Cross-Sections in the Planar N-Body Problem

Details Cross-Sections in the Planar N-Body Problem Cross-Sections in the Planar N-Body Problem

Oct 2000

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2000hscm.conf..250m&link_type=abstract

HAMILTONIAN SYSTEMS AND CELESTIAL MECHANICS (HAMSYS-98). Proceedings of the III International Symposium. Held 7-11 December 1998

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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 mR(M, ν). The topology of this manifold depends only on the masses M = (m1,..., mN) and the quantity ν = h|c|2. In spite of the central importance of this manifold in a classical dynamical problem, very little is known about the topology of mR(M, ν). In this note, we build on the topological analysis of Smale to describe the homology of mR(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.

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