Journal of Nonlinear Science

, Volume 6, Issue 6, pp 543–563 | Cite as

Symmetry methods in collisionless many-body problems

  • I. Stewart


We formulate an appropriate symmetry context for studying periodic solutions to equal-mass many-body problems in the plane and 3-space. In a technically tractable but unphysical case (attractive force a smooth function of squared distance, bodies permitted to coincide) we apply the equivariant Moser-Weinstein Theorem of Montaldiet al. to prove the existence of various symmetry classes of solutions. In so doing we expoit the direct product structure of the symmetry group and use recent results of Dionneet al. on ‘C-axial’ isotropy subgroups. Along the way we obtain a classification of C-axial subgroups of the symmetric group. The paper concludes with a speculative analysis of a three-dimensional solution to the 2n-body problem found by Davieset al. and some suggestion for further work.


Periodic Solution Periodic Orbit Hopf Bifurcation Symmetric Group Wreath Product 
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  1. [1]
    R. Abraham & J. E. Marsden.Foundations of Mechanics. Benjamin/Cummings, Reading, MA, 1985.Google Scholar
  2. [2]
    J. F. Adams,Lectures on Lie Groups. Benjamin/Cummings, New York, 1969.Google Scholar
  3. [3]
    J. Binney & S. Tremaine,Galactic Dynamics. Princeton Unuversity Press, Princeton, NJ, 1987.Google Scholar
  4. [4]
    T. Bröcker & T. tom Dieck.Representations of Compact Lie Groups. Springer-Verlag, New York, 1985.Google Scholar
  5. [5]
    J. J. Collins & I. Stewart. Hexapodal gaits and coupled nonlinear oscillator models,Biol. Cybernet. 68 (1993) 287–298.zbMATHCrossRefGoogle Scholar
  6. [6]
    I. Davies, A. Truman, & D. Williams. Classical periodic solutions of the equal-mass 2n-body problem, 2n-ion problem, and then-electron atom problem,Phys. Lett. A99 (1983) 15–18.MathSciNetCrossRefGoogle Scholar
  7. [7]
    B. Dionne, M. Golubitsky, & I. Stewart. Coupled cells with internal symmetry. Part I: wreath products,Nonlinearity,9 (1996) 559–574.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    B. Dionne, M. Golubitsky, & I. Stewart. Coupled cells with internal symmetry. Part 2: direct products,Nonlinearity,9 (1996) 575–599.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    M. Golubitsky, J. E. Marsden, I. Stewart & M. Dellnitz. The constrained Liapunov-Schmidt procedure and periodic orbits,Fields Inst. Commun. 4 (1995) 81–127.zbMATHMathSciNetGoogle Scholar
  10. [10]
    M. Golubitsky & I. Stewart. Hopf bifurcation in the presence of symmetry,Arch. Ratl. Mech. Anal. 87 (1985) 107–165.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    M. Golubitsky, I. Stewart, & B. Dionne. Coupled cells: wreath products and direct products, inDynamics, Bifurcation, and Symmetry, ed. P. Chossat. Proceedings, Cargèse 1993, NATO ASI Series C437, Kluwer, Dordrecht, 1994, 127–138.Google Scholar
  12. [12]
    M. Golubitsky, I. Stewart & D. G. Schaeffer,Singularities and Groups in Bifurcation Theory, Vol. 2, Springer-Verlag, New York, 1988.Google Scholar
  13. [13]
    M. Hall,The Theory of Groups. Macmillan, New York, 1959.Google Scholar
  14. [14]
    A. A. Kirillov,Elements of the Theory of Representations. Springer-Verlag, Berlin, 1976.Google Scholar
  15. [15]
    M. Kroon & I. N. Stewart. Detecting the symmetry of attractors for six oscillators coupled in a ring,Int. J. Bifurcations Chaos 5 (1995) 209–229.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    A. M. Liapunov. The general problems of the stability of motion, Doctoral Dissertation, University of Kharkhov 1892, published by Kharkhov Math. Soc. English transl. (transl. and ed. A. T. Fuller), Taylor and Francis, London, 1992.Google Scholar
  17. [17]
    K. R. Meyer. Periodic solutions of theN-body problem,J. Diff. Eq. 39 (1981) 2–38.zbMATHCrossRefGoogle Scholar
  18. [18]
    K. R. Meyer & D. S. Schmidt. Librations of central configurations and braided Saturn rings,Celest. Mech. Dyn. Astron. 55 (1993) 289–303.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    J. A. Montaldi, R. M. Roberts, & I. Stewart. Periodic solutions near equilibria of symmetric Hamiltonian systems,Phil. Trans. R. Soc. Lond. A325 (1988) 237–293.zbMATHMathSciNetGoogle Scholar
  20. [20]
    J. A. Montaldi, R. M. Roberts, & I. Stewart. Existence of nonlinear modes of symmetric Hamiltonian systems,Nonlinearity 3 (1990) 695–730.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    J. Moser. Periodic orbits near equilibrium and a theorem by Alan Weinstein,Commun. Pure Appl. Math. 29 (1976) 727–747.zbMATHGoogle Scholar
  22. [22]
    R. M. Roberts. Nonlinear normal modes of the spring pendulum, inPapers Presented to Christopher Zeeman, unpublished duplicated notes, Math. Inst. U. Warwick, June 1988, 207–216.Google Scholar
  23. [23]
    D. H. Sattinger. Branching in the presence of symmetry,CBMS-NSF Conference Notes 40, SIAM, Philadelphia, 1983, pp. 1–73.Google Scholar
  24. [24]
    V. S. Varadarajan.Lie Groups, Lie Algebras, and Their Representations, Graduate Texts in Math.102. Springer-Verlag, New York, 1984.Google Scholar
  25. [25]
    A. Weinstein. Normal modes for nonlinear Hamiltonian systems,Invent. Math. 20 (1973) 47–57zbMATHMathSciNetCrossRefGoogle Scholar
  26. [26]
    Z. Xia. Arnold diffusion and oscillatory solutions in the planar three-body problem,J. Diff. Eq. 110 (1994) 289–321.zbMATHCrossRefGoogle Scholar

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© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • I. Stewart
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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