View from the Pennines: Newtonian Choreographies

Brass Band competitions are still held in our village. Local bands play marches at the Band Room and hymns outside one of the village pubs. Their performances are marked by a panel of judges. I never get to hear the marches, but this year the almost disorderly procession of bands between the two venues reminded me of some of the new periodic orbits which have been found for the N−body problem. This orbit is shown in Figure 1a (11 on a 10 chain, see [3]). Eleven particles move from left to right and back again, weaving up and down, miraculously avoiding collisions at the crossing points of the path. Although this analogy should not be taken too far, several other marches of Newtonian particles are shown in Figure 1. The differential equations which determine the motion of N point masses subject to the forces of Newtonian gravity are easy to write down but hard to analyse. Most first year undergraduates learn that the two body problem is integrable and that the orbits can be described after noting some conservation laws and changing variables (does the transformation u = 1r ring a bell?). However, if there are more than two bodies then the system can have chaotic orbits. Given this level of complexity it is natural to restrict attention either to simplifications of the general case (perturbation theory for example), or to special solutions of the full problem.