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