Introduction to Hamiltonian dynamical systems and the N-body problem / Kenneth R. Meyer, Glen R. Hall, Dan Offin.

Format:

Book

Author/Creator:

Meyer, Kenneth R. (Kenneth Ray), 1937-

Contributor:

Hall, Glen R.

Offin, Daniel C. (Daniel Clyde), 1953-

Series:

Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 90.

Applied mathematical sciences (Springer-Verlag New York Inc.)

Language:

English

Subjects (All):

Hamiltonian systems.

Many-body problem.

Physical Description:

xiii, 399 pages : illustrations ; 24 cm.

Edition:

Second edition.

Place of Publication:

New York : Springer Science+Business Media, [2009]

Summary:

This text grew out of graduate level courses in mathematics, engineering and physics given at several universities. The courses took students who had some background in differential equations and lead them through a systematic grounding in the theory of Hamiltonian mechanics from a dynamical systems point of view. Topics covered include a detailed discussion of linear Hamiltonian systems, an introduction to variational calculus and the Maslov index, the basics of the symplectic group, an introduction to reduction, applications of Poincare's continuation to periodic solutions, the use of normal forms, applications of fixed point theorems and KAM theory. There is a special chapter devoted to finding symmetric periodic solutions by calculus of variations methods.

The main examples treated in this text are the N-body problem and various specialized problems like the restricted three-body problem. The theory of the N-body problem is used to illustrate the general theory. Some of the topics covered are the classical integrals and reduction, central configurations, the existence of periodic solutions by continuation and variational methods, stability and instability of the Lagrange triangular point.

Contents:

