Simple Choreographic Motions of N Bodies: A Preliminary Study
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 ). 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/r a , 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/r a , focusing on the fate of some simple choreographies guaranteed to exist for a≥2 which disappear as a tends to 1.
KeywordsPeriodic Solution Periodic Orbit Periodic Point Linear Chain Rotation Number
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