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

Published online by Cambridge University Press:  04 June 2001

CHRISTOPHER McCORD
Affiliation:
University of Cincinnati, Cincinnati, Ohio 45221-0025, USA (e-mail: [email protected])

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,...,mN) 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.

Type
Research Article
Copyright
2001 Cambridge University Press

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