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Celestial Mechanics and Dynamical Astronomy

, Volume 124, Issue 2, pp 145–153 | Cite as

Splendid isolation: local uniqueness of the centered co-circular relative equilibria in the N-body problem

  • Marshall Hampton
Original Article

Abstract

We study the neighborhood of the equal mass regular polygon relative equilibria in the N-body probem, and show that this relative equilibirum is isolated among the co-circular configurations (in which each point lies on a common circle) for which the center of mass is located at the center of the common circle. It is also isolated in the sense that a sufficiently small mass cannot be added to the common circle to form a \(N+1\)-body relative equilibrium. These results provide strong evidence for a conjecture that the equal mass regular polygon is the only co-circular relative equilibrium with its center of mass located at the center of the common circle.

Keywords

Central configurations Relative equilibria N-body problem Choreographies Equal mass regular polygon Co-circular configurations 

References

  1. Albouy, A., Cabral, H.E., Santos, A.A.: Some problems on the classical n-body problem. Celest. Mech. Dyn. Astron. 113, 369–375 (2012)CrossRefADSMathSciNetGoogle Scholar
  2. Barutello, V., Terracini, S.: Action minimizing orbits in the n-body problem with simple choreography constraint. Nonlinearity 17, 2015 (2004)CrossRefADSMathSciNetzbMATHGoogle Scholar
  3. Barutello, V., Ferrario, D.L., Terracini, S.: Symmetry groups of the planar three-body problem and action-minimizing trajectories. Arch. Ration. Mech. Anal. 190, 189–226 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  4. Checiner, A., Gerver, J., Montgomery, R., Simo, C.: Simple choreographies of \(N\) bodies: a preliminary study. In: Newton, P., Holmes, P., Weinstein, Alan (eds.) Geometry, Mechanics and Dynamics, pp. 287–308. Springer, New York (2002)CrossRefGoogle Scholar
  5. Chen, K.-C.: Action-minimizing orbits in the parallelogram four-body problem with equal masses. Arch. Ration. Mech. Anal. 158, 293–318 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  6. Chenciner, A.: New advances in celestial mechanics and Hamiltonian systems. Are there Perverse Choreographies?. Kluwer/Plenum, New York (2004)CrossRefGoogle Scholar
  7. Chenciner, A., Montgomery, R.: A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. Math. 153, 881–902 (2000)CrossRefMathSciNetGoogle Scholar
  8. Chenciner, A., Venturelli, A.: Minima de l’intégrale d’action du probléme Newtoniende 4 corps de masses égales dans \(R^3\): orbites ‘hip-hop’. Celest. Mech. Dyn. Astron. 77, 139–151 (2000)CrossRefADSMathSciNetzbMATHGoogle Scholar
  9. Cors, J.M., Hall, G.R., Roberts, G.E.: Uniqueness results for co-circular central configurations for power-law potentials. Phys. D Nonlinear Phenom. 280, 44–47 (2014)CrossRefADSMathSciNetGoogle Scholar
  10. Ferrario, D.J., Terracini, S.: On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. Math. 155, 305–362 (2004)CrossRefADSMathSciNetzbMATHGoogle Scholar
  11. Hampton, M.: Co-circular central configurations in the four-body problem. Equadiff 2003, 993–998 (2003)Google Scholar
  12. Helmholtz, H.: Uber Integrale der hydrodynamischen Gleichungen, Welche den Wirbelbewegungen entsprechen. Crelle’s J. für Math. 55, 25–55 (1858). English translation by P. G. Tait, P.G. On the integrals of the hydrodynamical equations which express vortex-motion, Philosophical Magazine (1867), 485–51CrossRefMathSciNetzbMATHGoogle Scholar
  13. Longley, W.R.: Some particular solutions in the problem of \(n\)-bodies. Bull. Am. Math. Soc. 7, 324–335 (1907)CrossRefMathSciNetGoogle Scholar
  14. Maxwell, J.C.: in Maxwell on Saturn’s rings. MIT press, Cambridge (1983)Google Scholar
  15. Moeckel, R.: Linear stability of relative equilibria with a dominant mass. J. Dyn. Diff. Equ. 6, 37–51 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  16. Newton, I.: Philosophi Naturalis Principia Mathematica. Royal Society, London (1687)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Minnesota, DuluthDuluthUSA

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