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Tables of Contents for Chaos in Classical and Quantum Mechanics
Chapter/Section Title
Page #
Page Count
Preface
vii
Introduction
1
4
The Mechanics of Lagrange
5
14
Newton's Equations According to Lagrange
6
1
The Variational Principle of Lagrange
7
3
Conservation of Energy
10
1
Example: Space Travel in a Given Time Interval: Lambert's Formula
11
3
The Second Variation
14
2
The Spreading Trajectories
16
3
The Mechanics of Hamilton and Jacobi
19
11
Phase Space and Its Hamiltonian
19
1
The Action Function S
20
2
The Variational Principle of Euler and Maupertuis
22
1
The Density of Trajectories on the Energy Surface
23
3
Example: S[pace Travel with a Given Energy
26
4
Integrable Systems
30
15
Constants of Motion and Poisson Brackets
30
2
Invariant Tori and Action-Angle Variables
32
2
Multiperiodic Motion
34
2
The Hydrogen Molecule Ion
36
2
Geodesics on a Triaxial Ellipsoid
38
3
The Toda Lattice
41
2
Integrable versus Separable
43
2
The Three-Body Problem: Moon-Earth-Sun
45
13
Reduction to Four Degrees of Freedom
45
3
Application in Atomic Physics and Chemistry
48
2
The Action-Angle Variables in the Lunar Observations
50
3
The Best Temporary Fit to a Kepler Ellipse
53
3
The Time-Dependent Hamiltonian
56
2
Three Methods of Solution
58
17
Variation of the Constants (Lagrange)
58
1
Canonical Transformation (Delaunay)
59
3
The Application of Canonical Transformations
62
1
Small Denominators and Other Difficulties
63
2
Hill's Periodic Orbit in the Three-Body Problem
65
5
The Motion of the Perigee and the Node
70
2
Displacements from the Periodic Orbit and Hill's Equation
72
3
Periodic Orbits
75
12
Potentials with Circular Symmetry
77
3
The Number of Periodic Orbits in an Integrable System
80
2
The Neighborhood of a Periodic Orbit
82
2
Elliptic, Parabolic, and Hyperbolic Periodic Orbits
84
3
The Surface of Section
87
12
The Invariant Two-Form
87
2
Integral In variants and Louisville Theorem
89
2
Area Conservation on the Surface of Section
91
2
The Theorem of Darboux
93
2
The Conjugation of Time and Energy in Phase Space
95
4
Models of the Galaxy and of Small Molecules
99
17
Stellar Trajectories in the Galaxy
100
2
The Henon-Heiles Potential
102
1
Numerical investigations
103
3
Some Analytic Results
106
3
Searching for Integrability with Kowalevskaya and Painleve
109
2
Discrete Area-Preserving Maps
111
5
Soft Chaos and the KAM Theorem
116
26
The Origin of Soft Chaos
116
2
Resonances in Celestial Mechanics
118
2
The Analogy with the Ordinary Pendulum
120
5
Islands of Stability and Overlapping Resonances
125
4
How Rational Are the Irrational Numbers?
129
3
The KAM Theorem
132
3
Homoclinic Points
135
3
The Lore of the Golden Mean
138
4
Entropy and Other Measures of Chaos
142
14
Abstact Dynamical Systems
143
2
Ergodicity, Mixing, and K-Systems
145
2
The Metric Entropy
147
2
The Automorphisms of the Torus
149
2
The Topological Entropy
151
3
Anosov Systems and Hard Chaos
154
2
The Anisotropic Kepler Problem
156
17
The Donor Impurity in a Semiconductor Crystal
156
3
Normalized Coordinates in the Anisotropic Kepler Problem
159
2
The Surface of Section
161
3
Construction of Stable and Unstable Manifolds
164
4
The Periodic Orbits in the Anisotropic Kepler Problem
168
3
Some Questions Concerning the AKP
171
2
The Transition from Classical to Quantum Mechanics
173
21
Are Classical Mechanics and Quantum Mechanics Compatible?
