ISBN: 3540589635
TITLE: Theory of Orbits
AUTHOR: Boccaletti, Pucacco
TOC:

Introduction - The Theory of Orbits from Epicycles to "Chaos" 1
Chapter I. Dynamics and Dynamical Systems - Quod Satis 15
A. Dynamical Systems and Newtonian Dynamics 16
1.1 Dynamical Systems: Generalities 16
1.2 Classification of Critical Points - Stability 20
1.3 The n-Dimensional Oscillator 26
B. Lagrangian Dynamics 32
1.4 Lagrange's Equations 32
1.5 Ignorable Variables and Integration of Lagrange's Equations 37
1.6 Noether's Theorem 43
1.7 An Application of Noether's Theorem: The n-Dimensional Oscillator 50
1.8 The Principle of Least Action in Jacobi Form 57
C. Hamiltonian Dynamics and Hamilton-Jacobi Theory 61
1.9 The Canonical Equations 61
1.10 The Integral Invariants - Liouville's Theorem 65
1.11 Poisson Brackets and Poisson's Theorem -The Generation of New Integrals 73
1.12 Canonical Transformations 76
1.13 Generating Functions -Infinitesimal Canonical Transformations 82
1.14 The Extended Phase Space 85
1.15 The Hamilton-Jacobi Equation and the Problem of Separability 90
1.16 Action-Angle Variables 98
1.17 Separable Multiperiodic Systems -Uniqueness of the Action-Angle Variables 106
1.18 Integrals in Involution - Liouville's Theorem for Integrable Systems 113
1.19 Lax's Method - The Painlev Property 117
Chapter 2. The Two-Body Problem 125
2.1 The Two-Body Problem and Kepler's Three Laws 126
2.2 The Laplace-Runge-Lenz Vector 136
2.3 Bertrand's Theorem and Related Questions 141
2.4 The Position of the Point on the Orbit 147
2.5 The Elements of the Orbit 156
2.6 The Problem of Regularization 162
2.7 Topology of the Two-Body Problem 171
Chapter 3. The N-Body Problem 177
3.1 Equations of Motion and the Existence Theorem 178
3.2 The Integrals of the Motion 184
3.3 The Singularities 192
3.4 Sundman's Theorem 198
3.5 The Evolution of the System for t -> infinity 200
3.6 The Virial Theorem 209
3.7 Particular Solutions of the N-Body Problem 216
3.8 Homographic Motions and Central Configurations 229
Chapter 4. The Three-Body Problem 237
4.1 The General Three-Body Problem 238
4.2 Existence of the Solution - Sundman and Levi-Civita Regularization 244
4.3 The Restricted Three-Body Problem 256
4.4 The Stability of the Equilibrium Solutions 265
4.5 The Delaunay Elements for the Restricted Three-Body Problem 272
4.6 The Regularization of the Restricted Three-Body Problem 279
4.7 Extensions and Generalizations of the Restricted Problem 284
Chapter 5. Orbits in Given Potentials 301
5.1 Introduction 302
5.2 Orbits in Spherically Symmetric Potentials 306
5.3 Orbits in Isochronal Potentials 316
5.4 Elliptical Coordinates and Stckel's Theorem 323
5.5 Planar Potentials 334
5.6 The Problem of Two Fixed Centres in the Plane 341
5.7 Axially Symmetric Potentials - Motion in the Potential of the Earth 349
5.8 Orbits in Triaxial Potentials 352
5.9 Configurational Invariants 359
Mathematical Appendix 363
A.1 Spherical Trigonometry 364
A.2 Curvilinear Coordinate Systems 365
A.3 Riemannian Geometry 370
Bibliographical Notes 375
Name Index 385
Subject Index 389
END
