Invariant tori in the lunar problem. (English) Zbl 1365.70010

Meyer, Kenneth R.; Palacián, Jesús F.; Yanguas, Patricia

Publ. Mat., Barc. 2014, Extra, 353-394 (2014).

In a previous paper the authors considered existence, stability and characteristic multipliers for periodic solutions of a Hamiltonian system that is a perturbation of a circle bundle flow. They obtained a reduced system that arises from the orbit space of the circle bundle by averaging the perturbation over the fibers. They then applied the results to the lunar problem of celestial mechanics (the restricted three-body problem wherein the particle of smallest mess is close to one of the two primary bodies).
In the current paper the authors sharpen the general stability theorems and show when a degenerate twist is sufficient to establish invariant tori near a periodic solution. Then they apply the more general results to show that the circular periodic solutions of the planar lunar problem are enclosed by invariant 2-tori. In the three-dimensional case, the authors show that there are invariant 3-tori enclosing the periodic solutions that are near circular equatorial or near rectilinear in the vertical axis.

Reviewer: William J. Satzer Jr. (St. Paul)

MSC:

70F10	\(n\)-body problems
70K50	Bifurcations and instability for nonlinear problems in mechanics
70K65	Averaging of perturbations for nonlinear problems in mechanics
37J15	Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37J40	Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
53D20	Momentum maps; symplectic reduction

Keywords:

KAM tori; averaging; symmetry reduction; orbit space; restricted three-body problem