Differential Equations: Theory and Applications with Maple

Differential Equations: Theory and Applications with Maple

David Betounes Differential Equations: Theory and Applications with Maple

David Betounes Mathematics Department University of Southern Mississippi Hattiesburg, MS 39406-5045 USA david.betounes@usm.edu Additional material to this book can be downloaded from http://extras.springer.com Library of Congress Cataloging-in-Publication Data Betounes, David. Differential equations : theory and applications : with Maple I David Betounes. p. em. Includes bibliographical references and index. ISBN 978-1-4757-4973-1 ISBN 978-1-4757-4971-7 (ebook) DOI 10.1007/978-1-4757-4971-7 I. Differential equations-data processing. 2. Maple (Computer file). I. Title. QA371.5D37B47 2000 515'.35'02855369-dc21 00-062030 Printed on acid-free paper. Maple is a registered trademark of Waterloo Maple, Inc. 2001 Springer Science+ Business Media New York Originally published by Springer-Verlag New York, Inc. in 2001 Softcover reprint of the hardcover 1st edition 2001 This Work consists of a printed book and a CD-ROM packaged with the book, both of which are protected by federal copyright law and international treaty. The book may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+ Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. For copyright information regarding the CD-ROM, please consult the printed information packaged with the CD-ROM in the back of this publication, and which is also stored as a "readme" file on the CD-ROM. Use of the printed version of this Work in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known, or hereafter developed, other than those uses expressly granted in the CD-ROM copyright notice and disclaimer information, is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Where those designations appear in the book and Springer-Verlag was aware of a trademark claim, the designations follow the capitalization style used by the manufacturer. Production managed by Steven Pisano; manufacturing supervised by Jerome Basma. Photocomposed pages prepared from the author's LaTeX files. 9 8 7 6 5 4 3 2 1 SPIN 10780636

Preface This book was written as a comprehensive introduction to the theory of ordinary differential equations with a focus on mechanics and dynamical systems as time-honored and important applications of this theory. Historically, these were the applications that spurred the development of the mathematical theory and in hindsight they are still the best applications for illustrating the concepts, ideas, and impact of the theory. While the book is intended for traditional graduate students in mathematics, the material is organized so that the book can also be used in a wider setting within today's modern university and society (see "Ways to Use the Book" below). In particular, it is hoped that interdisciplinary programs with courses that combine students in mathematics, physics, engineering, and other sciences can benefit from using this text. Working professionals in any of these fields should be able to profit too by study of this text. An important, but optional component of the book (based on the instructor's or reader's preferences) is its computer material. The book is one of the few graduate differential equations texts that use the computer to enhance the concepts and theory normally taught to first- and second-year graduate students in mathematics. I have made every attempt to blend together the traditional theoretical material on differential equations and the new, exciting techniques afforded by computer algebra systems (CAS), like Maple, Mathematica, or Matlab. The accompanying CD-ROM contains all the electronic material for mastering and enjoying the computer component of this book. Ways to Use the Book The book js desjgned for use in a one- or two-semester course (preferably two). The core material, which can be covered in a single course, v

Vl Preface consists of Chapters 1, 2, 3, 5, and 6 (maybe Chapter 7, depending on what you think is basic). The other chapters consist of applications of the core material to integrable systems (Chapters 4 and 8), Newtonian mechanics (Chapters 9 and 10), and Hamiltonian systems (Chapter 11). These applications can be covered in a sequel to a first course on the core material, or, depending on the depth with which the core material is treated, parts of these applications (like Chapter 11) can be squeezed into the first course. There is also much additional material in the appendices, ranging from background material on analysis and linear algebra (Appendices A and C) to additional theory on Lipschitz maps and a proof of the Hartman-Grobman Linearization Theorem (Appendix B). The CD ROM also serves as an extensive electronic appendix to supplement the text. The material is structured so that the book can be used in a number of different ways based on the instructor's preferences, the type of course, and the types of students. Basically the book can be used in one of three ways: theoretical emphasis applied emphasis combination of theoretical and applied emphasis Here the designation applied means, for the students, not being required to prove theorems in the text or results in the exercises. Besides using the first emphasis, I have also had success using the third emphasis in classes that have had math and physics students. For such classes, I would require that students understand major theorems and definitions, be able to prove some of the easier results, and work most of the non-theoretical exercises (especially the ones requiring a computer). Guide to the Chapters Chapters 1 and 2 are intended to develop the students' backgrounds and give them plenty of examples and experiences with particular systems of differential equations (DEs) before they begin studying the theory in Chapter 3. I have found this works well, because it gives students concrete exercises to study and work on while I am covering the existence and uniqueness results in Chapter 3. Chapter 3 is devoted both to proving existence and uniqueness results for systems of differential equations and to introducing the important concept

