| Description |
1 online resource (xiv, 341 pages) : illustrations |
| Series |
Astrophysics and space science library ; v. 345 |
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Astrophysics and space science library ; v. 345.
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| Contents |
1 The Hamilton-Jacobi Theory 1 -- 1.1 Canonical Pertubation Equations 1 -- 1.2 Hamilton's Principle 2 -- 1.2.1 Maupertuis' Least Action Principle 4 -- 1.2.2 Helmholtz Invariant 5 -- 1.3 Canonical Transformations 6 -- 1.4 Lagrange Brackets 9 -- 1.5 Poisson Brackets 11 -- 1.5.1 Reciprocity Relations 12 -- 1.6 The Extended Phase Space 13 -- 1.7 Gyroscopic Systems 15 -- 1.7.1 Gyroscopic Forces 15 -- 1.7.3 Rotating Frames 17 -- 1.7.4 Apparent Forces 17 -- 1.8 The Partial Differential Equation of Hamilton and Jacobi 18 -- 1.9 One-Dimensional Motion with a Generic Potential 20 -- 1.9.1 The Case m <0 23 -- 1.9.2 The Harmonic Oscillator 23 -- 1.10 Involution. Mayer's Lemma. Liouville's Theorem 24 -- 2 Angle-Action Variables. Separable Systems 29 -- 2.1 Periodic Motions 29 -- 2.1.1 Angle-Action Variables 30 -- 2.1.2 The Sign of the Action 32 -- 2.2 Direct Construction of Angle-Action Variables 33 -- 2.3 Actions in Multiperiodic Systems. Einstein's Theory 35 -- 2.4 Separable Multiperiodic Systems 37 -- 2.4.1 Uniformized Angles. Charlier's Theory 37 -- 2.4.2 The Actions 38 -- 2.4.3 Algorithms for Construction of the Angles 39 -- 2.4.4 Angle-Action Variables of H(q[subscript 1], p[subscript 1], [subscript 2] ..., p[subscript N]) 40 -- 2.4.5 Historical Postscript 42 -- 2.5 Simple Separable Systems 42 -- 2.5.1 Example: Central Motions 43 -- 2.5.2 Angle-Action Variables of Central Motions 44 -- 2.6 Kepler Motion 47 -- 2.7 Degeneracy 50 -- 2.7.1 Schwarzschild Transformation 51 -- 2.7.2 Delaunay Variables 52 -- 2.8 The Separable Cases of Liouville and Stackel 53 -- 2.8.1 Example: Liouville Systems 55 -- 2.8.2 Example: Stackel Systems 56 -- 2.8.3 Example: Central Motions 56 -- 2.9 Angle-Action Variables of a Quadratic Hamiltonian 57 -- 2.9.1 Gyroscopic Systems 60 -- 3 Classical Perturbation Theories 61 -- 3.1 The Problem of Delaunay 61 -- 3.2 The Poincare Theory 63 -- 3.2.1 Expansion of H[subscript 0] 65 -- 3.2.2 Expansion of H[subscript k] 66 -- 3.2.3 Perturbation Equations 67 -- 3.3 Averaging Rule 68 -- 3.3.1 Small Divisors. Non-Resonance Condition 69 -- 3.4 Degenerate Systems. The von Zeipel-Brouwer Theory 70 -- 3.4.1 Expansion of H[subscript *] 72 -- 3.4.2 von Zeipel-Brouwer Perturbation Equations 72 -- 3.4.3 The von Zeipel Averaging Rule 73 -- 3.5 Small Divisors and Resonance 74 -- 3.5.1 Elimination of the Non-Critical Short-Period Angles 74 -- 3.6 An Example -- Part I 77 -- 3.7 Linear Secular Theory 81 -- 3.8 An Example -- Part II 83 -- 3.9 Iterative Use of von Zeipel-Brouwer Operations 86 -- 3.10 Divergence of the Series. Poincare's Theorem 88 -- 3.11 Kolmogorov's Theorem 88 -- 3.11.1 Frequency Relocation 89 -- 3.11.2 Convergence 91 -- 3.11.3 Degenerate Systems 93 -- 3.11.4 Degeneracy in the Extended Phase Space 94 -- 3.12 Inversion of a Jacobian Transformation 94 -- 3.12.1 Lagrange Implicit Function Theorem 96 -- 3.13 Lindstedt's Direct Calculation of the Series 97 -- 4 Resonance 99 -- 4.1 The Method of Delaunay's Lunar Theory 99 -- 4.2 Introduction of the Square Root of the Small Parameter 101 -- 4.2.1 Garfinkel's Abnormal Resonance 103 -- 4.3 Delaunay Theory According to Poincare 103 -- 4.3.1 First-Approximation Solution 106 -- 4.4 Garfinkel's Ideal Resonance Problem 107 -- 4.4.1 Garfinkel-Jupp-Williams Integrals 109 -- 4.4.2 Circulation ([Characters not reproducible]> [Characters not reproducible]> 0) 110 -- 4.4.3 Libration ( 4.4.4 Asymptotic Motions (E = A[subscript *]) 114 -- 4.5 Angle-Action Variables of the Ideal Resonance Problem 115 -- 4.5.1 Circulation 115 -- 4.5.2 