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Chapter 1 The Hamilton-Jacobi Theory |
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1.1 Canonical Pertubation Equations 1 |
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1.2 Hamilton's Principle 2 |
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1.2.1 Maupertuis' Least Action Principle 4 |
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1.2.2 Helmholtz Invariant 5 |
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1.3 Canonical Transformations 6 |
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1.4 Lagrange Brackets 9 |
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1.5 Poisson Brackets 11 |
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1.5.1 Reciprocity Relations 12 |
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1.6 The Extended Phase Space 13 |
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1.7 Gyroscopic Systems 15 |
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1.7.1 Gyroscopic Forces 15 |
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1.7.2 Example 16 |
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1.7.3 Rotating Frames 17 |
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1.7.4 Apparent Forces 17 |
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1.8 The Partial Differential Equation of Hamilton and Jacobi 18 |
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1.9 One-Dimensional Motion with a Generic Potential 20 |
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1.9.1 The Case m<0 23 |
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1.9.2 The Harmonic Oscillator 23 |
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1.10 Involution. Mayer's Lemma. Liouville's Theorem 24 |
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Chapter 2 Angle-Action Variables. Separable Systems |
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2.1 Periodic Motions 29 |
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2.1.1 Angle--Action Variables 30 |
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2.1.2 The Sign of the Action 32 |
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2.2 Direct Construction of Angle--Action Variables 33 |
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2.3 Actions in Multiperiodic Systems. Einstein's Theory 35 |
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2.4 Separable Multiperiodic Systems 37 |
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2.4.2 The Actions 38 |
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2.4.3 Algorithms for Construction of the Angles 39 |
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2.4.4
Angle--Action Variables of H (q1, p1, p2,...,
pN)
40 |
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2.4.5 Historical Postscript 42 |
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2.5 Simple Separable Systems 42 |
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2.5.1 Example: Central Motions 43 |
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2.5.2 Angle--Action Variables of Central Motions 44 |
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2.6 Kepler Motion 47 |
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2.7 Degeneracy 50 |
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2.7.1 Schwarzschild Transformation 51 |
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2.7.2 Delaunay Variables 52 |
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2.8 The Separable Cases of Liouville and Stäckel 53 |
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2.8.1 Example: Liouville Systems 55 |
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2.8.2 Example: Stäckel Systems 56 |
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2.8.3 Example: Central Motions 56 |
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2.9 Angle--Action Variables of a Quadratic Hamiltonian 57 |
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2.9.1 Gyroscopic Systems 60 |
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Chapter 3 Classical Perturbation Theories |
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3.1 The Problem of Delaunay 61 |
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3.2 The Poincaré Theory 63 |
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3.2.1 Expansion of H 0 65 |
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3.2.2 Expansion of H k 66 |
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3.2.3 Perturbation Equations 67 |
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3.3 Averaging Rule 68 |
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3.3.1 Small Divisors. Non-Resonance Condition 69 |
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3.4 Degenerate Systems. The von Zeipel--Brouwer Theory 70 |
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3.4.1 Expansion of H * 72 |
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3.4.2 von Zeipel--Brouwer Perturbation Equations 72 |
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3.4.3 The von Zeipel Averaging Rule 73 |
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3.5 Small Divisors and Resonance 74 |
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3.5.1 Elimination of the Non-Critical Short-Period Angles 74 |
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3.6 An Example -- Part I 77 |
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3.7 Linear Secular Theory 81 |
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3.8 An Example -- Part II 83 |
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3.9 Iterative Use of von Zeipel--Brouwer Operations 86 |
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3.10 Divergence of the Series. Poincaré's Theorem 88 |
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3.11 Kolmogorov's Theorem 88 |
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3.11.1 Frequency Relocation 89 |
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3.11.2 Convergence 91 |
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3.11.3 Degenerate Systems 93 |
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3.11.4 Degeneracy in the Extended Phase Space 94 |
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3.12 Inversion of a Jacobian Transformation 94 |
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3.12.1 Lagrange Implicit Function Theorem 96 |
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3.12.2 Practical Considerations 96 |
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3.13 Lindstedt's Direct Calculation of the Series 97 |
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Chapter 4 Resonance |
