Chaos in Classical and Quantum Mechanics

Description

Cover Title page Preface Introduction 1. The Mechanics of Lagrange 1.1 Newton's Equations According to Lagrange 1.2 The Variational Principle of Lagrange 1.3 Conservation of Energy 1.4 Example: Space Travel in a Given Time Interval; Lambert's Formula 1.5 The Second Variation 1.6 The Spreading Trajectories 2. The Mechanics of Hamilton and Jacobi 2.1 Phase Space and Its Hamiltonian 2.2 The Action Function S 2.3 The Variational Principle of Euler and Maupertuis 2.4 The Density of Trajectories on the Energy Surface 2.5 Example: Space Travel with a Given Energy 3. Integrable Systems 3.1 Constants of Motion and Poisson Brackets 3.2 Invariant Tori and Action-Angle Variables 3.3 Multiperiodic Motion 3.4 The Hydrogen Molecule Ion 3.5 Geodesics on a Triaxial Ellipsoid 3.6 The Toda Lattice 3.7 Integrable versus Separable 4. The Three-Body Problem: Moon-Earth-Sun 4.1 Reduction to Four Degrees of Freedom 4.2 Applications in Atomic Physics and Chemistry 4.3 The Action-Angle Variables in the Lunar Observations 4.4 The Best Temporary Fit to a Kepler Ellipse 4.5 The Time-Dependent Hamiltonian 5. Three Methods of Solution 5.1 Variation of the Constants (Lagrange) 5.2 Canonical Transformations (Delaunay) 5.3 The Application of Canonical Transformations 5.4 Small Denominators and Other Difficulties 5.5 Hill's Periodic Orbit in the Three-Body Problem 5.6 The Motion of the Perigee and the Node 5.7 Displacements from the Periodic Orbit and Hill's Equation 6. Periodic Orbits 6.1 Potentials with Circular Symmetry 6.2 The Number of Periodic Orbits in an Integrable System 6.3 The Neighborhood of a Periodic Orbit 6.4 Elliptic, Parabolic, and Hyperbolic Periodic Orbits 7. The Surface of Section 7.1 The Invariant Two-Form 7.2 Integral Invariants and Liouville's Theorem 7.3 Area Conservation on the Surface of Section 7.4 The Theorem of Darboux 7.5 The Conjugation of Time and Energy in Phase Space 8. Models of the Galaxy and of Small Molecules 8.1 Stellar Trajectories in the Galaxy 8.2 The Hénon-Heiles Potential 8.3 Numerical Investigations 8.4 Some Analytic Results 8.5 Searching for Integrability with Kowalevskaya and Painlevé 8.6 Discrete Area-Preserving Maps 9. Soft Chaos and the KAM Theorem 9.1 The Origin of Soft Chaos 9.2 Resonances in Celestial Mechanics 9.3 The Analogy with the Ordinary Pendulum 9.4 Islands of Stability and Overlapping Resonances 9.5 How Rational Are the Irrational Numbers? 9.6 The KAM Theorem 9.7 Homoclinic Points 9.8 The Lore of the Golden Mean 10. Entropy and Other Measures of Chaos 10.1 Abstact Dynamical Systems 10.2 Ergodicity, Mixing, and K-Systems 10.3 The Metric Entropy 10.4 The Automorphisms of the Torus 10.5 The Topological Entropy 10.6 Anosov Systems and Hard Chaos 11. The Anisotropic Kepler Problem 11.1 The Donor Impurity in a Semiconductor Crystal 11.2 Normalized Coordinates in the Anisotropic Kepler Prob1em 11.3 The Surface of Section 11.4 Construction of Stable and Unstable Manifolds 11.5 The Periodic Orbits in the Anisotropic Kepler Problem 11.6 Some Questions Conceming the AKP 12. The Transition from Classical to Quantum Mechanics 12.1 Are Classical Mechanics and Quantum Mechanics Compatible? 