Libration 116 -- 4.5.3 Small-Amplitude Librations 117 -- 4.6 Morbidelli's Successive Elimination of Harmonics 118 -- 5 Lie Mappings 127 -- 5.1 Lie Transformations 127 -- 5.1.1 Infinitesimal Canonical Transformations 127 -- 5.2 Lie Derivatives 130 -- 5.3 Lie Series 131 -- 5.4 Inversion of a Lie Mapping 134 -- 5.5 Lie Series Expansions 135 -- 5.5.1 Lie Series Expansion of f 136 -- 5.5.2 Deprit's Recursion Formula 137 -- 6 Lie Series Perturbation Theory 139 -- 6.2 Lie Series Theory with Angle-Action Variables 140 -- 6.2.1 Averaging 142 -- 6.2.2 High-Order Theories 143 -- 6.3 Comparison to Poincare Theory. Example I 144 -- 6.4 Comparison to Poincare Theory. Example II 147 -- 6.5 Hori's General Theory. Hori Kernel and Averaging 151 -- 6.5.1 Cauchy-Darboux Theory of Characteristics 154 -- 6.6 Topology and Small Divisors 155 -- 6.6.1 Topological Constraint. The Rise of Small Divisors 156 -- 6.7 Hori's Formal First Integral 157 -- 6.8 "Average" Hamiltonians 158 -- 6.8.1 On Secular Theories and Proper Elements 159 -- 7 Non-Singular Canonical Variables 161 -- 7.1 Singularities of the Actions 161 -- 7.2 Poincare Non-Singular Variables 162 -- 7.3 The d'Alembert Property 164 -- 7.4 Regular Integrable Hamiltonians 165 -- 7.5 Lie Series Expansions About the Origin 167 -- 7.6 Lie Series Perturbation Theory in Non-Singular Variables 169 -- 7.6.1 Solutions Close to the Origin (Case J[subscript 1]<0) 172 -- 7.6.2 Angle-Action Variables of H[Characters not reproducible] (Case J[subscript 1]<0) 173 -- 7.7 The Non-Resonance Condition 173 -- 8 Lie Series Theory for Resonant Systems 181 -- 8.1 Bohlin's Problem (The Single-Resonance Problem) 181 -- 8.2 Outline of the Solution 182 -- 8.3 Functions Expansions 185 -- 8.4 Perturbation Equations 188 -- 8.5 Averaging 190 -- 8.7 Example with a Separated Hori Kernel 198 -- 8.8 One Degree of Freedom 204 -- 8.8.1 Garfinkel's Ideal Resonance Problem 204 -- 9 Single Resonance near a Singularity 209 -- 9.1 Resonances Near the Origin: Real and Virtual 209 -- 9.2 One Degree of Freedom 210 -- 9.3 Many Degrees of Freedom. One Single Resonance 213 -- 9.4 A First-Order Resonance Case Study 216 -- 9.4.1 The Hori Kernel 218 -- 9.4.2 First Perturbation Equation 219 -- 9.4.3 Averaging 220 -- 9.4.4 The Post-Harmonic Solution 221 -- 9.4.5 Secular Resonance 223 -- 9.4.6 Secondary Resonances 224 -- 9.4.7 Initial Conditions Diagram 225 -- 9.5 Sessin Transformation and Integral 227 -- 9.5.1 The Restricted (Asteroidal) Case 229 -- 10 Nonlinear Oscillators 231 -- 10.1 Quasiharmonic Hamiltonian Systems 231 -- 10.2 Formal Solutions. General Case 232 -- 10.3 Exact Commensurability of Frequencies (Resonance) 234 -- 10.4 Birkhoff Normalization 236 -- 10.4.1 A Formal Extension Including One Single Resonance 240 -- 10.4.2 The Comensurabilities of Lower Order 242 -- 10.5 The Restricted Three-Body Problem 242 -- 10.5.1 Equations of the Motion Around the Lagrangian Point L[superscript 4] 244 -- 10.5.2 Internal 2:1 Resonance 246 -- 10.5.3 Internal 3:1 Resonance 247 -- 10.5.4 Other Internal Resonances 249 -- 10.6 The Henon-Heiles Hamiltonian 250 -- 10.6.1 The Toda Lattice Hamiltonian 252 -- 10.7 Systems with Multiple Commensurabilities 253 -- 10.7.1 The Ford-Lunsford Hamiltonian. 1:2:3 Resonance 255 -- 10.8 Parametrically Excited Systems 255 -- 10.8.1 A Nonlinear Extension 260 -- A Bohlin Theory 263 -- A.1 Bohlin's Resonance Problem 263 -- A.2 Bohlin's Perturbation Equations 265 -- A.3 Poincare Singularity 268 -- A.4 An Extension of Delaunay Theory 269 -- B The Simple Pendulum 271 -- B.1 Equations of Motion 271 -- B.1.1 Circulation 273 -- B.1.2 Libration 274 -- B.1.3 The Separatrix 276 -- B.2 Angle-Action Variables of the Pendulum 277 -- B.2.1 Circulation 277 -- B.2.2 Libration 278 -- B.3 Small Oscillations of the Pendulum 279 -- B.3.1 Angle-Action Variables 280 -- B.4 Direct