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

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.}