Abstract
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.
Highlights
We construct a normal form suited to fast driven systems
On we focus on the motions of the averaged Hamiltonian (9), bypassing any quantitative statement concerning the averaging procedure, as this would lead much beyond the purposes of the paper3
To prove Lemma 2.1, we look for a change of coordinates which conjugates the vector–field X = N +P to a new vector–field X+ = N+ +P+, where P+ depends in the coordinates I at higher orders
Summary
To the readers who are familiar with Kolmogorov– Arnold–Moser (kam) or Nekhorossev theories, this kind of problems is well known: see [3, 38, 44, 22], or [9, 20, 29, 25, 47] for applications to realistic models Those are theories originally formulated for Hamiltonian vector–fields ( extended to more general ODEs), in particular, with n = m and the coordinate y absent. Theorem B Inside the region C there exists an open set W such that along any motion with initial datum in W , for all t with |t| ≤ tXex,W , the ratio between the absolute variations of the Euler integral E from time 0 to time t, for all |t| ≤ tXex,W , and the a–priori bound t In [23, 24] the result has been recently extended to the spatial version, often denoted scrtbp
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