Construction of Angle-Action Variables 281 -- B.5 The Neighborhood of the Pendulum Separatrix 283 -- B.5.1 Motion near the Separatrix 285 -- B.6 The Separatrix or Whisker Map 286 -- B.7 The Standard Map 288 -- C Andoyer Hamiltonian with k = 1 289 -- C.1 Andoyer Hamiltonians 289 -- C.2 Centers and Saddle Points 290 -- C.2.1 The Case k = 1 292 -- C.3 Morphogenesis 293 -- C.4 Width of the Libration Zone 296 -- C.5 Integration 298 -- C.5.1 The Case [Delta]> 0 301 -- C.5.2 The Case [Delta] <0 302 -- C.5.3 The Separatrices 303 -- C.5.4 The Angle [sigma] 304 -- C.6 Equilibrium Points 305 -- C.6.1 The Inner Circulations Center 306 -- C.6.2 The Libration Center 306 -- C.7 Proper Periods 306 -- C.7.1 Inner Circulations 307 -- C.7.2 Librations 307 -- C.8 The Angle Variable w 308 -- C.9 Small-Amplitude Librations 308 -- C.9.1 The Action [Lambda] 312 -- C.9.2 The New Hamiltonian 312 -- D Andoyer Hamiltonians with k [greater than or equal] 2 315 -- D.2 The Case k = 2 315 -- D.2.1 Morphogenesis 316 -- D.2.2 Width of the Libration Zone 318 -- D.3 The Case k = 3 320 -- D.3.1 Morphogenesis 321 -- D.3.2 Width of the Libration Zone 323 -- D.4 The Case k = 4 325 -- D.4.1 Morphogenesis 327 -- D.4.2 Width of the Libration Zone 327 -- D.5 Comparative Analysis 328 -- D.5.1 Virtual Resonances 329 |
| Summary |
Canonical Perturbation Theories, Degenerate Systems and Resonance presents the foundations of Hamiltonian Perturbation Theories used in Celestial Mechanics, emphasizing the Lie Series Theory and its application to degenerate systems and resonance. This book is the complete text on the subject including advanced topics in Hamiltonian Mechanics, Hori's Theory, and the classical theories of Poincaré, von Zeipel-Brouwer, and Delaunay. Also covered are Kolmogorov's frequency relocation method to avoid small divisors, the construction of action-angle variables for integrable systems, and a complete overview of some problems in Classical Mechanics. Sylvio Ferraz-Mello makes these ideas accessible not only to Astronomers, but also to those in the related fields of Physics and Mathematics |
| Analysis |
resonance hamiltonian methods in celestial mechanics and applications |
| Bibliography |
Includes bibliographical references (pages 331-335) and index |
| Notes |
Print version record |
| In |
Springer e-books |
| Subject |
Perturbation (Astronomy)
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Series, Lie.
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Hamiltonian systems.
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SCIENCE -- Astronomy.
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Series, Lie.
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Hamiltonian systems.
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Mecânica celeste.
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Perturbation (Astronomy)
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Physique.
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Astronomie.
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Hamiltonian systems
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Perturbation (Astronomy)
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Series, Lie
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Mecânica celeste.
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| Form |
Electronic book
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| ISBN |
9780387389059 |
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0387389059 |
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0387389008 |
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9780387389004 |
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6610865388 |
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9786610865383 |
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