Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (eBook)

Preface to the Second Edition 5
Preface to the First Edition 6
Contents 8
Hamiltonian Systems 13
Notation 13
Hamilton's Equations 14
The Poisson Bracket 15
The Harmonic Oscillator 17
The Forced Nonlinear Oscillator 18
The Elliptic Sine Function 19
General Newtonian System 21
A Pair of Harmonic Oscillators 22
Linear Flow on the Torus 26
Euler--Lagrange Equations 27
The Spherical Pendulum 33
The Kirchhoff Problem 34
Equations of Celestial Mechanics 38
The N-Body Problem 38
The Classical Integrals 39
Equilibrium Solutions 40
Central Configurations 41
The Lagrangian Solutions 42
The Euler-Moulton Solutions 44
Total Collapse 45
The 2-Body Problem 46
The Kepler Problem 47
Solving the Kepler Problem 48
The Restricted 3-Body Problem 49
Equilibria of the Restricted Problem 52
Hill's Regions 53
Linear Hamiltonian Systems 56
Preliminaries 56
Symplectic Linear Spaces 63
The Spectra of Hamiltonian and Symplectic Operators 67
Periodic Systems and Floquet--Lyapunov Theory 74
Topics in Linear Theory 80
Critical Points in the Restricted Problem 80
Parametric Stability 89
Logarithm of a Symplectic Matrix. 94
Functions of a Matrix 95
Logarithm of a Matrix 96
Symplectic Logarithm 98
Topology of Sp(2n,R) 99
Maslov Index and the Lagrangian Grassmannian 102
Spectral Decomposition 110
Normal Forms for Hamiltonian Matrices 114
Zero Eigenvalue 114
Pure Imaginary Eigenvalues 119
Exterior Algebra and Differential Forms 127
Exterior Algebra 127
The Symplectic Form 132
Tangent Vectors and Cotangent Vectors 132
Vector Fields and Differential Forms 135
Changing Coordinates and Darboux's Theorem 139
Integration and Stokes' Theorem 141
Symplectic Transformations 143
General Definitions 143
Rotating Coordinates 145
The Variational Equations 146
Poisson Brackets 147
Forms and Functions 148
The Symplectic Form 148
Generating Functions 148
Mathieu Transformations 150
Symplectic Scaling 150
Equations Near an Equilibrium Point 151
The Restricted 3-Body Problem 151
Hill's Lunar Problem 153
Special Coordinates 156
Jacobi Coordinates 156
The 2-Body Problem in Jacobi Coordinates 158
The 3-Body Problem in Jacobi Coordinates 159
Action--Angle Variables 159
d'Alembert Character 160
General Action--Angle Coordinates 161
Polar Coordinates 163
Kepler's Problem in Polar Coordinates 164
The 3-Body Problem in Jacobi--Polar Coordinates 165
Spherical Coordinates 166
Complex Coordinates 169
Levi--Civita Regularization 170
Delaunay and Poincaré Elements 172
Planar Delaunay Elements 172
Planar Poincaré Elements 174
Spatial Delaunay Elements 175
Pulsating Coordinates 176
Elliptic Problem 179
Geometric Theory 183
Introduction to Dynamical Systems 183
Discrete Dynamical Systems 187
Diffeomorphisms and Symplectomorphisms 187
The Henon Map 189
The Time Map 190
The Period Map 190
The Convex Billiards Table 191
A Linear Crystal Model 192
The Flow Box Theorem 194
Noether's Theorem and Reduction 199
Symmetries Imply Integrals 199
Reduction 200
Periodic Solutions and Cross-Sections 203
Equilibrium Points 203
Periodic Solutions 204
A Simple Example 207
Systems with Integrals 208
The Stable Manifold Theorem 210
Hyperbolic Systems 216
Shift Automorphism and Subshifts of Finite Type 216
Hyperbolic Structures 218
Examples of Hyperbolic Sets 219
The Shadowing Lemma 221
The Conley--Smale Theorem 221
Continuation of Solutions 225
Continuation Periodic Solutions 225
Lyapunov Center Theorem 227
Applications to the Euler and Lagrange points 228
Poincaré's Orbits 229
Hill's Orbits 230
Comets 232
From the Restricted to the Full Problem 233
Some Elliptic Orbits 235
Normal Forms 239
Normal Form Theorems 239
Normal Form at an Equilibrium Point 239
Normal Form at a Fixed Point 242
Forward Transformations 245
Near-Identity Symplectic Change of Variables 245
The Forward Algorithm 246
The Remainder Function 248
The Lie Transform Perturbation Algorithm 251
Example: Duffing's Equation 251
The General Algorithm 253
The General Perturbation Theorem 253
Normal Form at an Equilibrium 258
Normal Form at L4 265
Normal Forms for Periodic Systems 267
Bifurcations of Periodic Orbits 279
Bifurcations of Periodic Solutions 279
Extremal Fixed Points. 281
Period Doubling 282
k-Bifurcation Points 286
Duffing Revisited 290
k-Bifurcations in Duffing's Equation 293
Schmidt's Bridges 294
Bifurcations in the Restricted Problem 296
Bifurcation at L4 299
Variational Techniques 308
The N-Body and the Kepler Problem Revisited 309
Symmetry Reduction for Planar 3-Body Problem 312
Reduced Lagrangian Systems 315
Discrete Symmetry with Equal Masses 318
The Variational Principle 320
Isosceles 3-Body Problem 322
A Variational Problem for Symmetric Orbits 324
Instability of the Orbits and the Maslov Index 328
Remarks 334
Stability and KAM Theory 335
Lyapunov and Chetaev's Theorems 337
Moser's Invariant Curve Theorem 341
Arnold's Stability Theorem 344
1:2 Resonance 348
1:3 Resonance 350
1:1 Resonance 352
Stability of Fixed Points 355
Applications to the Restricted Problem 357
Invariant Curves for Small Mass 357
The Stability of Comet Orbits 358
Twist Maps and Invariant Circle 361
Introduction 361
Notations and Definitions 362
Elementary Properties of Orbits 366
Existence of Periodic Orbits 372
The Aubry--Mather Theorem 376
A Fixed-Point Theorem 376
Subsets of A 377
Nonmonotone Orbits Imply Monotone Orbits 380
Invariant Circles 385
Properties of Invariant Circles 385
Invariant Circles and Periodic Orbits 389
Relationship to the KAM Theorem 391
Applications 392
References 394
Index 402