A three dimensional investigation of two dimensional orbits

TL;DR

This study investigates the three-dimensional behavior of two-dimensional orbits in triaxial potentials, revealing that the third dimension influences stability and chaos, especially in the logarithmic potential.

Contribution

It demonstrates the importance of considering the third dimension in orbit analysis and explores the relationship between chaos, resonances, and integrals in such systems.

Findings

Instability increases with lower normal force components.

Partially chaotic orbits are found near resonances and elsewhere.

Normal action relates to certain integrals distinguishing orbit types.

Abstract

Orbits in the principal planes of triaxial potentials are known to be prone to unstable motion normal to those planes, so that three dimensional investigations of those orbits are needed even though they are two dimensional. We present here an investigation of such orbits in the well known logarithmic potential which shows that the third dimension must be taken into account when studying them and that the instability worsens for lower values of the forces normal to the plane. Partially chaotic orbits are present around resonances, but also in other regions. The action normal to the plane seems to be related to the isolating integral that distinguishes regular form partially chaotic orbits, but not to the integral that distinguishes partially from fully chaotic orbits.