Project Abstract

Project Overview

<p>1.0 INTRODUCTION<br>Since the 17th century, the N-body problem has held the attention of generations of astronomers and mathematicians. The problem is simple: given a<br>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<br>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<br>directly led to the development of new concepts and ideas in dynamics.<br>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<br>fast, user-friendly program which can quickly and reliably calculate trajectories from user input. The program will also calculate Poincaré maps, which will be<br>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<br>Dullin, Meiss and Sterling, and it is expected that the planar circular restricted three body problem will exhibit the same behaviour.<br>We begin with a discussion of the history of the problem in Chapter 2, using<br>Barrow-Green, Valtonen &amp; Karttunen and James as our primary sources. This background serves a dual purpose, neatly introducing many of the theoretical<br>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.<br>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 &amp;<br>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.<br>Chapter 4 deals with the biggest obstacle in any attempt to integrate trajectories of the N-body problem, regularising collision orbits. Although an elegant<br>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<br>and Thiele-Burrau transformations are discussed. An elegant numerical method for calculating Poincaré maps designed by Henón [20] is also presented.<br><br>1.1 BACKGROUND OF STUDY.<br>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.<br></p>