Sharp Nekhoroshev estimates for the three body problem around periodic orbits

TL;DR

This paper establishes sharp Nekhoroshev stability estimates for the planar three body problem using periodic averaging, with potential generalizations to other near-integrable Hamiltonian systems, and provides concrete examples.

Contribution

It introduces a Nekhoroshev-like stability result with sharp constants for the three body problem, utilizing periodic averaging techniques and analyzing the dependence on analyticity widths.

Findings

Derived sharp Nekhoroshev estimates for the three body problem.

Analyzed the dependence of stability constants on analyticity widths.

Provided concrete examples demonstrating the estimates in planetary and restricted cases.

Abstract

We construct a Nekhoroshev-like result of stability with sharp constants for the planar three body problem, both in the planetary and in the restricted circular case, by using the periodic averaging technique. Our constructions can be generalized to any near-integrable hamiltonian system whose unperturbed hamiltonian is quasi-convex. The dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account. This allows for a deep analytical understanding of the limits of such techniques in insuring Nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases. Finally, two examples with concrete values are considered, one for the planetary case and one for the restricted one.