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Preface V
The Hamilton-Jacobi Theory 1
Canonical Pertubation Equations 1
Hamilton's Principle 2
Maupertuis' Least Action Principle 4
Helmholtz Invariant 5
Canonical Transformations 6
Lagrange Brackets 9
Poisson Brackets 11
Reciprocity Relations 12
The Extended Phase Space 13
Gyroscopic Systems 15
Gyroscopic Forces 15
Example 16
Rotating Frames 17
Apparent Forces 17
The Partial Differential Equation of Hamilton and Jacobi 18
One-Dimensional Motion with a Generic Potential 20
The Case m < 0 23
The Harmonic Oscillator 23
Involution. Mayer's Lemma. Liouville's Theorem 24
Angle-Action Variables. Separable Systems 29
Periodic Motions 29
Angle-Action Variables 30
The Sign of the Action 32
Direct Construction of Angle-Action Variables 33
Actions in Multiperiodic Systems. Einstein's Theory 35
Separable Multiperiodic Systems 37
Uniformized Angles. Charlier's Theory 37
The Actions 38
Algorithms for Construction of the Angles 39
Angle-Action Variables of H(q[subscript 1], p[subscript 1], [subscript 2],..., p[subscript N]) 40
Historical Postscript 42
Simple Separable Systems 42
Example: Central Motions 43
Angle-Action Variables of Central Motions 44
Kepler Motion 47
Degeneracy 50
Schwarzschild Transformation 51
Delaunay Variables 52
The Separable Cases of Liouville and Stackel 53
Example: Liouville Systems 55
Example: Stackel Systems 56
Example: Central Motions 56
Angle-Action Variables of a Quadratic Hamiltonian 57
Gyroscopic Systems 60
Classical Perturbation Theories 61
The Problem of Delaunay 61
The Poincare Theory 63
Expansion of H[subscript 0] 65
Expansion of H[subscript k] 66
Perturbation Equations 67
Averaging Rule 68
Small Divisors. Non-Resonance Condition 69
Degenerate Systems. The von Zeipel-Brouwer Theory 70
Expansion of H[subscript *] 72
von Zeipel-Brouwer Perturbation Equations 72
The von Zeipel Averaging Rule 73
Small Divisors and Resonance 74
Elimination of the Non-Critical Short-Period Angles 74
An Example - Part I 77
Linear Secular Theory 81
An Example - Part II 83
Iterative Use of von Zeipel-Brouwer Operations 86
Divergence of the Series. Poincare's Theorem 88
Kolmogorov's Theorem 88
Frequency Relocation 89
Convergence 91
Degenerate Systems 93
Degeneracy in the Extended Phase Space 94
Inversion of a Jacobian Transformation 94
Lagrange Implicit Function Theorem 96
Practical Considerations 96
Lindstedt's Direct Calculation of the Series 97
Resonance 99
The Method of Delaunay's Lunar Theory 99
Introduction of the Square Root of the Small Parameter 101
Garfinkel's Abnormal Resonance 103
Delaunay Theory According to Poincare 103
First-Approximation Solution 106
Garfinkel's Ideal Resonance Problem 107
Garfinkel-Jupp-Williams Integrals 109
Circulation ([Characters not reproducible] > [Characters not reproducible] > 0) 110
Libration (|E|<
Asymptotic Motions (E = A[subscript *]) 114
Angle-Action Variables of the Ideal Resonance Problem 115
Circulation 115
Libration 116
Small-Amplitude Librations 117
Morbidelli's Successive Elimination of Harmonics 118
An Example 120
Lie Mappings 127
Lie Transformations 127
Infinitesimal Canonical Transformations 127
Lie Derivatives 130
Lie Series 131
Inversion of a Lie Mapping 134
Lie Series Expansions 135
Lie Series Expansion of f 136
Deprit's Recursion Formula 137
Lie Series Perturbation Theory 139
Introduction 139
Lie Series Theory with Angle-Action Variables 140
Averaging 142
High-Order Theories 143
Comparison to Poincare Theory. Example I 144
Comparison to Poincare Theory. Example II 147
Hori's General Theory. Hori Kernel and Averaging 151
Cauchy-Darboux Theory of Characteristics 154
