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The n-Centre Problem of Celestial Mechanics

Knauf A (2001)

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Publication Type: Journal article, Original article

Publication year: 2001

Journal

Publisher: European Mathematical Society

Book Volume: 4

Pages Range: 1-114

DOI: 10.1007/s100970100037

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-{\em 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.

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How to cite

APA:

Knauf, A. (2001). The n-Centre Problem of Celestial Mechanics. Journal of the European Mathematical Society, 4, 1-114. https://dx.doi.org/10.1007/s100970100037

MLA:

Knauf, Andreas. "The n-Centre Problem of Celestial Mechanics." Journal of the European Mathematical Society 4 (2001): 1-114.

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