CHAPTER ONE

1.0 INTRODUCTION

Since the 17th century, the N-body problem has held the attention of generationsof 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 often 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,

Birkhoff 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 often 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.

1.2 RESEARCH PROBLEMS

Three-body problem deals with the question of how three objects, with different masses, different initial positions and velocities, will move under one of the physical forces, such as gravity, Coulomb force and elasticity. The masses remain coliinear in an elliptical orbit around the center of the masses. And to integrate trajectories of the N-body problem

.1.3 RESEARCH OBJECTIVES

• To generate the splitting method for the Hamiltonian
• To regularize the collision orbits
• To develop the numerical integrator for the three body problem
• To determine how three objects will move under the physical for forces of either gravity, columb force or elasticity.

1.4 SIGNIFICANCE OF STUDY

This study gives a clear insight into the various ways in which the Hamiltonian method can be reduced using the step integrator. In particular, when determining trajectory of a probe from one planet to another, the probe will typically begin and end at the surface of the bodies, and so the calculations must be accurate there1. Another reason the integrator must be accurate near the primaries is the consideration of low energy trajectories which rely on the slingshot effect whereby a probe passes very close to planets and moons to achieve a “boost.

1.5 SCOPE OF STUDY

This project work covers the area of lagrange method for first order numerical analysis, lagrangian points, the Pythagoras consideration for configuration of masses between 3, 4, and 5 units places at the corresponding positions on the right angle. It also covers some part of the Hamiltonian mechanics, and the Eular’s solution for masses.

1.6 DEFINITION OF TERMS

A CANONICAL TRANSFORMATION: Is a change of the canonical coordinate (q,p,t)

(Q,P,T) that preserves the form of hamilton’s equation resulting from the transformed Hamilton maybe simply obtained by substituting the new coordinates for the old coordinate.

LAGRANGIAN POINT: are positions in an orbit configuration of two large bodies where a small object affected only by gravity can maintain a stable position relative to the two bodies.

2.1 REVIEW OF LITERATURE

The three body problem as we consider it arose very naturally from the work of Newton. In the early 17th century Kepler proposed his laws of planetary motion, describing the orbits of the planets around the sun as ellipses. Newton formalized these ideas in 1687’s Philosophiæ Naturalis Principia Mathematica [1], one of the most important works in the history of science. In particular, the formula for the gravitational force between any two point masses is given as

For two masses m1 and m2 separated by a distance of r, and G the universal gravitational constant. Some controversy remains whether this law should be attributed to Newton or Hooke, but it is acknowledged that both men made very significant contributions to the development of celestial mechanics.

Having justified the laws proposed by Kepler, Newton turned his attention to systems more complex than a Sun-Planet system. One of his main considerations was the Sun-Earth-Moon system. However, Newton’s work in this regard was plagued by difficulties, and he remarked “...[his] head never ached but with his studies on the moon”. It would not be until after Newton’s lifetime that any major progress was made on the three body problem.

In 1747, Alexis Clairaut announced he had successfully constructed a series approximation for the motion of the three masses. After some modification, his approximations accounted for the perigee of the moon (the point at which the moon

is nearest to earth), which had been an aim of Newton’s. In 1752 Clairaut won the St. Petersburg Academy prize for his work on the problem, and in 1759 the value of his approximations was amply demonstrated when Halley’s comet passed Earth within a month of what his equations had predicted, the margin of error he himself had prescribed.

Meanwhile, Leonhard Euler had also turned his attentions to the three body problem. Euler proposed considering the restricted three body problem, a simplification of the general problem where one of the bodies is taken to have negligible mass. When considered with a circular orbits for the two masses, this is also known as the Euler three body problem. Euler also used variation of parameters to study perturbations of the planetary motion.

At the same time as Euler, Joseph Lagrange made significant progress on the general three body problem. Lagrange’s major contributions to theory included reducing the problem from a system of differential equations of order 18 to a system of order 7, and describing two types of particular solutions to the general problem

. Also of major importance, not just to the three body problem but to the general theory of dynamic systems, was his development of Lagrangian Mechanics.

Unaware of Lagrange’s work, Carl Jacobi reduced the general problem to a sixth order system and the restricted problem to a fourth-order system. A constant of motion was found, known as Jacobi’s integral, and is the only known conserved quantity of the restricted problem. In 1878, George Hill demonstrated a very useful application of Jacobi’s integral, describing the regions of possible motion for the body of negligible mass.

Another contributor to the theory of the problem was Charles-Eugéne Delaunay.

By taking repeated canonical transformations of the problem, requiring a truly staggering number of calculations, Delaunay completely eliminated the secular terms of the problem. Taking over two decades to complete, Delaunay’s methods were published in 1846 but his final results could not be published until 1860 and

1867, when they were published in two large volumes of over nine hundred pages each. A key useful result of the work was the introduction of Delaunay variables, a set of canonical action angles which give the equations of motion in Hamiltonian form. Although Delaunay’s method was impractical at the time (the expressions involved were extremely complex and converged very slowly), the theory has been highly influential, not only in lunar theory but in fields such as quantum theory as well.

Henri Poincaré

The end of the “classical” period of work on the three body problem was marked by the extremely influential work of Henri Poincaré. In the late nineteenth century, King Oscar II of Sweden established a prize for solving the N-body problem (a more general form of the problem with N ratherthan 3 masses) on the advice of Gösta Mittag-Leffler, Karl

Weierstrass, and Charles Hermite.