Skip to main content

Symmetry methods in collisionless many-body problems

Summary

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.

This is a preview of subscription content, access via your institution.

References

  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.

    MATH  Article  Google 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.

    MathSciNet  Article  Google Scholar 

  7. [7]

    B. Dionne, M. Golubitsky, & I. Stewart. Coupled cells with internal symmetry. Part I: wreath products,Nonlinearity,9 (1996) 559–574.

    MATH  MathSciNet  Article  Google Scholar 

  8. [8]

    B. Dionne, M. Golubitsky, & I. Stewart. Coupled cells with internal symmetry. Part 2: direct products,Nonlinearity,9 (1996) 575–599.

    MATH  MathSciNet  Article  Google 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.

    MATH  MathSciNet  Google Scholar 

  10. [10]

    M. Golubitsky & I. Stewart. Hopf bifurcation in the presence of symmetry,Arch. Ratl. Mech. Anal. 87 (1985) 107–165.

    MATH  MathSciNet  Article  Google 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.

    MATH  MathSciNet  Article  Google 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.

    MATH  Article  Google 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.

    MATH  MathSciNet  Article  Google 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.

    MATH  MathSciNet  Google Scholar 

  20. [20]

    J. A. Montaldi, R. M. Roberts, & I. Stewart. Existence of nonlinear modes of symmetric Hamiltonian systems,Nonlinearity 3 (1990) 695–730.

    MATH  MathSciNet  Article  Google Scholar 

  21. [21]

    J. Moser. Periodic orbits near equilibrium and a theorem by Alan Weinstein,Commun. Pure Appl. Math. 29 (1976) 727–747.

    MATH  Google 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.

  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) 47–57

    MATH  MathSciNet  Article  Google Scholar 

  26. [26]

    Z. Xia. Arnold diffusion and oscillatory solutions in the planar three-body problem,J. Diff. Eq. 110 (1994) 289–321.

    MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

This paper is dedicated to the memory of Juan C. Simo

This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.

Communicated by Jerrold Marsden and Stephen Wiggins

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Stewart, I. Symmetry methods in collisionless many-body problems. J Nonlinear Sci 6, 543–563 (1996). https://doi.org/10.1007/BF02434056

Download citation

Keywords

  • Periodic Solution
  • Periodic Orbit
  • Hopf Bifurcation
  • Symmetric Group
  • Wreath Product