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

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.

17 pages. Related papers: [V.I. Arnold. Russ. Math. Surv. 1963], [P. Robutel. Cel. Mech Dys Astr. 1995], [J. Féjoz. Erg. Th. Dyn Syst.2004], [G. Pinzari. PhD Dissertation, 2009; arXiv:1309.7028], [L. Chierchia and G. Pinzari. Invent. Math. 2011]. Acknowledgments and microscopic changes added