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ISSN 1088-6850(e) ISSN 0002-9947(p)

From the restricted to the full three-body problem

Author(s): Kenneth R. Meyer; Dieter S. Schmidt
Journal: Trans. Amer. Math. Soc. 352 (2000), 2283-2299.
MSC (2000): Primary 70F05, 37N05
Posted: 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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