12.2 Changing Coordinates in the Action 12.3 Adding Actions and Multiplying Probabilities 12.4 Rutherford Scattering 12.5 The Classical Version of Quantum Mechanics 12.6 The Propagator in Momentum Space 12.7 The Classical Green's Function 12.8 The Hydrogen Atom in Momentum Space 13. The New World of Quantum Mechanics 13.1 The Liberation from Classical Chaos 13.2 The Time-Dependent Schrödinger Equation 13.3 The Stationary Schrödinger Equation 13.4 Feynman's Path Integral 13.5 Changing Coordinates in the Path Integral 13.6 The Classical Limit 14. The Quantization of Integrable Systems 14.1 Einstein's Picture of Bohr's Quantization Rules 14.2 Keller's Construction of Wave Functions and Maslov Indices 14.3 Transformation to Normal Forms 14.4 The Frequency Analysis of a Classical Trajectory 14.5 The Adiabatic Principle 14.6 Tunneling Between Tori 15. Wave Functions in Classically Chaotic Systems 15.1 The Eigenstates of an Integrable System 15.2 Patterns of Nodal Lines 15.3 Wave-Packet Dynamics 15.4 Wigner's Distribution Function in Phase Space 15.5 Correlation Lengths in Chaotic Wave Functions 15.6 Sears, or What Is Left of the Classieal Periodic Orbits 16. The Energy Spectrum of a Classically Chaotic System 16.1 The Spectrum as a Set of Numbers 16.2 The Density of States and Weyl's Formula 16.3 Measures for Spectral Fluctuations 16.4 The Spectrum of Random Matrices 16.5 The Density of States and Periodic Orbits 16.6 Level Clustering in the Regular Spectrum 16.7 The Fluctuations in the Irregular Spectrum 16.8 The Transition from the Regular to the Irregular Spectrum 16.9 Classical Chaos and Quantal Random Matrices 17. The Trace Formula 17.1 The Van Vleck Formula Revisited 17.2 The Classical Green's Function in Action-Angle Variables 17.3 The Trace Formula for Integrable Systems 17.4 The Trace Formula in Chaotic Dynamical Systems 17.5 The Mathematical Foundations of the Trace Formula 17.6 Extensions and Applications 17.7 Sum Rules and Correlations 17.8 Homogeneous Hamiltonians 17.9 The Riemann Zeta-Function 17.10 Discrete Symmetries and the Anisotropic Kepler Problem 17.11 From Periodic Orbits to Code Words 17.12 Transfer Matrices 18. The Diamagnetic Kepler Problem 18.1 The Hamiltonian in the Magnetic Field 18.2 Weak Magnetic Fields and the Third Integral 18.3 Strong Fields and Landau Levels 18.4 Scaling the Energy and the Magnetic Field 18.5 Calculation of the Oscillator Strengths 18.6 The Chaotic Speetrum in Terms of Closed Orbits 19. Motion on a Surface of Constant Negative Curvature 19.1 Mechanics in a Riemannian Space 19.2 Poincaré's Model of Hyperbolic Geometry 19.3 The Construction of Polygons and Tilings 19.4 The Geodesics on a Double Torus 19.5 Selberg's Trace Formula 19.6 Computations on the Double Torus 19.7 Surfaces in Contact with the Outside World 19.8 Seattering on a Surface of Constant Negative Curvature 19.9 Chaos in Quantum-Mechanical Scattering 19.10 The Classical Interpretation of the Quantal Scattering 20. Scattering Problems, Coding, and Multifractal Invariant Measures 20.1 Electron Seattering in a Muffin-Tin Potential 20.2 The Coding of Geodesics on a Singular Polygon 20.3 The Geometry of the Continued Fractions 20.4 A New Measure in Phase Space Based on the Coding 20.5 Invariant Multifractal Measures in Phase Space 20.6 Multifractals in the Anisotropic Kepler Problem 20.7 Bundling versus Pruning a Binary Tree References Index