ISBN: 3540603557
TITEL: Theory of Orbits Vol.2
AUTHOR: Boccaletti / Pucacco
TOC:

Chapter 6. Classical Perturbation Theory in Celestial Mechanics. The Equations of Planetary Motion 1
6.1 Introduction 2
6.2 Perturbation of the Coordinates 7
6.3 The Gauss Method for the Secular Terms 11
6.4 Perturbation of the Elements 14
6.4.1 Preliminary Considerations - Poisson's Equations 14
6.4.2 Variation of the Elements 16
6.5 Application of the Method of Variation of the Elements 24
6.6 Lindstedt's Device 33
6.7 Some Remarks on the Perturbative Expansions So Far Discussed 37
6.8 An Outline of the Averaging Method 40
6.9 Other Problems Concerning the Solar System 44
6.10 Motion Around an "Oblate" Planet 45
6.11 Kepler's Problem with Drag 50
6.11.1 Application of the Averaging Method 50
6.11.2 The First Integrals by the Direct Method 53
6.11.3 The First Integrals from Noether's Theorem 59
Chapter 7. Canonical Perturbation Theory 69
7.1 Jacobi's Equations - First-Order Canonical Theory 69
7.2 Canonical Perturbation Theory to Any Order - Von Zeipel's Method 74
7.3 Systems with One Degree of Freedom 80
7.4 Systems with Several Degrees of Freedom 89
7.5 The KAM Theorem 99
7.6 The Destruction of the Tori - Arithmetic Properties and Applications 106
7.7 The Twist Map and the KAM Theorem 112
Chapter 8. Lie Transform Perturbation Theory 125
8.1 Deprit's Method 125
8.1.1 Lie Series 126
8.1.2 The Lie Transform 132
8.1.3 The Relation with the Classical Canonical Theory 139
8.2 Kamel's Generalized Method 141
8.2.1 The Procedure 141
8.2.2 An Application 148
8.3 Application of Deprit's Method to Hamiltonian Systems 152
8.3.1 The Procedure 152
8.3.2 The Pendulum 154
8.3.3 The Parametrically Driven Non-linear Oscillator 157
8.3.4 The Perturbed Isochronal Potential 159
8.4 The Dragt and Finn Method 161
8.5 Normalization 167
8.5.1 Normal Forms - Generalities 168
8.5.2 Hamiltonian Normal Forms 171
8.6 Superconvergent Series and Kolmogorov's Technique 175
Chapter 9. The Theory of Adiabatic Invariants 179
9.1 Adiabatic Invariants 180
9.2 Adiabatic Invariants for One-Degree-of-Freedom Systems 183
9.3 Adiabatic Invariants, the Averaging Method and Noether's Theorem 189
9.4 Adiabatic Invariants and the Lie Transform Method 195
9.5 The Limits of Validity of Adiabatic Invariants 198
9.5.1 Stability of Linear Differential Systems with Periodic Coefficients 198
9.5.2 The Mathieu Equation 202
9.5.3 Perpetual Stability of Non-linear Periodic Systems 206
9.6 The Motion of Many-Degrees-of-Freedom Systems on Different Time Scales 209
9.6.1 The Equations of Motion of a Charged Particle in an Electromagnetit Field 
209
9.6.2 The Motion of the Guiding Centre 212
9.6.3 The Longitudinal Invariant 216
9.6.4 The Invariant Flux 218
9.6.5 Final Remarks 219
9.7 Adiabatic Invariants for One-Degree-of-Freedom Systems: A Retrospect 223
9.8 Adiabatic Invariants for Many-Degrees-of-Freedom Hamiltonian Systems: A 
Summary 225
9.9 Astronomical Applications of Adiabatic Invariants 227
9.9.1 Kepler's Problem with Variable Mass 227
9.9.2 The Problems of Stellar Dynamits 229
9.9.3 The Adiabatic Invariants in the Earth's Magnetosphere 231
Chapter 10. Periodic Orbits and Resonances 235
10.1 Periodic Orbits - The Restricted Three-Body Problem 236
10.1.1 The Method of Analytic Continuation 237
10.1.2 Application of the Analytic Continuation Method to the Restricted Three-
Body Problem 240
10.2 The Role of Periodic Motion 243
10.3 Local Results in the Vicinity of an Equilibrium Point 245
10.3.1 An Application to the Equilibrium Solutions of the Restricted Three-Body 
Problem 248
10.4 An Elementary Outline of the Bifurcation Theory of the Equilibrium 
Positions and Periodic Solutions 250
10.5 The Bifurcation Theory for Hamiltonian Systems and the Case of L_4 and L_5 
261
10.6 Global Results 268
10.6.1 The Existence of Periodic Solutions 268
10.6.2 Direct Methods for the Determination of Periodic Solutions 271
10.7 The Resonances in Celestial Mechanics 278
10.7.1 Orbital Resonances 280
10.7.2 Spin-Orbit Resonances 287
10.7.3 From Resonances to Chaos 294
Chapter 11. Chaos 297
11.1 Introduction to Chaos 297
11.2 Hints of Stochasticity 301
11.3 Weak Chaos 307
11.3.1 The Smale Horseshoe 307
11.3.2 Symbolic Dynamits and the Invariant Set 313
11.3.3 The Smale-Birkhoff-Moser Homoclinic Theorem 316
11.4 Melnikov's Method 325
11.4.1 The Melnikov Function 325
11.4.2 The Pendulum Coupled to the Linear Oscillator 332
11.4.3 Perturbations of Stckel Potentials 335
11.5 The Transition to Global Stochasticity 340
11.6 Hard Chaos 348
Chapter 12. Numerical Experiments 355
12.1 Stochasticity and Numerital Errors 361
12.2 Elementary Numerical Computation 361
12.2.1 Runge-Kutta and Related Routines 364
12.2.2 Symplectic Codes 365
12.2.3 Application to Kepler's Problem 373
12.3 Numerical Chaoticity Indicators 373
12.3.1 Numerical Computation of the Surfaces of Section 373
12.3.2 Lyapunov's Exponents 378
12.3.3 Geometric Detection of Chaos 383
12.4 Chaos in N-Body Systems 384
12.4.1 Exponential Instability of N-Body Systems 385
12.4.2 Chaos in the Restricted Three-Body System 389
12.4.3 Chaos in the Solar System 392
12.5 Epilogue 396
Bibliographical Notes 399
Name Index 409
Subject Index 415
Corrections to Volume 1 421
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