Analytic Lagrangian tori for the planetary many-body problem

TL;DR

This paper proves the existence of a positive measure set of smooth Lagrangian invariant tori in the planetary many-body problem using KAM theory, extending previous results to the real-analytic setting.

Contribution

It establishes the existence of Lagrangian invariant tori in the planetary problem within the real-analytic category, building on and completing earlier smooth results.

Findings

Existence of positive measure set of invariant tori proven

Extension of results to real-analytic class

Utilization of Rüssmann's KAM theory for the proof

Abstract

In 2004, F\'ejoz [D\'emonstration du 'th\'eor\'eme d'Arnold' sur la stabilit\'e du syst\`eme plan\'etaire (d'apr\`es M. Herman). Ergod. Th. & Dynam. Sys. 24(5) (2004), 1521-1582], completing investigations of Herman's [D\'emonstration d'un th\'eor\'eme de V.I. Arnold. S\'eminaire de Syst\'emes Dynamiques et manuscripts, 1998], gave a complete proof of 'Arnold's Theorem' [V. I. Arnol'd. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963), 91-192] on the planetary many-body problem, establishing, in particular, the existence of a positive measure set of smooth (C\infty) Lagrangian invariant tori for the planetary many-body problem. Here, using R\"u{\ss}mann's 2001 KAM theory [H. R\"u{\ss}mann. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. R. & C. Dynamics 2(6) (2001), 119-203], we prove the above result in the real-analytic class.