Rigorous estimates for the relegation algorithm

Abstract

Abstract. We revisit the relegation algorithm by Deprit et al. (2001) in the light of the rigorous Nekhoroshev's like theory. This relatively recent algorithm is nowadays widely used for implementing closed form analytic perturbation theories, as it generalises the classical Birkhoff normalisation algorithm. The algorithm, here briefly explained by means of Lie transformations, has been so far introduced and used in a formal way, i.e. without providing any rigorous convergence or asymptotic estimates. The overall aim of this paper is to find such quantitative estimates and to show how the results about stability over exponentially long times can be recovered in a simple and effective way, at least in the non-resonant case.

On the continuation of degenerate periodic orbits via normal form: full dimensional resonant tori

“…We give here a sketch of the proof of Proposition 1.1. The proof is based on standard arguments in the Lie series theory, that we recall here, referring to, e.g., [7,11,26], for more details.…”

On the continuation of degenerate periodic orbits via normal form: lower dimensional resonant tori

“…The key estimates that allow to complete the proof are reported in Lemma 4.4 in Section 4. We do not report here all the (tedious) details since similar results have been already published in, e.g., [7,9,10,11,12,39,34].…”

Closed form perturbation theory in the restricted three-body problem without relegation

“…Remark 6. Once having computed the Poisson bracket by applying formula (19), we substitute φ with e sin u in all produced terms depending on the equation of the center.…”

“…However, it is obvious that, even if ω 1 << ω 2 , the method may not converge if the coefficients k 1 , k 2 are such that k 1 ω 1 /k 2 ω 2 ≥ 1. We refer to [19] for more details about the convergence of the relegation algorithm.…”