On the co--existence of maximal and whiskered tori for the planetary three--body problem

TL;DR

This paper investigates the coexistence of stable and unstable quasi-periodic tori in the three-body problem using KAM theory and alternative coordinate systems, revealing complex phase space structures.

Contribution

It introduces a novel approach employing two non-smooth coordinate systems to demonstrate coexistence of different types of tori in the three-body problem.

Findings

Existence of coexisting stable and unstable tori in phase space.

Application of Nekhorossev and Mischenko-Fomenko theorem to celestial mechanics.

Use of alternative canonical coordinates to classical reduction methods.

Abstract

In this paper we discuss about the possibility of {\it coexistence} of stable and unstable quasi--periodic {\sc kam} tori in a region of phase space of the three-body problem. The {argument of proof} goes along {{\sc kam} theory and, especially,} the production of two non smoothly related systems of canonical coordinates in the same region of the phase space, the possibility of which is foreseen, for `properly--degenerate' systems, by a theorem of Nekhorossev and Mi{\v{s}}{\v{c}}enko and Fomenko. The two coordinate systems are alternative to the classical reduction of the nodes by Jacobi, described, e.g., in~[V.I.~Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, 18, 85 (1963); p. 141].