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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
73/2000

The n-Centre Problem of Celestial Mechanics

Andreas Knauf

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 (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 smooth 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.

Received:
Nov 6, 2000
Published:
Nov 6, 2000

Related publications

inJournal
2002 Repository Open Access
Andreas Knauf

The n-centre problem of celestial mechanics for large energies

In: Journal of the European Mathematical Society, 4 (2002) 1, pp. 1-114