On the Homology of the Integral Manifolds in the Planar N-body Problem

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 intgrals 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 m R (M;). The topology of this manifold depends only on the masses M = (m 1 ; : : : ; m N) and the quantity = ?hc 2. In spite of the central importance of this manifold in a classical dynamical problem, very little is known about the topology of m R (M;). In this note, we build on the topological analysis of Smale to describe the homology of m R (M;). A variety of homological results are presented, including the computation of the homology groups for very large for all M; and for all for three masses, and for four equal masses. 1. Introduction This note is part of a recent series of investigations 11, 12, 13] of the topology of the integral manifolds in the N-body problem. The N-body problem refers to the motion of N point masses, moving under their mutual gravitational attraction. In the planar N-body problem, the particles are all constrained to a plane. As a Hamiltonian system, the dynamics conserves the quantities of center of mass, linear and angular momentum, and energy. A level set of these rst integrals is the so-called integral manifold m. The system also admits a rotational symmetry about the angular momentum vector, and the resulting quotient manifold is the reduced integral manifold m R. The (reduced) integral manifold is an open algebraic set of dimension 4N?6 (resp. 4N?7), with the precise structure depending on the values m 1 ; : : : m N of the masses, the angular momentum c and the energy h. As the topology of the underlying integral manifold clearly innuences the dynamics, it is natural to try to understand that topology, and its dependence on the masses, angular momentum and energy. It is a classical result that, for xed masses and non-zero angular momentum, the topology of the integral manifold depends only on the