Project Abstract

Abstract
The study of the Planar Circular Restricted Three Body Problem (PCR3BP) is a fascinating area of research in celestial mechanics that involves analyzing the motion of a small mass particle under the gravitational influence of two primary bodies moving in circular orbits around their center of mass. This problem serves as a simplified model for understanding the dynamics of various celestial systems such as the Earth-Moon-Spacecraft system, the Sun-Earth-Moon system, and other similar configurations. The PCR3BP has been widely studied due to its relevance in space missions, satellite dynamics, and theoretical celestial mechanics. Researchers have developed analytical and numerical techniques to investigate the behavior of test particles in this dynamical system. By considering the motion of the test particle in the rotating reference frame with the same angular velocity as the primary bodies, the equations of motion can be simplified, leading to a more manageable mathematical framework for analysis. One of the key features of the PCR3BP is the presence of libration points, also known as Lagrange points, where the gravitational forces of the two primary bodies create regions of equilibrium for the test particle. These Lagrange points play a crucial role in understanding the stability and dynamics of the system. The five Lagrange points, labeled L1 to L5, have distinct characteristics in terms of stability and energy requirements for maintaining a stationary position relative to the primaries. Researchers have explored various aspects of the PCR3BP, including the stability of periodic orbits, the existence of invariant manifolds, and the effect of perturbations on the motion of test particles. Numerical simulations have provided insights into the long-term behavior of particles in the vicinity of Lagrange points, revealing complex trajectories and dynamical phenomena such as chaotic motion and resonant interactions. Furthermore, studies have investigated the application of the PCR3BP to space mission design, trajectory planning, and spacecraft maneuvers. By leveraging the analytical solutions and numerical simulations of this problem, researchers can optimize trajectories for interplanetary missions, lunar missions, and satellite deployments. In conclusion, the Planar Circular Restricted Three Body Problem offers a rich field for theoretical and applied research in celestial mechanics. Understanding the dynamics of test particles in this gravitational system not only enhances our knowledge of fundamental astrodynamics principles but also has practical implications for space exploration and mission planning.

Project 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.