 | Palais R.S. Differential equations, mechanics, and computation / R.S.Palais, R.A.Palais. - Providence: American Mathematical Society; Princeton: Institute for Advanced Study, 2009. - xiii, 313 p.: ill. - (Student mathematical library. IAS/Park City mathematical subseries; vol.51). - Bibliogr.: p.307-309. - Ind.: p.311-313. - ISBN 978-0-8218-2138-1
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IAS/Park City Mathematics Institute ............................ ix
Preface ........................................................ xi
Acknowledgments .............................................. xiii
Introduction .................................................... 1
Chapter 1 Differential Equations and Their Solutions ........... 5
1.1 First-Order ODE: Existence and Uniqueness .................. 5
1.2 Euler's Method ............................................ 16
1.3 Stationary Points and Closed Orbits ....................... 19
1.4 Continuity with Respect to Initial Conditions ............. 22
1.5 Chaos-Or a Butterfly Spoils Laplace's Dream ............... 25
1.6 Analytic ODE and Their Solutions .......................... 31
1.7 Invariance Properties of Flows ............................ 33
Chapter 2 Linear Differential Equations ....................... 37
2.1 First-Order Linear ODE .................................... 37
2.2 Nonautonomous First-Order Linear ODE ...................... 48
2.3 Coupled and Uncoupled Harmonic Operators .................. 50
2.4 Inhomogeneous Linear Differential Equations ............... 52
2.5 Asymptotic Stability of Nonlinear ODE ..................... 53
2.6 Forced Harmonic Oscillators ............................... 55
2.7 Exponential Growth and Ecological Models .................. 56
Chapter 3 Second-Order ODE and the Calculus of Variations ..... 63
3.1 Tangent Vectors and the Tangent Bundle .................... 63
3.2 Second-Order Differential Equations ....................... 66
3.3 The Calculus of Variations ................................ 68
3.4 The Euler-Lagrange Equations .............................. 70
3.5 Conservation Laws for Euler-Lagrange Equations ............ 73
3.6 Two Classic Examples ...................................... 75
3.7 Derivation of the Euler-Lagrange Equations ................ 78
3.8 More General Variations ................................... 80
3.9 The Theorem of E. Noether ................................. 81
3.10 Lagrangians Defining the Same Functionals ................. 82
3.11 Riemannian Metrics and Geodesies .......................... 85
3.12 A Preview of Classical Mechanics .......................... 86
Chapter 4 Newtonian Mechanics ................................. 91
4.1 Introduction .............................................. 91
4.2 Newton's Laws of Motion ................................... 92
4.3 Newtonian Kinematics ...................................... 96
4.4 Classical Mechanics as a Physical Theory .................. 99
4.5 Potential Functions and Conservation of Energy ........... 106
4.6 One-Dimensional Systems .................................. 111
4.7 The Third Law and Conservation Principles ................ 118
4.8 Synthesis and Analysis of Newtonian Systems .............. 122
4.9 Linear Systems and Harmonic Oscillators .................. 124
4.10. Small Oscillations about Equilibrium .................... 126
Chapter 5 Numerical Methods .................................. 133
5.1 Introduction ............................................. 133
5.2 Fundamental Examples and Their Behavior .................. 144
5.3 Summary of Method Behavior on Model Problems ............. 169
5.4 Paired Methods: Error, Step-Size, Order Control .......... 177
5.5 Behavior of Example Methods on a Model 2x2 System ........ 180
5.6 Stiff Systems and the Method of Lines .................... 187
5.7 Convergence Analysis: Euler's Method ..................... 213
Appendix A. Linear Algebra and Analysis ....................... 225
A.l. Metric and Normed Spaces ............................ 225
A.2. Inner-Product Spaces ................................ 227
Appendix B. The Magic of Iteration ............................ 233
B.l. The Banach Contraction Principle .................... 233
B.2. Newton's Method ..................................... 238
B.3. The Inverse Function Theorem ........................ 240
B.4. The Existence and Uniqueness Theorem for ODE ........ 241
Appendix C. Vector Fields as Differential Operators ........... 243
Appendix D. Coordinate Systems and Canonical Forms ............ 247
D.l. Local Coordinates ................................... 247
D.2. Some Canonical Forms ................................ 250
Appendix E. Parametrized Curves and Arclength ................. 255
Appendix F. Smoothness with Respect to Initial Conditions ..... 257
Appendix G. Canonical Form for Linear Operators ............... 259
G.l. The Spectral Theorem ................................ 259
Appendix H. Runge-Kutta Methods ............................... 263
Appendix I. Multistep Methods ................................. 281
Appendix J. Iterative Interpolation and Its Error ............. 303
Bibliography .................................................. 307
Index ......................................................... 311
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