On the averaging principle for one-frequency systems. Seminorm estimates for the error

TL;DR

This paper generalizes error estimates for the averaging method in one-frequency systems using seminorms, allowing for more flexible and coordinate-specific error analysis, with applications to rigid bodies and satellite motion.

Contribution

It introduces a new approach to error estimation in averaging methods that applies seminorms, extending previous Euclidean norm-based results and enabling coordinate-specific error bounds.

Findings

Generalized error estimates using seminorms.

Application to rigid body dynamics with damping.

Framework for satellite motion analysis in a companion paper.

Abstract

We extend some previous results of our work [1] on the error of the averaging method, in the one-frequency case. The new error estimates apply to any separating family of seminorms on the space of the actions; they generalize our previous estimates in terms of the Euclidean norm. For example, one can use the new approach to get separate error estimates for each action coordinate. An application to rigid body under damping is presented. In a companion paper [2], the same method will be applied to the motion of a satellite around an oblate planet.