174
2
Changing Coordinates in the Action
176
2
Adding Actions and Multiplying Probabilities
178
2
Rutherford Scattering
180
4
The Classical Version of Quantum Mechanics
184
2
The Propagator in Momentum Space
186
2
The Classical Green's Function
188
2
The Hydrogen Atom in Momentum Space
190
4
The New World of Quantum Mechanics
194
13
The Liberation from Classical Chaos
194
2
The Time-Dependent Schrodinger Equation
196
2
The Stationary Schrodinger Equation
198
2
Feynman's Path Integral
200
2
Changing Coordinates in the Path Integral
202
2
The Classical Limit
204
3
The Quantization of Integrable Systems
207
24
Einstein's Picture of Bohr's Quantization Rules
208
3
Keller's Construction of Wave Function and Maslov Indices
211
4
Transformation to Normal Forms
215
5
The Frequency Analysis of a Classical Trajectory
220
4
The Adiabatic Principle
224
3
Tunneling Between Tori
227
4
Wave Functions in Classically Chaotic Systems
231
23
The Eigenstates of an Integrable System
232
1
Patterns of Nodal Lines
233
5
Wave-Packet Dyanmics
238
3
Wigner's Distribution Function in Phase Space
241
6
Correlation Lengths in Chaotic Wave Functions
247
2
Scars, or What Is Left of the Classical Periodic Orbits
249
5
The Energy Spectrum of a Classically Chaotic System
254
28
The Spectrum as a Set of Numbers
255
2
The Density of States and Weyl's Formula
257
4
Measures for Spectral Fluctuations
261
2
The Spectrum of Random Matrices
263
3
The Density of States and Periodic Orbits
266
4
Level Clustering in the Regular Spectrum
270
3
The Fluctuations in the Irregular Spectrum
273
2
The Transition from the Regular to the Irregular Spectrum
275
3
Classical Chaos and Quantal Random matrices
278
4
The Trace Formula
282
40
The Van Vleck Formula Revisited
283
2
The Classical Green's Function in Action-Angle Variables
285
2
The Trace Formula for Integrable Systems
287
4
The Trace Formula in Chaotic Dynamical Systems
291
4
The Mathematical Foundations of the Trace Formula
295
3
Extensions and Applications
298
3
Sum Rules and Correlations
301
4
Homogeneous Hamiltonians
305
2
The Riemann Zeta-Function
307
5
Discrete Symmetries and the Anisotropic Kepler Problem
312
2
From Periodic Orbits to Code Words
314
3
Transfer Matrices
317
5
The Diamagnetic Kepler Problem
322
18
The Hamiltonian in the Magnetic Field
323
2
Weak Magnetic Fields and the Third Integral
325
1
Strong Fields and Landau Levels
326
3
Scaling the Energy and the Magnetic Field
329
3
Calculation of the Oscillator Strengths
332
4
The Chaotic Spectrum in Terms of Closed Orbits
336
4
Motion on a Surface of Constant Negative Curvature
340
43
Mechanics in a Riemannian Space
341
4
Poincare's Model of Hyperbolic Geometry
345
3
The Construction of Polygons and Tilings
348
6
The Geodesics on a Double Torus
354
4
Selberg's Trace Formula
358
5
Computations on the Double Torus
363
6
Surfaces in Contact with the Outside World
369
5
Scattering on a Surface of Constant Negative Curvature
374
3
Chaos in Quantum-Mechanical Scattering
377
2
The Classical Interpretation of the Quantal Scattering
379
4
Scattering Problems, Coding, and Multifractal Invariant Measures
383
27
Election Scattering in a Muffin-Tin Potential
384
5
The Coding of Geodesics on a Singular Polygon
389
4
The Geometry of the Continued Fractions
393
2
A New Measure in Phase Space Base on the Coding
395
3
Invariant Multifractal Measures in Phase Space
398
4
Multifractals in the Anisotropic Kepler Problem
402
5
Bundling versus Pruning a Binary Tree
407
3
References
410
17
Index
427