Effective stability estimates close to resonances with applications to rotational dynamics

TL;DR

This paper develops optimized Nekhoroshev-like estimates and perturbation techniques to analyze long-term stability near resonances in nearly-integrable Hamiltonian systems, with applications to celestial rotational dynamics.

Contribution

It introduces an optimization algorithm for stability parameters and applies perturbation theory to improve stability bounds near resonances in celestial mechanics models.

Findings

Effective stability bounds are established near resonances.

Optimization of parameters extends the stability time.

Applications demonstrate stability of orbits close to main resonances.

Abstract

We consider nearly-integrable Hamiltonian systems defined over a non-resonant domain. In the neighborhood of resonances, we use Nekhoroshev-like estimates to provide effective stability bounds for the action variables over long time. The applicability conditions of these estimates allow some freedom in the choice of parameters. Hence, we develop an optimization algorithm for choosing parameters that maximize the stability time. To further improve the stability estimates, we use perturbation theory to reduce the norm of the perturbing function. We implement this procedure (effective stability estimates and perturbation theory) to analyze the stability of sequences of irrational (Diophantine) frequencies converging to frequencies corresponding to resonances. We consider two applications to models describing problems of rotational dynamics in Celestial Mechanics: the spin-orbit problem, described by a 1D time-dependent Hamiltonian, and the spin-spin-orbit model, described by a 2D time-dependent Hamiltonian. We show stability results for orbits close to the main resonances associated with such models.