On the homology of the integral manifolds in the planar N-body problem

Abstract

In the planar N-body problem, N point masses move in the plane under their mutual gravitational attraction. It is classical that the dynamics of this motion conserves the integrals of motion: center of mass, linear momentum, angular momentum c, and energy h. Further, the motion has a rotational symmetry. The dynamics thus takes place on a (4N-7)-dimensional open manifold, known as the reduced integral manifold [mfr ]_R(M,ν). The topology of this manifold depends only on the masses M = (m₁,...,m_N) and the quantity ν = -hc². In spite of the central importance of this manifold in a classical dynamical problem, very little is known about the topology of [mfr ]_R(M,ν). In this note, we build on the topological analysis of Smale to describe the homology of [mfr ]_R(M,ν). A variety of homological results are presented, including the computation of the homology groups for ν very large for all M; and for all ν for three masses, and for four equal masses.