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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

From the restricted to the full three-body problem


Authors: Kenneth R. Meyer and Dieter S. Schmidt
Journal: Trans. Amer. Math. Soc. 352 (2000), 2283-2299
MSC (2000): Primary 70F05, 37N05
Published electronically: February 16, 2000
MathSciNet review: 1694376
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Abstract:

The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem.

For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.


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Additional Information

Kenneth R. Meyer
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221–0025
Email: ken.meyer@uc.edu

Dieter S. Schmidt
Affiliation: Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, Cincinnati, Ohio 45221–0030
Email: dieter.schmidt@uc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02542-3
PII: S 0002-9947(00)02542-3
Keywords: Restricted three--body problem, three--body problem, reduction, symplectic scaling, normal forms, KAM theory
Received by editor(s): July 22, 1997
Received by editor(s) in revised form: January 20, 1998
Published electronically: February 16, 2000
Additional Notes: This research was partially supported by grants from the National Science Foundation and the Taft Foundation.
Dedicated: To Hugh Turrittin on his ninety birthday
Article copyright: © Copyright 2000 American Mathematical Society