Chazy-Type Asymptotics and Hyperbolic Scattering for the n-Body Problem

We study solutions of the Newtonian n -body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as t → + ∞ or as t → - ∞ . In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium poi...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2020-10, Vol.238 (1), p.255-297
Main Authors: Duignan, Nathan, Moeckel, Richard, Montgomery, Richard, Yu, Guowei
Format: Article
Language:English
Subjects:
Asymptotic properties
Classical Mechanics
Complex Systems
Fluid- and Aerodynamics
Infinity
Manifolds
Mathematical analysis
Mathematical and Computational Physics
Physics
Physics and Astronomy
Scattering
Theoretical
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Summary:We study solutions of the Newtonian n -body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as t → + ∞ or as t → - ∞ . In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold “at infinity”. We show that the flow near these manifolds can be analytically linearized and use this to give a new proof of Chazy’s classical asymptotic formulas. We also address the scattering problem, namely: for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related? After proving some basic theorems about this scattering relation, we use perturbations of our manifold at infinity to study scattering “near infinity”, that is, when the bodies stay far apart and interact only weakly.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-020-01542-2