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