Differential equations : theory and applications / by David Betounes.

Format:

Book

Author/Creator:

Betounes, David.

Contributor:

Anne and Joseph Trachtman Memorial Book Fund.

Language:

English

Subjects (All):

Differential equations--Data processing.

Differential equations.

Maple (Computer file).

Mechanics.

Hamiltonian systems.

Dynamics.

Vector fields.

Physical Description:

xiv, 626 pages : illustrations ; 25 cm

Edition:

Second edition.

Place of Publication:

New York : Springer, [2010]

Summary:

The book provides a comprehensive introduction to the theory of ordinary differential equations at the graduate level and includes applications to Newtonian and Hamiltonian mechanics. It not only has a large number of examples and computer graphics, but also has a complete collection of proofs for the major theorems, ranging from the usual existence and uniqueness results to the Hartman-Grobman linearization theorem and the Jordan canonical form theorem.

The book can be used almost exclusively in the traditional way for graduate math courses, or it can be used in an applied way for interdisciplinary courses involving physics, engineering, and other science majors. For this reason an extensive computer component using Maple is provided on Springer's website.

This new edition has been extensively revised throughout, particularly the chapters on linear systems, stability theory and Hamiltonian systems.

The computer component is an in-depth supplement and complement to the material in the text and contains an introduction to discrete dynamical systems and iterated maps, special-purpose Maple code for animating phase portraits, stair diagrams, N-body motions, and rigid-body motions, and numerous tutorial Maple worksheets pertaining to all aspects of using Maple to study the topics in the text.

Contents:

1 Introduction 1

1.1 Examples of Dynamical Systems 1

1.2 Vector Fields and Dynamical Systems 16

1.3 Nonautonomous Systems 26

1.4 Fixed Points 29

1.5 Reduction to 1st-Order, Autonomous 30

1.6 Summary 34

2 Techniques, Concepts and Examples 37

2.1 Euler's Numerical Method 38

2.1.1 The Geometric View 38

2.1.2 The Analytical View 40

2.2 Gradient Vector Fields 43

2.3 Fixed Points and Stability 50

2.4 Limit Cycles 55

2.5 The Two-Body Problem 60

2.5.1 Jacobi Coordinates 62

2.5.2 The Central Force Problem 63

2.6 Summary 77

3 Existence and Uniqueness: The Flow Map 79

3.1 Picard Iteration 82

3.2 Existence and Uniqueness Theorems 86

3.3 Maximum Interval of Existence 96

3.4 The Flow Generated by a Time-Dependent Vector Field 100

3.5 The Flow for Autonomous Systems 108

3.6 Summary 117

4 Linear Systems 119

4.1 Existence and Uniqueness for Linear Systems 124

4.2 The Fundamental Matrix and the Flow 128

4.3 Homogeneous, Constant Coefficient Systems 154

4.4 The Geometry of the Integral Curves 161

4.4.1 Real Eigenvalues 163

4.4.2 Complex Eigenvalues 174

4.5 Canonical Systems 201

4.5.1 Diagonalizable Matrices 204

4.5.2 Complex Diagonalizable Matrices 208

4.5.3 The Nondiagonalizable Case: Jordan Forms 209

4.6 Summary 218

5 Linearization & Transformation 221

5.1 Linearization 221

5.2 Transforming Systems of DEs 237

5.2.1 The Spherical Coordinate Transformation 243

5.2.2 Some Results on Differentiable Equivalence 248

5.3 The Linearization and Flow Box Theorems 258

6 Stability Theory 267

6.1 Stability of Fixed Points 269

6.2 Linear Stability of Fixed Points 272

6.2.1 Computation of the Matrix Exponential for Jordan Forms 272

6.3 Nonlinear Stability 283

6.4 Liapunov Functions 285

6.5 Stability of Periodic Solutions 296

7 Integrable Systems 333

7.1 First Integrals (Constants of the Motion) 334

7.2 Integrable Systems in the Plane 339

7.3 Integrable Systems in 3-D 345

7.4 Integrable Systems in Higher Dimensions 359

8 Newtonian Mechanics 371

8.1 The N-Body Problem 372

8.1.1 Fixed Points 375

8.1.2 Initial Conditions 376

8.1.3 Conservation Laws 377

8.1.4 Stability of Conservative Systems 385

8.2 Euler's Method and the N-body Problem 394

8.2.1 Discrete Conservation Laws 403

8.3 The Central Force Problem Revisited 412

8.3.1 Effective Potentials 415

8.3.2 Qualitative Analysis 416

8.3.3 Linearization and Stability 420

8.3.4 Circular Orbits 421

8.3.5 Analytical Solution 423

8.4 Rigid-Body Motions 436

8.4.1 The Rigid-Body Differential Equations 443

8.4.2 Kinetic Energy and Moments of Inertia 450

8.4.3 The Degenerate Case 458

8.4.4 Euler's Equation 459

8.4.5 The General Solution of Euler's Equation 463

9 Hamiltonian Systems 475

9.1 1-Dimensional Hamiltonian Systems 478

9.1.1 Conservation of Energy 481

9.2 Conservation Laws and Poisson Brackets 489

9.3 Lie Brackets and Arnold's Theorem 502

9.3.1 Arnold's Theorem 505

9.4 Liouville's Theorem 528.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

9781441911629

1441911626

1441911634

9781441911636

OCLC:

467779993

Publisher Number:

99937141416