Abstract
A “simple choreography” for an N-body problem is a periodic solution in which all N masses trace the same curve without colliding. We shall require all masses to be equal and the phase shift between consecutive bodies to be constant. The first 3-body choreography for the Newtonian potential, after Lagrange’s equilateral solution, was proved to exist by Chenciner and Montgomery in December 1999 (Chenciner and Montgomery [2000]). In this paper we prove the existence of planar N-body simple choreographies with arbitrary complexity and/or symmetry, and any number N of masses, provided the potential is of strong force type (behaving like 1/ra, a≥2 as r→0). The existence of simple choreographies for the Newtonian potential is harder to prove, and we fall short of this goal. Instead, we present the results of a numerical study of the simple Newtonian choreographies, and of the evolution with respect to a of some simple choreographies generated by the potentials 1/ra, focusing on the fate of some simple choreographies guaranteed to exist for a≥2 which disappear as a tends to 1.
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To Jerry Marsden on the occasion of his 60th birthday
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© 2002 Springer-Verlag New York, Inc.
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Chenciner, A., Gerver, J., Montgomery, R., Simó, C. (2002). Simple Choreographic Motions of N Bodies: A Preliminary Study. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-21791-6_9
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DOI: https://doi.org/10.1007/0-387-21791-6_9
Publisher Name: Springer, New York, NY
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