Symmetry Methods in Collisionless Many-Body Problems
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 Montaldi et 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 Dionne et 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 Davies et al. and some suggestions for further work.
KeywordsPeriodic Solution Periodic Orbit Hopf Bifurcation Symmetric Group Wreath Product
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- R. Abraham & J. E. Marsden. Foundations of Mechanics. Benjamin/Cummings, Reading, MA, 1985.Google Scholar
- J. F. Adams, Lectures on Lie Groups. Benjamin/Cummings, New York, 1969.Google Scholar
- M. Golubitsky, I. Stewart, & B. Dionne. Coupled cells: wreath products and direct products, in Dynamics, Bifurcation, and Symmetry, ed. P. Chossat. Proceedings, Cargèse 1993, NATO AST Series C 437, Kluwer, Dordrecht, 1994, 127–138.Google Scholar
- A. M. Liapunov. The general problems of the stability of motion, Doctoral Dissertation, University of Kharkhov 1892, published by Kharkhov Math. Soc. English trans]. (transi. and ed. A. T. Fuller), Taylor and Francis, London, 1992.Google Scholar
- R. M. Roberts. Nonlinear normal modes of the spring pendulum, in Papers Presented to Christopher Zeeman, unpublished duplicated notes, Math. Inst. U. Warwick, June 1988, 207–216.Google Scholar
- V. S. Varadarajan. Lie Groups, Lie Algebras, and Their Representations, Graduate Texts in Math. 102. Springer-Verlag, New York, 1984.Google Scholar