Geography of Order and Chaos in Mechanics: Investigations of Quasi-Integrable Systems with Analytical, Numerical, and Graphical Tools (Progress in Mathematical Physics Book 64)

This original monograph aims to explore the dynamics in the particular but very important and significant case of quasi-integrable Hamiltonian systems, or integrable systems slightly perturbed by other forces. With both analytic and numerical methods, the book studies several of these systems—including for example the hydrogen atom or the solar system, with the associated Arnold web—through modern tools such as the frequency modified fourier transform, wavelets, and the frequency modulation indicator. Meanwhile, it draws heavily on the more standard KAM and Nekhoroshev theorems. __Geography of Order and Chaos in Mechanics__ will be a valuable resource for professional researchers and certain advanced undergraduate students in mathematics and physics, but mostly will be an exceptional reference for Ph.D. students with an interest in perturbation theory. Geography of Order and Chaos in Mechanics 3 Preface 7 Contents 11 List of Figures 15 CHAPTER 1 Introductory Survey 19 1.1 Configuration Space and Lagrangian Dynamics 20 1.2 Symplectic Manifolds 21 1.3 Phase Space and Hamiltonian Dynamics 22 1.4 The Liouville and Arnold Theorems 23 1.5 Quasi-Integrable Hamiltonian Systems and KAM Theorem 25 1.6 Geography of the Phase Space 29 1.7 Numerical Tools 30 1.8 The Perturbed Kepler Problem 32 1.9 The Multi-Body Gravitational Problem 33 CHAPTER 2 Analytical Mechanics and Integrable Systems 35 2.1 Differential Geometry 35 2.1.1 Differentiable Manifolds 36 Tangent and Cotangent Spaces 38 Push-forward and Pull-back 40 2.1.2 Tensors and Forms 42 Forms and Exterior Derivative 43 Lie Derivative 46 2.1.3 Riemannian, Symplectic, and Poisson Manifolds 48 Riemannian Manifolds 48 Symplectic Manifolds 50 Poisson Manifolds 52 2.2 Lie Groups and Lie Algebras 54 2.2.1 Definition and Properties 54 2.2.2 Adjoint and Coadjoint Representation 58 2.2.3 Action of a Lie Group on a Manifold 60 2.2.4 Classification of Lie Groups and Lie Algebras 63 2.3 Lagrangian and Hamiltonian Mechanics 66 2.3.1 Lagrange Equations 67 2.3.2 Hamilton Principle 69 2.3.3 Noether Theorem 70 2.3.4 From Lagrange to Hamilton 72 2.3.5 Canonical Transformations 74 2.3.6 Hamilton–Jacobi Equation 75 2.3.7 Symmetries and Reduction 76 2.3.8 Liouville Theorem 81 2.3.9 Arnold Theorem 86 2.3.10 Action-Angle Variables: Examples 91 2.A Appendix: The Problem of two Fixed Centers 96 CHAPTER 3 Perturbation Theory 100 3.1 Formal Expansions 101 3.1.1 Lie Series and Formal Canonical Transformations 101 3.1.2 Homological Equation and Its Formal Solution 104 3.2 Perpetual Stability and KAM Theorem 108 3.2.1 Cauchy Inequality 110 3.2.2 Convergence of Lie Series 112 3.2.3 Homological Equation and Its Solution 114 3.2.4 KAM Theorem (According to Kolmogorov) 117 3.2.5 KAM Theorem (According to Arnold) 127 3.2.6 Isoenergetic KAM Theorem 129 3.3 Exponentially Long Stability and Nekhoroshev Theorem 130 3.4 Geography of the Phase Space 137 3.5 Elliptic Equilibrium Points 140 3.A Appendix: Results from Diophantine Theory 142 3.B Appendix: Homoclinic Tangle and Chaos 144 CHAPTER 4 Numerical Tools I: ODE Integration 148 4.1 Cauchy Problem 149 4.2 Euler Method 150 4.3 Runge–Kutta Methods 152 4.4 Gragg–Bulirsch–Stoer Method 154 4.5 Adams–Bashforth–Moulton Methods 155 4.6 Geometric Methods 157 4.7 What Methods Are in the MATLAB Programs? 161 ode113 162 ode45 162 IRK–Gauss 162 Dop853 162 Odex 162 Tom 162 RungeKutta4 163 RungeKutta5 163 gni_irk2 163 gni_lmm2 163 gni_comp 163 CHAPTER 5 Numerical Tools II: Detecting Order, Chaos, and Resonances 164 5.1 Poincaré Section 165 5.2 Variational Equation Methods 166 5.2.1 The Largest Lyapunov Exponents 167 5.2.2 The Fast Lyapunov Indicator 167 5.2.3 Other Methods 168 5.3 Fourier Transform Methods 169 5.3.1 Fast Fourier Transform (FFT) 170 5.3.2 Frequency Modified Fourier Transform (FMFT) 171 5.3.3 Wavelets and Time-Frequency Analysis 173 5.3.4 Frequency Modulation Indicator (FMI) 179 5.4 Some Examples 182 5.4.1 Symplectic Maps and the POINCARE Program 183 5.4.2 A Test Hamiltonian and the HAMILTON Program 185 5.4.3 The Lagrange Points L4 and L5 and the LAGRANGE Program 186 CHAPTER 6 The Kepler Problem 191 6.1 Basics on the Kepler Problem 191 6.1.1 Conformal Group and Geodesic Flow on the Sphere 195 6.1.2 The Moser–Souriau Regularization 199 6.1.3 The Kustaanheimo–Stiefel Regularization 201 6.1.4 Action-Angle Variables 204 Delaunay Variables 206 Pauli Variables 207 6.2 The Perturbed Kepler Problem 209 6.3 Normal Form of the Perturbed Kepler Problem 211 6.4 Numerical Integration 216 6.4.1 Perturbative