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

We construct a normal form suited to fast driven systems. We call so systems including actions I, angles ψ, and one fast coordinate y, moving under the action of a vector-field N depending only on I and y and with vanishing I-components. In the 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 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 Euler integral have exponentially small variations in a short time, closely to collisions.