Thesis Abstract

Abstract
The Planar Circular Restricted Three-Body Problem (PCR3BP) is a classical problem in celestial mechanics that involves the motion of three bodies under mutual gravitational attraction, where one of the bodies is considered negligible compared to the other two. This problem has been extensively studied due to its relevance in understanding various astronomical phenomena such as the motion of satellites, spacecraft, and celestial bodies in the solar system. In this study, we investigate the dynamics of the PCR3BP with a focus on the planar case, where all bodies lie in the same plane. The primary objective is to analyze the motion of a test particle under the influence of two primary bodies, which are assumed to move in circular orbits around their common center of mass. The dynamics of the system are described by a set of nonlinear differential equations that govern the motion of the test particle in the rotating reference frame of the primaries. We explore the equilibrium points of the system where the test particle remains stationary relative to the primaries. These points provide valuable insights into the stability and behavior of the system. By linearizing the equations of motion around these equilibrium points, we investigate the stability properties using linear stability analysis techniques. The stability of these points is crucial for understanding the long-term behavior of the system and predicting the trajectories of test particles. Furthermore, we examine the formation of periodic orbits in the PCR3BP, which represent closed trajectories that repeat over time. These periodic orbits play a significant role in understanding the global dynamics of the system and identifying regions of stability and chaos. Using numerical simulations, we illustrate different types of periodic orbits and their stability characteristics in the planar PCR3BP. Overall, this study contributes to the theoretical understanding of the dynamics of the Planar Circular Restricted Three-Body Problem and provides insights into the complex behavior of three-body systems in celestial mechanics. The results obtained from this research have implications for various applications in astrodynamics, space mission design, and the study of natural celestial phenomena.

Thesis Overview

1.0 INTRODUCTION
Since the 17th century, the N-body problem has held the attention of generations of astronomers and mathematicians. The problem is simple: given a
collection of N celestial bodies (be they planets, asteroids, stars, black holes) interacting with each other through gravitational forces, what will their trajectories be? For
N = 2, the problem has been solved for centuries; for N _ 3, the problem still has no solution in any meaningful sense. As the theory and vocabulary of dynamics have evolved, so too has the analysis of the problem, and indeed the study of the problem has oen
directly led to the development of new concepts and ideas in dynamics.
In this thesis, we consider the planar circular restricted three body problem, a specific case of the N-body problem for N = 3. The primary goal is to develop a
fast, user-friendly program which can quickly and reliably calculate trajectories from user input. The program will also calculate Poincaré maps, which will be
used to analyse the system for various parameter values. We then hope to verify the existence of a particular bifurcation called the twistless bifurcation for orbits near the Lagrangian points. The twistless bifurcation was found for a general system by
Dullin, Meiss and Sterling, and it is expected that the planar circular restricted three body problem will exhibit the same behaviour.
We begin with a discussion of the history of the problem in Chapter 2, using
Barrow-Green, Valtonen & Karttunen and James as our primary sources. This background serves a dual purpose, neatly introducing many of the theoretical
concepts used to analyse the problem. We discuss several “particular solutions” which illustrate useful ideas and dynamics, and give a summary of the theory of Lagrangian and Hamiltonian mechanics.
In Chapter 3, the solution to the two body problem is presented, and the dynamics for the three body problem are derived. Following Koon, Lo, Marsden &
Ross, we take a Hamiltonian approach to the problem. Other physical considerations such as the Hill region and Lagrangian points are introduced. Also defined are the Poincaré map and extended phase space.
Chapter 4 deals with the biggest obstacle in any attempt to integrate trajectories of the N-body problem, regularising collision orbits. Although an elegant
split-step integrator can be found for the problem, regularising transforms are still required. The discussion of these transformations follows from Szebehely [16], but are here derived in the context of Hamiltonian mechanics. The Levi-Civita, Birkho
and Thiele-Burrau transformations are discussed. An elegant numerical method for calculating Poincaré maps designed by Henón [20] is also presented.

1.1 BACKGROUND OF STUDY.
The study and theory of the three body problem has developed over the last four centuries concurrent to (and one catalysing) the general theory of dynamical systems. It is therefore natural to explore the history of the problem, not only for context and insight but to introduce key approaches and techniques to be utilized in the project.