Asymptotic Methods in Resonance Analytical Dynamics

Asymptotic Methods in Resonance Analytical Dynamics......Page 1
Introduction to the Series......Page 3
Preface......Page 5
Acknowledgments......Page 13
Contents......Page 14
1.0 Introduction......Page 18
1.1 Main Symbols......Page 20
1.2 Asymptotic Series and their Properties......Page 22
Theorem 1.3.1......Page 25
1.4 Geometric Interpretation of Solutions of Oscillating Systems......Page 29
1.5 On the Method of Characteristics for Quasi-Linear First-Order Partial Differential Equations: Method of Characteristics......Page 36
1.6 Iterative Variant of the Poincare-Lyapunov Small Parameter Method......Page 37
1.6.1 Simple iterations......Page 38
1.6.2 Iterations with quadratic convergence......Page 41
1.7 Comments and References......Page 45
2.0 Introduction......Page 46
2.1 Mathematical Averaging Principle......Page 49
2.2 Classification of Systems of Differential Equations, where Resonance are Possible......Page 55
2.3 The Basis of the Asymptotic Theory for Locally Nonresonance Systems......Page 58
2.4 Initial Conditions for Comparison Equations......Page 70
2.5 Averaging Operator for Time-Independent Disturbances......Page 74
2.6 Asymptotic Theory of Systems with Their Paths Passing through Resonance Points......Page 78
2.7 The Algorithim of Joining of Resonance and Nonresonance Path Sections......Page 83
2.8 Periodic and Quasi-Periodic Oscillations in the Van der Pol Oscillator System......Page 84
2.9 Study of Multifrequency Systems with Their Solutions Not Remaining Close to Resonance Points......Page 105
2.10 Study of Multifrequency Systems Belonging to Class II......Page 116
2.11 Multifrequency Systems with their Solutions Not Leaving the Neighborhood of a Resonance Point......Page 123
2.12 Comments and References......Page 129
3.1 Newtonian Three-Body Problem......Page 132
3.2 The Problem of Justification of the Averaging Principle in the Bounded Newtonian Three-Body Problem......Page 142
3.3 Construction of Explicit Solutions of Averaged Differential Equations of the Bounded Three-Body Problem in the Case of Resonance......Page 149
3.4 Quasi-Periodic Solutions of Resonance Hamiltonian Systems......Page 154
3.5 Motion of a Geostationary Satellite......Page 164
3.6 Averaging Method in the Theory of Partial Differential Equations......Page 172
3.7 Energy Method of Construction of Amplitude-Phase Equations......Page 181
3.8 Averaging Method and Maximum Principle in Boundary Value Problems......Page 187
3.9 Comments and References......Page 197
4.0 Introduction......Page 199
4.1 Construction of Lyapunov Transform for a Linear System with Periodic Coefficients......Page 200
4.1.1 Construction of matrices L (t) and W by means of series......Page 201
4.1.2 Construction of matrices L (t) and W by means of iterations......Page 202
4.1.3 Interpolation formulae for Lyapunov transform matrices......Page 204
4.2.1 Noncritical case......Page 207
4.2.2 Example......Page 209
4.2.3 Critical cases......Page 211
4.3 Direct Numerical-Analytic Method of Construction of Periodic Solutions......Page 212
4.4 Construction of Periodic Solutions in Hill\'s Problem of Lunar Motion......Page 218
4.5.1 Algorithm construction......Page 222
4.5.2 Computational layout......Page 225
4.5.3 Quick-Basic program......Page 228
4.5.4 Comments on the program......Page 235
4.6.1 Initial differential equations of the problem......Page 236
4.6.2 Basic equations for the coefficients of the sought solution......Page 238
4.6.3 Construction of a solution by the method of simple iterations......Page 240
4.6.4 Construction of a solution by the method of iterations with quadratic convergence......Page 244
4.7 Numerical-Analytic Implementation of Krylov-Bogolyubov Transform......Page 252
4.8 Comments and References......Page 257
References......Page 259