Canonical Perturbation Theories: Degenerate Systems and Resonance / Edition 1

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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 Poincare, 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.
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Editorial Reviews

From the Publisher

From the reviews:

"The book consists of 10 chapters and four appendices. … Undoubtedly, the monograph will be useful for postgraduate students." (Sergei Georgievich Zhuravlev, Zentralblatt MATH, Vol. 1122 (24), 2007)

"The book by Sylvio Ferraz-Mello provides a deep insight in perturbation theories. … the mathematical presentation of the different facets is always accompanied by a number of case studies and examples. Such peculiarity makes the book unique … . Thanks to the … author, the book is recommended for both teaching and research purposes: the didactical and self-contained aspects address the book to advanced undergraduate students; the researchers in Dynamical Systems and Celestial Mechanics will certainly enjoy the excellent and deep presentation of perturbation theories." (Alessandra Celletti, Celestial Mechanics and Dynamical Astronomy, Vol. 100 (4), 2008)

"The purpose of this book is … ‘to deal with the perturbation theories used in Celestial Mechanics, but they should be presented in a universal way, so as to be understandable by investigators and students from related fields of science.’ In other words, one does not need a special background in astronomy to read and understand the book. … a large number of the examples will be of interest to people working in Hamiltonian theory. … a valuable source of material on conservative systems." (Ferdinand Verhulst, SIAM Review, Vol. 50 (2), 2008)

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Product Details

  • ISBN-13: 9780387389004
  • Publisher: Springer New York
  • Publication date: 1/16/2007
  • Series: Astrophysics and Space Science Library Series, #345
  • Edition description: 2007
  • Edition number: 1
  • Pages: 346
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.81 (d)

Table of Contents

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|<<br>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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