TITLE:
Simple Choreographic Motions of N Bodies: A Preliminary Study

AUTHORS:
Alain Chenciner
Astronomie et Syst\`emes Dynamiques, IMCCE, UMR 8028 du CNRS,
77, avenue Denfert--Rochereau, 75014 Paris, France
e-mail: chencine@bdl.fr
D\'epartement de Math\'ematiques, Univ. Paris VII--Denis Diderot,
16, rue Clisson, 75013 Paris, France
e-mail: chencine@bdl.fr

Joseph Gerver
Department of Mathematics, Rutgers Univ., Camden, NJ 08102, USA
e-mail: gerver@crab.rutgers.edu

Richard Montgomery
Department of Mathematics, Univ. of California at Santa Cruz,
Santa Cruz, CA 95064, USA
e-mail: rmont@math.ucsc.edu

Carles Sim\'o
Departament de Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona,
Gran Via, 585, Barcelona 08007, Spain
e-mail: carles@maia.ub.es

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. 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 $1/r^a$, $a\ge 2$ as $r\to
0$). 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 $a \ge 2$ which disappear as $a$
tends to $1$.
