On the use of the autonomous Birkhoff equations in Lie series perturbation theory

TL;DR

This paper introduces a Lie transformation algorithm tailored for autonomous Birkhoff systems, demonstrating its effectiveness in simplifying Hamiltonian systems and applying it to the restricted three-body problem.

Contribution

The paper develops a novel Lie transformation method specifically for autonomous Birkhoff systems, enhancing perturbation theory techniques for Hamiltonian dynamics.

Findings

Algorithm effectively normalizes restricted three-body problem examples

Suitable for problems requiring non-canonical variables

Improves asymptotic integration in Hamiltonian dynamics

Abstract

In this article, we present the Lie transformation algorithm for autonomous Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a symplectic structure of the general form. Two examples of normalization in the restricted three-body problem are given to illustrate the application of the algorithm in perturbation theory. The efficiency of this algorithm for problems of asymptotic integration in dynamics is discussed for the case where there is a need to use non-canonical variables in the phase space.