Method 217 6.4.2 Non-Perturbative Method 218 6.5 Reduction under Axial Symmetry 218 CHAPTER 7 The KEPLER Program 226 7.1 First Window: Single Orbit Analysis 226 7.1.1 Perturbation Hamiltonian 226 Zeeman Effect 227 Stark Effect 228 Quadratic Zeeman Effect 228 Euler Problem 228 Oscillator 229 p_2 230 Satellite 230 Hill Potential 230 Inverse Quadratic and Cubic Potentials 230 Anisotropic 230 7.1.2 Initial Conditions 231 7.1.3 ODE Solver 231 7.1.4 Revolution Number 231 7.1.5 Regularization Method 232 Space and Momentum Trajectories 232 Kepler Parameters vs False Time 232 Kepler Parameters vs Time 232 Time vs False Time 232 Integration Errors 232 False Time vs Step Number 233 User Functions 233 7.1.6 Standard Method 233 7.1.7 Action-Angle Variables and Time-Frequency Analysis on KAM Tori 234 7.2 Second Window: “Global Analysis: Poincaré Section” 236 7.3 Third Window: “Global Analysis: Rotated Delaunay” 237 7.4 Fourth Window: “Global Analysis: Rotated Pauli” 239 7.5 Menu 239 7.5.1 File 239 7.5.2 Window 240 7.5.3 Figure 241 No degree symbol 241 Large marker size 241 Save figure as. . . 241 Mouse track 241 Scroll plot 241 Magnifying glass 242 7.5.4 Normal Form 242 7.5.5 Help 242 CHAPTER 8 Some Perturbed Keplerian Systems 243 8.1 The Stark–Quadratic–Zeeman (SQZ) Problem 243 8.1.1 First Order Normal Form 244 8.1.2 Second Order Nonintegrable Normal Form 246 8.1.3 Second Order Integrable Normal Form 248 8.1.4 The Quadratic Zeeman (QZ) Problem 252 8.1.5 The Parallel (SQZp) Problem 259 8.1.6 The Crossed (SQZc) Problem 260 8.1.7 The Generic (SQZ) Problem 269 8.2 The Non-Planar Circular Restricted Three-Body (CR3B) Problem 269 8.3 Satellite about an Oblate Primary 276 CHAPTER 9 The Multi-Body Gravitational Problem 279 9.1 Global Planar Three-Body Problem 280 9.1.1 The 2-Dimensional Secular Problem 282 9.1.2 Reduction of the Symplectic Manifold S2×S2 under the SO(2)-action 285 9.1.3 The Reduced Motion 289 (i) Analytical integration of the approximate Hamiltonian 291 (ii) Numerical integration of the exact Hamiltonian 296 9.2 The 3-Dimensional Planetary Problem 298 9.2.1 SO(3) and Poincaré Variables 299 9.2.2 The Secular Planetary Problem 304 9.2.3 Linear Approximation: Lagrange–Laplace Theory 309 9.3 The LAPLACE Program 311 9.3.1 First Panel: Initial Conditions 312 9.3.2 Second Panel: Integration 312 9.3.3 Third Panel: Plot and Frequency Analysis 313 9.3.4 Fourth Panel: Frequency Modulation Indicator 314 9.3.5 Menu 314 9.4 Some Examples 315 CHAPTER 10 Final Remarks and Perspectives 319 Numerical Detection of the Frequency Modulation 319 Arnold Diffusion 320 Leaving an Angle in the Normalized Hamiltonian 320 Herman’s Resonance 320 Transition State Theory 320 APPENDIX A What is in the CD? 323 1) PhSpGeo.zip 323 2) Visualize_3D.zip 329 3) Matlab figures 329 4) Euler.mw 329 APPENDIX B If one has no access to MATLAB 331 Bibliography 333 Index 341 This original monograph aims to explore the dynamics in the particular but very important and significant case of quasi-integrable Hamiltonian systems, or integrable systems slightly perturbed by other forces. With both analytic and numerical methods, the book studies several of these systems--including for example the hydrogen atom or the solar system, with the associated Arnold web--through modern tools such as the frequency-modified fourier transform, wavelets, and the frequency-modulation indicator. Meanwhile, it draws heavily on the more standard KAM and Nekhoroshev theorems. Geography of Order and Chaos in Mechanics contains many figures that illuminate its concepts in novel ways, but perhaps its most useful feature is its inclusion of software to reproduce the various numerical experiments. The graphical user interfaces of five supplied MATLAB programs allows readers without any knowledge of computer programming to visualize and experiment with the distribution of order, chaos and resonances in various Hamiltonian systems. This monograph will be a valuable resource for professional researchers and certain advanced undergraduate students in mathematics and physics, but mostly will be an exceptional reference for Ph. D. students with an interest in perturbation theory Front Matter....Pages i-xviii Introductory Survey....Pages 1-16 Analytical Mechanics and Integrable Systems....Pages 17-81 Perturbation Theory....Pages 83-130 Numerical Tools I: ODE Integration....Pages 131-146 Numerical Tools II: Detecting Order, Chaos, and Resonances....Pages 147-173 The Kepler Problem....Pages 175-209 The KEPLER Program....Pages 211-227 Some Perturbed Keplerian Systems....Pages 229-264 The Multi-Body Gravitational Problem....Pages 265-304 Final Remarks and Perspectives....Pages 305-308 Back Matter....Pages 309-332