1 Hamiltonian Systems 1

1.2 Hamilton's Equations 2

1.3 The Poisson Bracket 3

1.4 The Harmonic Oscillator 5

1.5 The Forced Nonlinear Oscillator 6

1.6 The Elliptic Sine Function 7

1.7 General Newtonian System 9

1.8 A Pair of Harmonic Oscillators 10

1.9 Linear Flow on the Torus 14

1.10 Euler-Lagrange Equations 15

1.11 The Spherical Pendulum 21

1.12 The Kirchhoff Problem 22

2 Equations of Celestial Mechanics 27

2.1 The N-Body Problem 27

2.1.1 The Classical Integrals 28

2.1.2 Equilibrium Solutions 29

2.1.3 Central Configurations 30

2.1.4 The Lagrangian Solutions 31

2.1.5 The Euler-Moulton Solutions 33

2.1.6 Total Collapse 34

2.2 The 2-Body Problem 35

2.2.1 The Kepler Problem 36

2.2.2 Solving the Kepler Problem 37

2.3 The Restricted 3-Body Problem 38

2.3.1 Equilibria of the Restricted Problem 41

2.3.2 Hill's Regions 42

3 Linear Hamiltonian Systems 45

3.2 Symplectic Linear Spaces 52

3.3 The Spectra of Hamiltonian and Symplectic Operators 56

3.4 Periodic Systems and Floquet-Lyapunov Theory 63

4 Topics in Linear Theory 69

4.1 Critical Points in the Restricted Problem 69

4.2 Parametric Stability 78

4.3 Logarithm of a Symplectic Matrix 83

4.3.1 Functions of a Matrix 84

4.3.2 Logarithm of a Matrix 85

4.3.3 Symplectic Logarithm 87

4.4 Topology of Sp(2n, R) 88

4.5 Maslov Index and the Lagrangian Grassmannian 91

4.6 Spectral Decomposition 99

4.7 Normal Forms for Hamiltonian Matrices 103

4.7.1 Zero Eigenvalue 103

4.7.2 Pure Imaginary Eigenvalues 108

5 Exterior Algebra and Differential Forms 117

5.1 Exterior Algebra 117

5.2 The Symplectic Form 122

5.3 Tangent Vectors and Cotangent Vectors 122

5.4 Vector Fields and Differential Forms 125

5.5 Changing Coordinates and Darboux's Theorem 129

5.6 Integration and Stokes' Theorem 131

6 Symplectic Transformations 133

6.1 General Definitions 133

6.1.1 Rotating Coordinates 135

6.1.2 The Variational Equations 136

6.1.3 Poisson Brackets 137

6.2 Forms and Functions 138

6.2.1 The Symplectic Form 138

6.2.2 Generating Functions 138

6.2.3 Mathieu Transformations 140

6.3 Symplectic Scaling 140

6.3.1 Equations Near an Equilibrium Point 141

6.3.2 The Restricted 3-Body Problem 141

6.3.3 Hill's Lunar Problem 143

7 Special Coordinates 147

7.1 Jacobi Coordinates 147

7.1.1 The 2-Body Problem in Jacobi Coordinates 149

7.1.2 The 3-Body Problem in Jacobi Coordinates 150

7.2 Action-Angle Variables 150

7.2.1 d'Alembert Character 151

7.3 General Action-Angle Coordinates 152

7.4 Polar Coordinates 154

7.4.1 Kepler's Problem in Polar Coordinates 155

7.4.2 The 3-Body Problem in Jacobi-Polar Coordinates 156

7.5 Spherical Coordinates 157

7.6 Complex Coordinates 160

7.6.1 Levi-Civita Regularization 161

7.7 Delaunay and Poincare Elements 163

7.7.1 Planar Delaunay Elements 163

7.7.2 Planar Poincare Elements 165

7.7.3 Spatial Delaunay Elements 166

7.8 Pulsating Coordinates 167

7.8.1 Elliptic Problem 170

8 Geometric Theory 175

8.1 Introduction to Dynamical Systems 175

8.2 Discrete Dynamical Systems 179

8.2.1 Diffeomorphisms and Symplectomorphisms 179

8.2.2 The Henon Map 181

8.2.3 The Time [tau] Map 182

8.2.4 The Period Map 182

8.2.5 The Convex Billiards Table 183

8.2.6 A Linear Crystal Model 184

8.3 The Flow Box Theorem 186

8.4 Noether's Theorem and Reduction 191

8.4.1 Symmetries Imply Integrals 191

8.4.2 Reduction 192

8.5 Periodic Solutions and Cross-Sections 195

8.5.1 Equilibrium Points 195

8.5.2 Periodic Solutions 196

8.5.4 Systems with Integrals 200

8.6 The Stable Manifold Theorem 202

8.7 Hyperbolic Systems 208

8.7.1 Shift Automorphism and Subshifts of Finite Type 208

8.7.2 Hyperbolic Structures 210

8.7.3 Examples of Hyperbolic Sets 211

8.7.4 The Shadowing Lemma 213

8.7.5 The Conley-Smale Theorem 213

9 Continuation of Solutions 217

9.1 Continuation Periodic Solutions 217

9.2 Lyapunov Center Theorem 219

9.2.1 Applications to the Euler and Lagrange points 220

9.3 Poincare's Orbits 221

9.4 Hill's Orbits 222

9.5 Comets 224

9.6 From the Restricted to the Full Problem 225

9.7 Some Elliptic Orbits 227

10 Normal Forms 231

10.1 Normal Form Theorems 231

10.1.1 Normal Form at an Equilibrium Point 231

10.1.2 Normal Form at a Fixed Point 234

10.2 Forward Transformations 237

10.2.1 Near-Identity Symplectic Change of Variables 237

10.2.2 The Forward Algorithm 238

10.2.3 The Remainder Function 240

10.3 The Lie Transform Perturbation Algorithm 243

10.3.1 Example: Duffing's Equation 243

10.3.2 The General Algorithm 245

10.3.3 The General Perturbation Theorem 245

10.4 Normal Form at an Equilibrium 250

10.5 Normal Form at L[subscript 4] 257

10.6 Normal Forms for Periodic Systems 259

11 Bifurcations of Periodic Orbits 271

11.1 Bifurcations of Periodic Solutions 271

11.1.1 Extremal Fixed Points 273

11.1.2 Period Doubling 274

11.1.3 k-Bifurcation Points 278

11.2 Duffing Revisited 282

11.2.1 k-Bifurcations in Duffing's Equation 285

11.3 Schmidt's Bridges 286

11.4 Bifurcations in the Restricted Problem 288

11.5 Bifurcation at L[subscript 4] 291

12 Variational Techniques 301

12.1 The N-Body and the Kepler Problem Revisited 302

12.2 Symmetry Reduction for Planar 3-Body Problem 305

12.3 Reduced Lagrangian Systems 308

12.4 Discrete Symmetry with Equal Masses 311

12.5 The Variational Principle 313

12.6 Isosceles 3-Body Problem 315

12.7 A Variational Problem for Symmetric Orbits 317

12.8 Instability of the Orbits and the Maslov Index 321

12.9 Remarks 327

13 Stability and KAM Theory 329

13.1 Lyapunov and Chetaev's Theorems 331

13.2 Moser's Invariant Curve Theorem 335

13.3 Arnold's Stability Theorem 338

13.4 1:2 Resonance 342

13.5 1:3 Resonance 344

13.6 1:1 Resonance 346

13.7 Stability of Fixed Points 349

13.8 Applications to the Restricted Problem 351

13.8.1 Invariant Curves for Small Mass 351

13.8.2 The Stability of Comet Orbits 352

14 Twist Maps and Invariant Circle 355

14.2 Notations and Definitions 356

14.3 Elementary Properties of Orbits 360

14.4 Existence of Periodic Orbits 366

14.5 The Aubry-Mather Theorem 370

14.5.1 A Fixed-Point Theorem 370

14.5.2 Subsets of A 371

14.5.3 Nonmonotone Orbits Imply Monotone Orbits 374

14.6 Invariant Circles 379

14.6.1 Properties of Invariant Circles 379

14.6.2 Invariant Circles and Periodic Orbits 383

14.6.3 Relationship to the KAM Theorem 385

14.7 Applications 386.

Notes:

Includes bibliographical references (pages [389]-396) and index.

ISBN:

9780387097237

0387097236

OCLC:

298342882