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.
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References
R. Abraham & J. E. Marsden.Foundations of Mechanics. Benjamin/Cummings, Reading, MA, 1985.
J. F. Adams,Lectures on Lie Groups. Benjamin/Cummings, New York, 1969.
J. Binney & S. Tremaine,Galactic Dynamics. Princeton Unuversity Press, Princeton, NJ, 1987.
T. Bröcker & T. tom Dieck.Representations of Compact Lie Groups. Springer-Verlag, New York, 1985.
J. J. Collins & I. Stewart. Hexapodal gaits and coupled nonlinear oscillator models,Biol. Cybernet. 68 (1993) 287–298.
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.
B. Dionne, M. Golubitsky, & I. Stewart. Coupled cells with internal symmetry. Part I: wreath products,Nonlinearity,9 (1996) 559–574.
B. Dionne, M. Golubitsky, & I. Stewart. Coupled cells with internal symmetry. Part 2: direct products,Nonlinearity,9 (1996) 575–599.
M. Golubitsky, J. E. Marsden, I. Stewart & M. Dellnitz. The constrained Liapunov-Schmidt procedure and periodic orbits,Fields Inst. Commun. 4 (1995) 81–127.
M. Golubitsky & I. Stewart. Hopf bifurcation in the presence of symmetry,Arch. Ratl. Mech. Anal. 87 (1985) 107–165.
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.
M. Golubitsky, I. Stewart & D. G. Schaeffer,Singularities and Groups in Bifurcation Theory, Vol. 2, Springer-Verlag, New York, 1988.
M. Hall,The Theory of Groups. Macmillan, New York, 1959.
A. A. Kirillov,Elements of the Theory of Representations. Springer-Verlag, Berlin, 1976.
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.
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.
K. R. Meyer. Periodic solutions of theN-body problem,J. Diff. Eq. 39 (1981) 2–38.
K. R. Meyer & D. S. Schmidt. Librations of central configurations and braided Saturn rings,Celest. Mech. Dyn. Astron. 55 (1993) 289–303.
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.
J. A. Montaldi, R. M. Roberts, & I. Stewart. Existence of nonlinear modes of symmetric Hamiltonian systems,Nonlinearity 3 (1990) 695–730.
J. Moser. Periodic orbits near equilibrium and a theorem by Alan Weinstein,Commun. Pure Appl. Math. 29 (1976) 727–747.
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.
D. H. Sattinger. Branching in the presence of symmetry,CBMS-NSF Conference Notes 40, SIAM, Philadelphia, 1983, pp. 1–73.
V. S. Varadarajan.Lie Groups, Lie Algebras, and Their Representations, Graduate Texts in Math.102. Springer-Verlag, New York, 1984.
A. Weinstein. Normal modes for nonlinear Hamiltonian systems,Invent. Math. 20 (1973) 47–57
Z. Xia. Arnold diffusion and oscillatory solutions in the planar three-body problem,J. Diff. Eq. 110 (1994) 289–321.
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Communicated by Jerrold Marsden and Stephen Wiggins
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.
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Stewart, I. Symmetry methods in collisionless many-body problems. J Nonlinear Sci 6, 543–563 (1996). https://doi.org/10.1007/BF02434056
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DOI: https://doi.org/10.1007/BF02434056