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Simple Choreographic Motions of N Bodies: A Preliminary Study

In: Geometry, Mechanics, and Dynamics

Author

Listed:
  • Alain Chenciner
  • Joseph Gerver
  • Richard Montgomery
  • Carles Simó

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/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.

Suggested Citation

  • Alain Chenciner & Joseph Gerver & Richard Montgomery & Carles Simó, 2002. "Simple Choreographic Motions of N Bodies: A Preliminary Study," Springer Books, in: Paul Newton & Philip Holmes & Alan Weinstein (ed.), Geometry, Mechanics, and Dynamics, chapter 9, pages 287-308, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-21791-8_9
    DOI: 10.1007/0-387-21791-6_9
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