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4.1 The Method of Delaunay's Lunar Theory 99 |
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4.2 Introduction of the Square Root of the Small Parameter 101 |
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4.2.1 Garfinkel's Abnormal Resonance 103 |
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4.3 Delaunay Theory According to Poincaré 103 |
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4.3.1 First-Approximation Solution 106 |
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4.4 Garfinkel's Ideal Resonance Problem 107 |
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4.4.1 Garfinkel-Jupp-Williams Integrals 109 |
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4.4.2 Circulation (En*11 > A*n*11 > 0) 110 |
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4.4.3 Libration (|E|<|A*|) 112 |
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4.4.4 Asymptotic Motions (E=A*) 114 |
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4.5 Angle--Action Variables of the Ideal Resonance Problem 115 |
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4.5.1 Circulation 115 |
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4.5.2 Libration 116 |
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4.5.3 Small-Amplitude Librations 117 |
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4.6 Morbidelli's Successive Elimination of Harmonics 118 |
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4.6.1 An Example 120 |
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Chapter 5 Lie Mappings |
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5.1 Lie Transformations 127 |
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5.1.1 Infinitesimal Canonical Transformations 127 |
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5.2 Lie Derivatives 130 |
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5.3 Lie Series 131 |
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5.4 Inversion of a Lie Mapping 134 |
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5.5 Lie Series Expansions 135 |
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5.5.1 Lie Series Expansion of f 136 |
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5.5.2 Deprit's Recursion Formula 137 |
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Chapter 6 Lie Series Perturbation Theory |
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6.1 Introduction 139 |
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6.2 Lie Series Theory with Angle-Action Variables 140 |
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6.2.1 Averaging 142 |
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6.2.2 High-Order Theories 143 |
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6.3 Comparison to Poincaré Theory. Example I 144 |
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6.4 Comparison to Poincaré Theory. Example II 147 |
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6.5 Hori's General Theory. Hori Kernel and Averaging 151 |
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6.5.1 Cauchy--Darboux Theory of Characteristics 154 |
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6.6 Topology and Small Divisors 155 |
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6.6.1 Topological Constraint. The Rise of Small Divisors 156 |
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6.7 Hori's Formal First Integral 157 |
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6.8 “Average” Hamiltonians 158 |
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6.8.1 On Secular Theories and Proper Elements 159 |
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Chapter 7 Non-Singular Canonical Variables |
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7.1 Singularities of the Actions 161 |
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7.2 Poincaré Non-Singular Variables 162 |
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7.3 The d'Alembert Property 164 |
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7.4 Regular Integrable Hamiltonians 165 |
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7.5 Lie Series Expansions about the Origin 167 |
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7.6 Lie Series Perturbation Theory in Non-Singular Variables 169 |
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7.6.1 Solutions Close to the Origin (Case J1<0) 172 |
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7.6.2 Angle-Action Variables of H *2 (Case J1<0) 173 |
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7.7 The Non-Resonance Condition 173 |
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7.8 Example 175 |
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Chapter 8 Lie Series Theory for Resonant Systems |
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8.1 Bohlin's Problem (The Single-Resonance Problem) 181 |
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8.2 Outline of the Solution 182 |
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8.3 Functions Expansions 185 |
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8.4 Perturbation Equations 188 |
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8.5 Averaging 190 |
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8.6 An Example 190 |
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8.7 Example with a Separated Hori Kernel 198 |
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8.8 One Degree of Freedom 204 |
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8.8.1 Garfinkel's Ideal Resonance Problem 204 |
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Chapter 9 Single Resonance near a Singularity |
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9.1 Resonances near the Origin: Real and Virtual 209 |
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9.2 One Degree of Freedom 210 |
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9.3 Many Degrees of Freedom. One Single Resonance 213 |
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9.4 A First-Order Resonance Case Study 216 |
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9.4.1 The Hori Kernel 218 |
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9.4.2 First Perturbation Equation 219 |
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9.4.3 Averaging 220 |
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9.4.4 The Post-Harmonic Solution 221 |
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9.4.5 Secular Resonance 223 |
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9.4.6 Secondary Resonances 224 |
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9.4.7 Initial Conditions Diagram 225 |
