Canonical coordinates for the planetary problem

Abstract: In 1963, V. I. Arnold stated his celebrated Thorem on the Stability of Planetary Motions. The general proof of his wonderful statement (that he provided completely only for the particular case of three bodies constrained in a plane) turned out to be more difficult than expected and was next completed by J. Laskar, P. Robutel, M. Herman, J. F\'ejoz, L. Chierchia and the author. We refer the reader to the technical papers \cite{arnold63, laskarR95, rob95, maligeRL02, herman09, fej04, pinzari-th09, ChierchiaPi11b} for detailed information; to \cite{fejoz13, chierchia13, chierchiaPi14}, or the introduction of \cite{pinzari13} for reviews. The complete understanding of Arnold's Theorem relied on an analytic part and a geometric one, both highly non trivial. Of such two aspects, the analytic part was basically settled out since \cite{arnold63} (notwithstanding refinements next given in \cite{fejoz04, chierchiaPi10}). The geometrical aspects were instead mostly unexplored after his 1963's paper and have been only recently clarified \cite{pinzari-th09, chierchiaPi11b}. In fact, switching from the three--body case to the many--body one needed to develop new constructions, because of a dramatic degeneracy due to its invariance by rotations, which, if not suitably treated, prevents the application of Arnold's 1963's strategy. The purpose of this note is to provide a historical survey of this latter part. We shall describe previous classical approaches going back to Delaunay, Poincar\'e, Jacobi and point out more recent progresses, based on the papers \cite{pinzari-th09, chierchiaPi11a, chierchiaPi11b, chierchiaPi11c, pinzari13, pinzari14}. In the final part of the paper we discuss a set of coordinates recently found by the author which reduces completely its integrals, puts the unperturbed part in Keplerian form, preserves the symmetry by reflections and is regular for zero inclinations.