The Integral Manifolds of the N Body Problem

Abstract

The “N-Body Problem” refers to the study of the dynamics of a system of N point masses, moving under their mutual gravitational attraction. For such a system, it is classical that there are conserved quantities of center of mass, linear momentum, angular momentum and energy. The level sets $${\mathfrak {M}}(c,h)$$ of these conserved quantities are parameterized by the angular momentum c and the energy h, and are known as the integral manifolds. In spite of their fundamental importance in the N-body problem, relatively little is known about the global structure of these spaces. In particular, for non-zero angular momentum, the manifolds exhibit a variety of discontinuities and complexities at collinear configurations. These discontinuities are removed by the introduction of a blow-up construction, which is then exploited to provide a description of the integral manifolds, in the sense of a reduction formula for the homology groups of the manifolds. That reduction formula is then applied to investigate the topology of the integral manifolds for energies just below zero. For four masses, the homology groups for this energy range are fully determined; for arbitrary N, the homology with real coefficients is determined. Comparison to the values for positive energies confirms that for all masses, the topology of the integral manifold changes at zero energy.