Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

The n-centre problem of celestial mechanics for large energies

The n-centre problem of celestial mechanics for large energies We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold. Whereas for n=1 there are no bounded orbits, and for n=2 there is just one closed orbit, for n≥3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set. Then we set up scattering theory, including symbolic dynamics of the scattering orbits and differential cross section estimates. The theory includes the n–centre problem of celestial mechanics, and prepares for a geometric understanding of a class of restricted n-body problems. To allow for applications in semiclassical molecular scattering, we include an additional smooth (electronic) potential which is arbitrary except its Coulombic decay at infinity. Up to a (optimal) relative error of order 1/E, all estimates are independent of that potential but only depend on the relative positions and strengths of the centres. Finally we show that different, non-universal, phenomena occur for collinear configurations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

The n-centre problem of celestial mechanics for large energies

Loading next page...
 
/lp/springer-journals/the-n-centre-problem-of-celestial-mechanics-for-large-energies-MOEsPWkGOe
Publisher
Springer Journals
Copyright
Copyright © 2001 by Springer-Verlag Berlin Heidelberg & EMS
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/s100970100037
Publisher site
See Article on Publisher Site

Abstract

We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold. Whereas for n=1 there are no bounded orbits, and for n=2 there is just one closed orbit, for n≥3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set. Then we set up scattering theory, including symbolic dynamics of the scattering orbits and differential cross section estimates. The theory includes the n–centre problem of celestial mechanics, and prepares for a geometric understanding of a class of restricted n-body problems. To allow for applications in semiclassical molecular scattering, we include an additional smooth (electronic) potential which is arbitrary except its Coulombic decay at infinity. Up to a (optimal) relative error of order 1/E, all estimates are independent of that potential but only depend on the relative positions and strengths of the centres. Finally we show that different, non-universal, phenomena occur for collinear configurations.

Journal

Journal of the European Mathematical SocietySpringer Journals

Published: Mar 1, 2002

There are no references for this article.