Canonical Perturbation Theories: Degenerate Systems and Resonance (Astrophysics and Space Science Library (345))

The book is written mainly to advanced graduate and post-graduate students following courses in Perturbation Theory and Celestial Mechanics. It is also intended to serve as a guide in research work and is written in a very explicit way: all perturbation theories are given with details allowing its immediate application to real problems. In addition, they are followed by examples showing all steps of their application. The book is not intended to explore the mathematics of Hamiltonian Systems, but may be useful to mathematicians in a great deal of circumstances as a reference on the practical application of the theories. In the same way, it may be a source book on the problems of degeneracy and small divisors, which affect the use of perturbation theories as well in Celestial Mechanics as in Physics. 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 [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]