Chaos in Classical and Quantum Mechanics

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