Exponential stability of fast driven systems, with an application to celestial mechanics

TL;DR

This paper develops a normal form for fast driven systems, demonstrating exponential stability of actions without small denominators, and applies this to show minimal variation in the three-body problem near collisions.

Contribution

It introduces a normal form for fast driven systems that avoids small denominators and trapping arguments, extending stability results to celestial mechanics.

Findings

Actions exhibit exponentially small variations under perturbations.

Normal form construction avoids small denominator issues.

Application to three-body problem shows stability near collisions.

Abstract

We construct a normal form suited to {\it fast driven systems}. We call so systems including actions ${\rm I}$, angles {$\psi$}, and one fast coordinate $y$, moving under the action of a vector--field $N$ depending only on ${\rm I}$ and $y$ and with vanishing ${\rm I}$--components. {In absence of the coordinate $y$, such systems have been extensively investigated and it is known that, after a small perturbing term is switched on, the normalised actions ${\rm I}$ turn to have exponentially small variations compared to the size of the perturbation. We obtain the same result of the classical situation, with the additional benefit that } no trapping argument is needed, as no small denominator arises. {We use the result to prove that, in the three--body problem, the level sets of a certain function called {\it Euler integral} have exponentially small variations in a short time, closely to collisions.}