# Theory of Orbit

### The Restricted Problem of Three Bodies

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Theory of Orbits: The Restricted Problem of Three Bodies is a 10-chapter text that covers the significance of the restricted problem of three bodies in analytical dynamics, celestial mechanics, and space dynamics. The introductory part looks into the use of three essentially different approaches to dynamics, namely, the qualitative, the quantitative, and the formalistic. The opening chapters consider the formulation of equations of motion in inertial and in rotating coordinate systems, as well as the reductions of the problem of three bodies and the corresponding streamline analogies. These topics are followed by discussions on the regularization and writing of equations of motion in a singularity-free systems; the principal qualitative aspect of the restricted problem of the curves of zero velocity; and the motion and nonlinear stability in the neighborhood of libration points. This text further explores the principles of Hamiltonian dynamics and its application to the restricted problem in the extended phase space. A chapter treats the problem of two bodies in a rotating coordinate system and treats periodic orbits in the restricted problem. Another chapter focuses on the comparison of the lunar and interplanetary orbits in the Soviet and American literature. The concluding chapter is devoted to modifications of the restricted problem, such as the elliptic, three-dimensional, and Hill’s problem. This book is an invaluable source for astronomers, engineers, and mathematicians.

## Table of Contents

Foreword

Preface

Acknowledgments

Introduction

Chapter 1. Description of the Restricted Problem

1.1 Introduction

1.2 Statement of the Problem and Equations of Motion in a Sidereal System

1.3 An Invariant Relation and the Total Energy of the System

1.4 Equations of Motion in a Synodic Coordinate System and the Jacobian Integral

1.5 Equations of Motion in Dimensionless Coordinates

1.6 Summary of the Equations of Motion

1.7 A Method of Finding the Potential

1.8 Relation to the General Problem of Three Bodies

1.9 Classification and Modifications of the Restricted Problem

1.10 Applications

1.11 Notes

1.12 References

Chapter 2. Reduction

2.1 Introduction

2.2 Reduction to the Third Order

2.3 Reduction to the Second Order

2.4 Four-Dimensional Streamline Analogy

2.5 Three-Dimensional Streamline Analogy

2.6 General Remarks on the Relation between Hamiltonian Systems and Flow in the Phase Space

2.7 Two-Dimensional Streamline Analogy

2.8 Further General Remarks

2.9 Applications

2.10 Notes

2.11 References

Chapter 3. Regularization

3.1 Introduction

3.2 Regularization of Collision Orbits in the Problem of Two Bodies

3.3 Regularization of the General Problem of Two Bodies

3.4 Local Regularization of the Restricted Problem

3.5 Birkhoff's Global Regularization of the Restricted Problem

3.6 The Thiele-Burrau Transformation

3.7 Lemaître's Regularization

3.8 Generalization of Global Regularization Methods

3.9 Comparison of Global Regularizations

3.10 The Existence of Solutions

3.11 Applications

3.12 Notes

3.13 References

Chapter 4. Totality of Solutions

4.1 Introduction

4.2 The Manifold of the States of Motion

4.3 Singularities of the Manifold of the States of Motion

4.4 Computation of the Location of the Collinear Points

4.5 Description of Appendixes I, II, and III for the Characteristics of the Collinear Equilibrium Points

4.6 Analysis of the Function Ω(X, Y)

4.7 Regions of Motion

4.8 Applications

4.9 Notes

4.10 References

Appendix I. First Collinear Equilibrium Point

Appendix II. Second Collinear Equilibrium Point

Appendix III. Third Collinear Equilibrium Point

Appendix IV. Values of the Jacobian Constant

Chapter 5. Motion near the Equilibrium Points

5.1 Introduction

5.2 Stability of Linear and Nonlinear Systems

5.3 Motion around the Collinear Points

5.4 Motion around the Equilateral Points

5.5 Nonlinear Phenomena around the Equilibrium Points

5.6 Applications

5.7 Notes

5.8 References

Appendix I. Roots of the Characteristic Equation at the Collinear Points

Appendix II. Roots of the Characteristic Equation for Stable Solution at the Triangular Points

Appendix III. Roots of the Characteristic Equation for Unstable Solution at the Triangular Points

Chapter 6. Hamiltonian Dynamics in the Extended Phase Space

6.1 Introduction

6.2 Equations of Motion

6.3 Canonical Transformation in the Phase Space

6.4 The Extended Phase Space

6.5 Examples of Hamiltonian Dynamics in the Extended Phase Space

6.6 Generalized Time Transformations

6.7 Notes

6.8 References

Chapter 7. Canonical Transformations of the Restricted Problem

7.1 Introduction

7.2 Canonical Transformation from Sidereal to Synodic Coordinates

7.3 Interpretation of the New Variables

7.4 Change of Origin by Means of Canonical Transformations

7.5 Transformation from Rectangular Synodic to Polar Coordinates

7.6 Transformation from Polar Coordinates to Delaunay's Variables

7.7 Modifications of Delaunay's Elements

7.8 Regularization with Canonical Variables

7.9 Notes

7.10 References

Chapter 8. Periodic Orbits

8.1 Introduction

8.2 Definitions

8.3 Surface Transformations and Representation on a Torus

8.4 Analytic Continuation

8.5 The Problem of Two Bodies in a Rotating Coordinate System

8.6 The Restricted Problem with Small Mass Parameter

8.7 Whittaker's Criterion for the Existence of Periodic Orbits

8.8 Characteristic Exponents

8.9 Notes

8.10 References

Chapter 9. Numerical Explorations

9.1 Introduction

9.2 Organization of Existing Numerical Results

9.3 Relation between Various Systems

9.4 The Copenhagen Category

9.5 Periodic Lunar Orbits

9.6 Motion around the Triangular Libration Points

9.7 Lunar Trajectories

9.8 Applications to Binary Systems

9.9 Additional Periodic Orbits

9.10 Stability

9.11 Notes

9.12 References

Chapter 10. Modifications of the Restricted Problem

10.1 Introduction

10.2 The Three-Dimensional Restricted Problem

10.3 The Elliptic Restricted Problem

10.4 Hill's Problem

10.5 Additional Modifications

10.6 Notes

10.7 References

List of Major Reference Books

Author Index

Subject Index

## Product details

- No. of pages: 684
- Language: English
- Copyright: © Academic Press 1967
- Published: January 1, 1967
- Imprint: Academic Press
- eBook ISBN: 9780323143462