Rigorous estimates for the relegation algorithm

TL;DR

This paper provides rigorous quantitative estimates for the relegation algorithm, enhancing its theoretical foundation and demonstrating stability over long times in non-resonant cases using Nekhoroshev-like theory.

Contribution

It introduces rigorous convergence and stability estimates for the relegation algorithm, bridging formal methods with quantitative stability analysis.

Findings

Established explicit bounds for the algorithm's convergence.

Demonstrated long-term stability in non-resonant cases.

Connected the algorithm with Nekhoroshev-like stability theory.

Abstract

We revisit the relegation algorithm by Deprit et al. (Celest. Mech. Dyn. Astron. 79:157-182, 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.