Quantitative KAM theory, with an application to the three-body problem

TL;DR

This paper develops quantitative KAM theory to prove the persistence of quasi-periodic motions in perturbed systems and applies it to demonstrate the coexistence of stable and whiskered tori in the three-body problem, a first in physical systems.

Contribution

It introduces new quantitative KAM theorems for degenerate systems and applies them to establish the coexistence of different types of tori in the three-body problem.

Findings

Existence of a small phase space set with coexisting tori.

Quantitative estimates of the density of these motions.

First proof of stable and whiskered tori coexistence in a physical system.

Abstract

Based on quantitative ``{\sc kam} theory'', we state and prove two theorems about the continuation of maximal and whiskered quasi--periodic motions to slightly perturbed systems exhibiting proper degeneracy. Next, we apply such results to prove that, in the three--body problem, there is a small set in phase space where it is possible to detect both such families of tori. We also estimate the density of such motions in proper ambient spaces. Up to our knowledge, this is the first proof of co--existence of stable and whiskered tori in a physical system.