Elliptic Two-Dimensional Invariant Tori for the Planetary Three-Body Problem

Satellite orbits design using frequency analysis

“…An essential ingredient that is also lacking for a complete proof, is the demonstration that the final periodic or quasiperiodic orbit that we are looking for really exists in the full perturbed problem. Significant progress on this particular question related to the KAM theorem have however been made recently by Biasco et al (2003Biasco et al ( , 2006, showing that this orbit nearly surely exists at least for the ðN þ 1Þ-body problem.…”

KAM tori for N-body problems: a brief history

“…are symplectic and analytic near circular, non-co-planar motions (for details, see, e.g., Biasco et al 2003). Denote by Then, for any + > − > 0 there exists r > 0 such that the planar Poincaré variables are symplectic and analytic on the domain − < i < + for 0 i n, (λ 1 , .…”

Flow reconstruction and invariant tori in the spatial three-body problem

“…The existence of quasi-periodic motions for almost all values of the ratio of the semi-major axis and almost all values of the mutual inclination up to about one degree is proved. Biasco, Chierchia and Valdinoci [5] deal with the case of lower-dimensional tori, proving the existence of two-dimensional KAM tori in the spatial three-body problem. Féjoz [22] (following Herman) gives a complete proof of 'Arnold's Theorem' on the planetary N -body problem, establishing the existence of a positive measure set of smooth Lagrangian invariant tori.…”