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 $1$.