From the restricted to the full three–body problem
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- by Kenneth R. Meyer and Dieter S. Schmidt PDF
- Trans. Amer. Math. Soc. 352 (2000), 2283-2299 Request permission
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
- 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.
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 2283-2299
- MSC (2000): Primary 70F05, 37N05
- DOI: https://doi.org/10.1090/S0002-9947-00-02542-3
- MathSciNet review: 1694376
Dedicated: To Hugh Turrittin on his ninety birthday