Simple Choreographies of the Planar Newtonian N-Body Problem

In the N-body problem, a simple choreography is a periodic solution, where all masses chase each other on a single loop. In this paper we prove that for the planar Newtonian N-body problem with equal masses, N ae 3, there are at least 2 (N-3) + 2([(N-3)/2]) different main simple choreographies. This confirms a conjecture given by Chenciner et al. (Geometry, mechanics, and dynamics. Springer, New York, pp 287-308, 2002). All the simple choreoagraphies we prove belong to the linear chain family.