# The $n$-centre problem of celestial mechanics for large energies

Journal of the European Mathematical Society (2002)

- Volume: 004, Issue: 1, page 1-114
- ISSN: 1435-9855

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topKnauf, Andreas. "The $n$-centre problem of celestial mechanics for large energies." Journal of the European Mathematical Society 004.1 (2002): 1-114. <http://eudml.org/doc/277244>.

@article{Knauf2002,

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 $n\ge 3$ 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.},

author = {Knauf, Andreas},

journal = {Journal of the European Mathematical Society},

keywords = {$n$-centre problem; three-dimensional motion; large energy; singularities; symbolic dynamics; Hausdorff dimension; topological entropy; hyperbolic sets; celestial mechanics; -centre problem; three-dimensional motion; large energy; singularities; symbolic dynamics; Hausdorff dimension; topological entropy; hyperbolic sets; celestial mechanics; molecular scattering; electronic potential},

language = {eng},

number = {1},

pages = {1-114},

publisher = {European Mathematical Society Publishing House},

title = {The $n$-centre problem of celestial mechanics for large energies},

url = {http://eudml.org/doc/277244},

volume = {004},

year = {2002},

}

TY - JOUR

AU - Knauf, Andreas

TI - The $n$-centre problem of celestial mechanics for large energies

JO - Journal of the European Mathematical Society

PY - 2002

PB - European Mathematical Society Publishing House

VL - 004

IS - 1

SP - 1

EP - 114

AB - 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 $n\ge 3$ 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.

LA - eng

KW - $n$-centre problem; three-dimensional motion; large energy; singularities; symbolic dynamics; Hausdorff dimension; topological entropy; hyperbolic sets; celestial mechanics; -centre problem; three-dimensional motion; large energy; singularities; symbolic dynamics; Hausdorff dimension; topological entropy; hyperbolic sets; celestial mechanics; molecular scattering; electronic potential

UR - http://eudml.org/doc/277244

ER -

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