On the Kolmogorov set for Many-Body Problems

TL;DR

This paper provides the first direct proof of Arnold's Planetary Theorem by rediscovering and applying a symplectic set of action-angle variables, leading to a rigorous understanding of the stability of planetary motions.

Contribution

The author introduces a new explicit symplectic set of action-angle variables for planetary systems, enabling a direct proof of Arnold's Planetary Theorem.

Findings

First direct proof of Arnold's Planetary Theorem

Rediscovery of Deprit's reduction of rotation invariance

Development of regularized planetary variables

Abstract

I defended my PhD Thesis in Rome, Universit\`a Roma Tre, on April, 23, 2009, under the direction of Professor Luigi Chierchia. The judging committee was composed by Professors M. Berti, A. Celletti, C. Falcolini, J. F\'ejoz. Professors M. Berti and J. F\'ejoz refereed my thesis. The main result of my thesis is the first direct proof (the first general proof was given in [J. F\'ejoz, ETDS, 2004]) of a famous statement by V. I. Arnold (1963), usually referred to as "Arnold's Planetary Theorem". My proof of Arnold's Planetary Theorem relies on the rediscovery, during the year 2008, of a symplectic set of action-angle variables (described in \S 4 of my thesis) which perform explicitly the reduction of rotation invariance of the system. Indeed, even though in a different form, they had been previously considered by [F. Boigey, Cel. Mech. Dyn. Astr., 1982] and [A. Deprit, Cel. Mech. Dyn. Astr., 1983]. The version I found in 2008 corresponds to the "planetary" form of Boigey-Deprit variables, since it includes the elliptic elements of the instantaneous ellipses of the planets around the sun and for this reason is especially fitted to this problem . I then regularized "my" planetary variables to include co-planar and co-circular motions. This regularization leads to a set of mixed action-angle and rectangular variables analogous to Poincar\'e' variables but better fitted to rotation invariance of the system, since they exhibit a cyclic couple of conjugated variables. I finally applied my regularized variables to the problem, checked non-trivial torsion and obtained the proof of the theorem. I wish to thank J. F\'ejoz for mentioning my contribution to the proof of Arnold's Theorem, and especially my rediscovery of Deprit's reduction, in his paper [J. Fejoz, DCDS-A, 2013].