Preface ......................................................... v
1. Introduction ................................................. 1
1.1. Examples of Dynamical Systems ........................... 1
1.2. Vector Fields and Dynamical Systems .................... 14
1.3. Nonautonomous Systems .................................. 23
1.4. Fixed Points ........................................... 26
1.5. Reduction to lst-Order, Autonomous ..................... 27
1.6. Summary ................................................ 32
2. Techniques, Concepts, and Examples .......................... 33
2.1. Euler's Numerical Method ............................... 34
2.1.1. The Geometric View .............................. 34
2.1.2. The Analytical View ............................. 36
2.2. Gradient Vector Fields ................................. 39
2.3. Fixed Points and Stability ............................. 45
2.4. Limit Cycles ........................................... 51
2.5. The Two-Body Problem ................................... 55
2.5.1. Jacobi Coordinates .............................. 57
2.5.2. The Central Force Problem ....................... 59
2.6. Summary ................................................ 72
3. Existence and Uniqueness: The Flow Map ...................... 75
3.1. Picard Iteration ....................................... 78
3.2. Existence and Uniqueness Theorems ...................... 82
3.3. Maximum Interval of Existence .......................... 92
3.4. The Flow Generated by a Time-Dependent Vector Field .... 95
3.5. The Flow for Autonomous Systems ....................... 104
3.6. Summary ............................................... 112
4. One-Dimensional Systems .................................... 115
4.1. Autonomous, One-Dimensional Systems ................... 116
4.1.1. Construction of the Flow for 1-D,
Autonomous Systems ............................. 123
4.2. Separable Differential Equations ...................... 128
4.3. Integrable Differential Equations ..................... 135
4.4. Homogeneous Differential Equations .................... 147
4.5. Linear and Bernoulli Differential Equations ........... 151
4.6. Summary ............................................... 155
5. Linear Systems ............................................. 157
5.1. Existence and Uniqueness for Linear Systems ........... 162
5.2. The Fundamental Matrix and the Flow ................... 165
5.3. Homogeneous, Constant Coefficient Systems ............. 174
5.4. The Geometry of the Integral Curves ................... 180
5.4.1. Real Eigenvalues ............................... 182
5.4.2. Complex Eigenvalues ............................ 192
5.5. Canonical Systems ..................................... 211
5.5.1. Diagonalizable Matrices ........................ 214
5.5.2. Complex Diagonalizable Matrices ................ 217
5.5.3. The Nondiagonalizable Case: Jordan Forms ....... 219
5.6. Summary ............................................... 227
6. Linearization and Transformation ........................... 231
6.1. Linearization ......................................... 231
6.2. Transforming Systems of DEs ........................... 247
6.2.1. The Spherical Coordinate Transformation ........ 253
6.2.2. Some Results on Differentiable Equivalence ..... 257
6.3. The Linearization and Flow Box Theorems ............... 266
7. Stability Theory ........................................... 275
7.1. Stability of Fixed Points ............................. 276
7.2. Linear Stability of Fixed Points ...................... 279
7.2.1. Computation of the Matrix Exponential for
Jordan Forms ................................... 280
7.3. Nonlinear Stability ................................... 290
7.4. Liapunov Functions .................................... 292
7.5. Stability of Periodic Solutions ....................... 303
8. Integrable Systems ......................................... 323
8.1. First Integrals (Constants of the Motion) ............. 324
8.2. Integrable Systems in the Plane ....................... 329
8.3. Integrable Systems in 3-D ............................. 334
8.4. Integrable Systems in Higher Dimensions ............... 348
9. Newtonian Mechanics ........................................ 361
9.1. The N-Body Problem .................................... 362
9.1.1. Fixed Points ................................... 365
9.1.2. Initial Conditions ............................. 366
9.1.3. Conservation Laws .............................. 366
9.1.4. Stability of Conservative Systems .............. 374
9.2. Euler's Method and the N-body Problem ................. 384
9.2.1. Discrete Conservation Laws ..................... 392
9.3. The Central Force Problem Revisited ................... 401
9.3.1. Effective Potentials ........................... 404
9.3.2. Qualitative Analysis ........................... 405
9.3.3. Linearization and Stability .................... 409
9.3.4. Circular Orbits ................................ 410
9.3.5. Analytical Solution ............................ 412
9.4. Rigid-Body Motions .................................... 424
9.4.1. The Rigid-Body Differential Equations .......... 432
9.4.2. Kinetic Energy and Moments of Inertia .......... 438
9.4.3. The Degenerate Case ............................ 446
9.4.4. Euler's Equation ............................... 447
9.4.5. The General Solution of Euler's Equation ....... 451
10.Motion on a Submanifold .................................... 463
10.1.Motion on a Stationary Submanifold .................... 464
10.1.1.Motion Constrained to a Curve .................. 471
10.1.2.Motion Constrained to a Surface ................ 476
10.2.Geometry of Submanifolds .............................. 484
10.3.Conservation of Energy ................................ 493
10.4.Fixed Points and Stability ............................ 495
10.5.Motion on a Given Curve ............................... 502
10.6.Motion on a Given Surface ............................. 513
10.6.1.Surfaces of Revolution ......................... 520
10.6.2.Visualization of Motion on a Given Surface ..... 526
10.7 Motion Constrained to a Moving Submanifold ............ 531
11.Hamiltonian Systems ........................................ 541
11.1.1-Dimensional Hamiltonian Systems ..................... 544
11.1.1.Conservation of Energy ......................... 547
11.2.Conservation Laws and Poisson Brackets ................ 551
11.3.Lie Brackets and Arnold's Theorem ..................... 565
11.3.1.Arnold's Theorem ............................... 567
11.4.Liouville's Theorem ................................... 582
A. Elementary Analysis ........................................ 589
A.l. Multivariable Calculus ................................ 589
A.2. The Chain Rule ........................................ 595
A.3. The Inverse and Implicit Function Theorems ............ 596
A.4. Taylor's Theorem and The Hessian ...................... 602
A.5. The Change of Variables Formula ....................... 606
В. Lipschitz Maps and Linearization ........................... 607
B.l. Norms ................................................. 608
B.2. Lipschitz Functions ................................... 609
B.3. The Contraction Mapping Principle ..................... 613
B.4. The Linearization Theorem ............................. 619
С. Linear Algebra ............................................. 633
C.l. Vector Spaces and Direct Sums ......................... 633
C.2. Bilinear Forms ........................................ 636
C.3. Inner Product Spaces .................................. 638
C.4. The Principal Axes Theorem ............................ 642
C.5. Generalized Eigenspaces ............................... 645
C.6. Matrix Analysis ....................................... 656
C.6.1. Power Series with Matrix Coefficients ......... 662
D. CD-ROM Contents ............................................ 665
Bibliography .................................................. 669
Index ......................................................... 675
|
