Stability, periodic orbits and KAM tori in the dynamics of the three fixed centers problem

TL;DR

This paper analyzes the dynamics of a particle influenced by three fixed primaries arranged in an equilateral triangle, identifying conditions for stable orbits and KAM tori using Hamiltonian normalization and symmetry reduction.

Contribution

It introduces a Hamiltonian normalization approach and symmetry reduction to study stability and invariant structures in the three fixed centers problem.

Findings

Conditions for existence of circular periodic orbits identified

Presence of KAM tori indicating long-term stability

Analysis of relative equilibria in reduced spaces

Abstract

We investigate the motion in space of an infinitesimal particle in the gravitational field generated by three primary bodies positioned at the vertices of a fixed equilateral triangle. We assume that the distances between the primaries are small compared to their separation from the particle. By applying a Lie-Deprit normalization, we simplify the Hamiltonian, relegating both the mean anomaly and the argument of periapisis to third-order terms or higher. After reducing out the symmetries associated with the Kepler flow and the central action of the angular momentum, we examine the relative equilibria in the first and second reduced spaces. We are able to identify the conditions for the existence of circular periodic orbits and KAM tori, thus providing insight into the system's long-term stability and dynamic structure.