. J. 133, 656–664 (2007)

    Article  Google Scholar 

  7. Hall, G.R.: Central configurations in the planar 1 + n body problem (preprint)

  8. Moeckel R.: Linear stability of relative equilibria with a dominant mass. J. Dyn. Diff. Equ. 6(1), 37–51 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Casasayas J., Llibre J., Nunes A.: Central configurations of the planar 1 + N-body problem. Celes. Mech. Dyn. Astron. 60, 273–288 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Salo H., Yoder C.F.: The dynamics of coorbital satellite systems. Astron. Astrophys. 205, 309–327 (1988)

    Google Scholar 

  11. Cors J.M., Llibre J., Olle M.: Central configurations of the planar coorbital satellite problem. Celest. Mech. Dyn. Astron. 89, 319–342 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Albouy A., Fu Y.: Relative equilibria of four identical satellites. Proc. R. Soc. A. 465, 2633–2645 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Renner S., Sicardy B.: Stationary configurations for co-orbital satellites with small arbitrary masses. Celest. Mech. Dyn. Astron. 88, 397–414 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Corbera M., Cors J.M., Llibre J.: On the central configurations of the planar 1 + 3-body problem. Celest. Mech. Dyn. Astron. 109, 27–43 (2011)

    Article  MathSciNet  Google Scholar 

  15. Gray R.M.: Toeplitz and circulant matrices: a review. Found. Trends Commun. Inf. Theory 2(3), 155–239 (2006)

    Article  Google Scholar 

  16. Meyer K.R., Schmidt K.R.: Librations of central configurations and braided Saturn rings. Celest. Mech. Dyn. Astron. 55, 289–303 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pascual F.G.: On periodic perturbations of uniform motion of Maxwell’s planetary ring. J. Dyn. Diff. Equ. 10, 47–72 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Purple Mountain Observatory, Chinese Academy of Sciences, No. 2, West Beijing Road, Nanjing, 210008, China

    Xingbo Xu

  2. Graduate University of Chinese Academy of Sciences, Beijing, 100039, China

    Xingbo Xu

  3. Observatoire de Paris, CNRS-UMR 8028, 77, Avenue Denfert-Rochereau, 75014, Paris, France

    Xingbo Xu

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  1. Xingbo Xu
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Xu, X. Linear Stability of the n-gon Relative Equilibria of the (1 + n)-Body Problem. Qual. Theory Dyn. Syst. 12, 255–271 (2013). https://doi.org/10.1007/s12346-012-0089-6

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  • DOI: https://doi.org/10.1007/s12346-012-0089-6

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