Global Kolmogorov tori in the planetary N-body problem. Announcement of result

TL;DR

This paper proves the existence of a positive measure set of quasi-periodic motions in the spatial planetary N-body problem, extending previous results and showing continuity with the planar case as inclinations vanish.

Contribution

It improves prior results by establishing quasi-periodic motions in the spatial problem with full symmetry reduction, independent of eccentricities and inclinations.

Findings

Existence of positive measure set of quasi-periodic motions in spatial N-body problem.

Continuity of these motions with the planar problem as inclinations tend to zero.

Full SO(3) symmetry reduction reveals a quasi-integrable structure.

Abstract

We improve a result in [L. Chierchia and G. Pinzari, Invent. Math. 2011] by proving the existence of a positive measure set of $(3n-2)$--dimensional quasi--periodic motions in the spacial, planetary $(1+n)$--body problem away from co--planar, circular motions. We also prove that such quasi--periodic motions reach with continuity corresponding $(2n-1)$--dimensional ones of the planar problem, once the mutual inclinations go to zero (this is related to a speculation in [V. I. Arnold. Russ. Math. Surv. 1963]). The main tool is a full reduction of the SO(3)--symmetry, which, in particular, retains symmetry by reflections and highlights a quasi--integrable structure, with a small remainder, independently of eccentricities and inclinations.