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9.5 Sessin Transformation and Integral 227 |
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9.5.1 The Restricted (Asteroidal) Case 229 |
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Chapter 10 Nonlinear Oscillators |
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10.1 Quasiharmonic Hamiltonian Systems 231 |
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10.2 Formal Solutions. General Case 232 |
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10.3 Exact Commensurability of Frequencies (Resonance) 234 |
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10.4 Birkhoff Normalization 236 |
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10.4.1 A Formal Extension Including One Single Resonance 240 |
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10.4.2 The Comensurabilities of Lower Order 242 |
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10.5 The Restricted Three-Body Problem 242 |
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10.5.1 Equations of the Motion around the Lagrangian Point L 4 244 |
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10.5.2 Internal 2:1 Resonance 246 |
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10.5.3 Internal 3:1 Resonance 247 |
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10.5.4 Other Internal Resonances 249 |
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10.6 The Hénon--Heiles Hamiltonian 250 |
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10.6.1 The Toda Lattice Hamiltonian 252 |
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10.7 Systems with Multiple Commensurabilities 53 |
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10.7.1 The Ford--Lunsford Hamiltonian. 1:2:3 Resonance 255 |
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10.8 Parametrically Excited Systems 255 |
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10.8.1 A Nonlinear Extension 260 |
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App. A Bohlin Theory |
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A.1 Bohlin's Resonance Problem 263 |
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A.2 Bohlin's Perturbation Equations 265 |
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A.3 Poincaré Singularity 268 |
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A.4 An Extension of Delaunay Theory 269 |
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App. B The Simple Pendulum |
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B.1 Equations of Motion 271 |
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B.1.1 Circulation 273 |
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B.1.2 Libration 274 |
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B.1.3 The Separatrix 276 |
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B.2 Angle-Action Variables of the Pendulum 277 |
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B.2.1 Circulation 277 |
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B.2.2 Libration 278 |
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B.3 Small Oscillations of the Pendulum 279 |
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B.3.1 Angle--Action Variables 280 |
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B.4 Direct Construction of Angle--Action Variables 281 |
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B.5 The Neighborhood of the Pendulum Separatrix 283 |
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B.5.1 Motion near the Separatrix 285 |
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B.6 The Separatrix or Whisker Map 286 |
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B.7 The Standard Map 288 |
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App. C Andoyer Hamiltonian with k=1 |
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C.1 Andoyer Hamiltonians 289 |
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C.2 Centers and Saddle Points 290 |
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C.2.1 The Case k=1 292 |
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C.3 Morphogenesis 293 |
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C.4 Width of the Libration Zone 296 |
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C.5 Integration 298 |
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C.5.1 The Case D>0 301 |
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C.5.2 The Case D<0 302 |
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C.5.3 The Separatrices 303 |
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C.5.4 The Angle s 304 |
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C.6 Equilibrium Points 305 |
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C.6.1 The Inner Circulations Center 306 |
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C.6.2 The Libration Center 306 |
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C.7 Proper Periods 306 |
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C.7.1 Inner Circulations 307 |
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C.7.2 Librations 307 |
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C.8 The Angle Variable w 308 |
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C.9 Small-Amplitude Librations 308 |
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C.9.1 The Action L 312 |
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C.9.2 The New Hamiltonian 312 |
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App. D Andoyer Hamiltonians with k≥2 |
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D.1 Introduction 315 |
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D.2 The Case k=2 315 |
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D.2.1 Morphogenesis 316 |
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D.2.2 Width of the Libration Zone 318 |
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D.3 The Case k=3 320 |
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D.3.1 Morphogenesis 321 |
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D.3.2 Width of the Libration Zone 323 |
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D.4 The Case k=4 325 |
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D.4.1 Morphogenesis 327 |
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D.4.2 Width of the Libration Zone 327 |
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D.5 Comparative Analysis 328 |
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D.5.1 Virtual Resonances 329 |
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References 331 |
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Index 337 |