Topology and Small Divisors 155
Topological Constraint. The Rise of Small Divisors 156
Hori's Formal First Integral 157
"Average" Hamiltonians 158
On Secular Theories and Proper Elements 159
Non-Singular Canonical Variables 161
Singularities of the Actions 161
Poincare Non-Singular Variables 162
The d'Alembert Property 164
Regular Integrable Hamiltonians 165
Lie Series Expansions About the Origin 167
Lie Series Perturbation Theory in Non-Singular Variables 169
Solutions Close to the Origin (Case J[subscript 1]<0) 172
Angle-Action Variables of H[Characters not reproducible] (Case J[subscript 1]< 0) 173
The Non-Resonance Condition 173
Example 175
Lie Series Theory for Resonant Systems 181
Bohlin's Problem (The Single-Resonance Problem) 181
Outline of the Solution 182
Functions Expansions 185
Perturbation Equations 188
Averaging 190
An Example 190
Example with a Separated Hori Kernel 198
One Degree of Freedom 204
Garfinkel's Ideal Resonance Problem 204
Single Resonance near a Singularity 209
Resonances Near the Origin: Real and Virtual 209
One Degree of Freedom 210
Many Degrees of Freedom. One Single Resonance 213
A First-Order Resonance Case Study 216
The Hori Kernel 218
First Perturbation Equation 219
Averaging 220
The Post-Harmonic Solution 221
Secular Resonance 223
Secondary Resonances 224
Initial Conditions Diagram 225
Sessin Transformation and Integral 227
The Restricted (Asteroidal) Case 229
Nonlinear Oscillators 231
Quasiharmonic Hamiltonian Systems 231
Formal Solutions. General Case 232
Exact Commensurability of Frequencies (Resonance) 234
Birkhoff Normalization 236
A Formal Extension Including One Single Resonance 240
The Comensurabilities of Lower Order 242
The Restricted Three-Body Problem 242
Equations of the Motion Around the Lagrangian Point L[superscript 4] 244
Internal 2:1 Resonance 246
Internal 3:1 Resonance 247
Other Internal Resonances 249
The Henon-Heiles Hamiltonian 250
The Toda Lattice Hamiltonian 252
Systems with Multiple Commensurabilities 253
The Ford-Lunsford Hamiltonian. 1:2:3 Resonance 255
Parametrically Excited Systems 255
A Nonlinear Extension 260
Bohlin Theory 263
Bohlin's Resonance Problem 263
Bohlin's Perturbation Equations 265
Poincare Singularity 268
An Extension of Delaunay Theory 269
The Simple Pendulum 271
Equations of Motion 271
Circulation 273
Libration 274
The Separatrix 276
Angle-Action Variables of the Pendulum 277
Circulation 277
Libration 278
Small Oscillations of the Pendulum 279
Angle-Action Variables 280
Direct Construction of Angle-Action Variables 281
The Neighborhood of the Pendulum Separatrix 283
Motion near the Separatrix 285
The Separatrix or Whisker Map 286
The Standard Map 288
Andoyer Hamiltonian with k = 1 289
Andoyer Hamiltonians 289
Centers and Saddle Points 290
The Case k = 1 292
Morphogenesis 293
Width of the Libration Zone 296
Integration 298
The Case [Delta] > 0 301
The Case [Delta] < 0 302
The Separatrices 303
The Angle [sigma] 304
Equilibrium Points 305
The Inner Circulations Center 306
The Libration Center 306
Proper Periods 306
Inner Circulations 307
Librations 307
The Angle Variable w 308
Small-Amplitude Librations 308
The Action [Lambda] 312
The New Hamiltonian 312
Andoyer Hamiltonians with k [greater than or equal] 2 315
Introduction 315
The Case k = 2 315
Morphogenesis 316
Width of the Libration Zone 318
The Case k = 3 320
Morphogenesis 321
Width of the Libration Zone 323
The Case k = 4 325
Morphogenesis 327
Width of the Libration Zone 327
Comparative Analysis 328
Virtual Resonances 329
References 331
Index 337
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