Preface vii of the flow generated by the vector field associated with such systems. Additional material on Lipshitz maps and Gronwall's inequality is presented in Appendix B. Chapter 4 applies the general theory to specific types of DEs involving a single function-the types usually studied in undergradute differential equations courses. If your time schedule permits, this is a good chapter for your students to see the rigorous treatment of the undergraduate results and to realize that these special types are "solvable" precisely because they are integrable differential equations. Chapter 5 presents the basic theory for linear systems of differential equations, and this material, given its heavy dependence on concepts from linear algebra, can take awhile to cover. Some might argue that this material ought to be in Chapter 1, because the theory for linear systems is the simplest and most detailed. However, I have found, over the years developing this book, that starting with this material can put too much emphasis on the linear theory and can tend to consume half of the semester in doing so. Chapter 6 describes the linearization technique for analyzing the behavior of a nonlinear system in a neighborhood of a hyperbolic fixed point. The proof of the validity of the technique (the Hartman-Grobman Theorem) is contained in Appendix B. Another, perhaps more important, purpose of the chapter is the introduction of the concept of transforming vector fields, and thus of transforming systems of differential equations. Indeed, this concept is the basis for classifying equivalent systems-topologically, diferentiably, linearly equivalent-and helps clarify the basis of the Linearization Theorem. Chapter 7 covers a number of results connected with the stability of systems of differential equations. The standard result for the stability of fixed points for linear systems in terms of the eigenvalues of their coefficient matrices leads, via the Linearization Theorem, to the corresponding result for nonlinear systems. The basic theorem on Liapunov stability, determined by a Liapunov function, is discussed and shown to be most useful in the chapters on mechanics and Hamiltonian systems. A brief introduction to the stability of periodic solutions, characteristic multipliers, and the Poincare map is also provided as an illustration of the analogies and differences between the stability techniques for fixed points and closed integral curves (cycles). Chapter 8 is a brief introduction to the topic of integrable systems, which is a special case of the more general theory for integrable systems of partial differential equations (in particular, Pffafian systems). The ideas are very

viii Preface simple, geometrically oriented, and are particularly suited to study with computer graphics. Chapter 9 begins the application of the theory to the topic of Newtonian mechanics and, together with Chapters 10 and 11, can serve as a short course on mechanics and dynamical systems. A large part of Chapter 9 deals with rigid-body motion, which serves to illustrate a number of the concepts studied for linear systems and integrable systems. Chapter 10 studies the motion of objects constrained to move on specified curves, surfaces, and generally submanifolds of Rn. The material provides insight into how systems of differential equations on abstract manifolds (Riemannian or semi-riemannian) are influenced by the underlying differential geometry of the manifold. This chapter can serve as an elementary introduction to, and motivation for, a number of concepts from differential geometry. Chapter 11 comprises the elementary theory of Hamiltonian systems and includes proofs of Arnold's Theorem, the Transport Theorem, Liouville's Theorem, and the Poincare Recurrence Theorem. This chapter is independent of Chapters 9 and 10, but certainly can serve as an important complement to those chapters. Because of the independence there is a certain amount of repetition of ideas (conservation laws, first integrals, sketching integral curves for 1-D systems). However, if your students studied the prior chapters, this can help reinforce their learning process. Additional Features There are several features of the book that were specifically included to enhance the study and comprehension of the theory, ideas, and concepts. They include the following: Proofs: Complete proofs for almost every major result are provided either in the text or in the appendices (with the exception of the Inverse Function Theorem, Taylor's Theorem, and the change of variables formula). Minor results are often proved or the proofs are assigned as exercises. Even if the book is used in an applied way, without an emphasis on proofs, students may at some later point in their careers become more interested in, and in fact need, the additional understanding of the theory that the proofs provide. Blackboard Drawings: An extensive number of figures is provided to either illustrate and enhance the concepts or to exhibit the often complex nature of the solutions of differential equations. Most of these

Preface ix have been done with Corel Draw and Maple. However, the text has a number of hand-drawn figures, which are reproduced so as to appear as blackboard sketches. These are meant to convey the belief that many aspects of visualization are still easiest and best done by hand. The CD-ROM: The electronic material on the CD-ROM is provided to complement and supplement the material in the text. It is a major resource for the book. Many of the pertinent examples in the text that use Maple are on the CD-ROM, in the form of Maple worksheets, along with extensions and additional commentary on these examples. These can be beneficial to students in working related computer exercises and can greatly reduce the amount of time spent on these exercises. An important part of the CD-ROM is the supplementary material it contains on discrete dynamical systems, or the theory of iterated maps, which has many analogies, similarities, and direct relations to systems of differential equations. However, to eliminate confusion, to add flexibility of use, and to save space, all the theory, applications, and examples of this subject have been relegated to the CD-ROM. This can serve as material for a short course in itself. The CD-ROM also contains many worksheets that are tutorial in nature (like how to plot phase portraits in Maple). There is also some special-purpose Maple code for performing tasks such as ( 1) plotting the curves of intersection of a family of surfaces with a given surface, (2) plotting integral curves dynamically as they are traced out in time (an extension of Maple's DEplot command), (3) implementating the Euler numerical scheme for the planar N-body problem, ( 4) animating rigid-body motions, (5) animating the motion of a body constrained to a given curve or surface, and (6) animating discrete dynamical systems in dimensions 1 and 2. The CD-ROM material is organized by chapters, CDChapter 1- CD Chapter 10, corresponding to the chapters in the text (Chapters 4 and 11 are not included). You can access all the Maple worksheets constituting a given chapter by opening the table of contents worksheet, cdtoc.mws, and using the hyperlinks there. Appendix D has the table of contents for the CD-ROM.

Contents Preface 1 Introduction 1.1 Examples of Dynamical Systems 1. 2 Vector Fields and Dynamical Systems 1.3 Nonautonomous Systems.... 1.4 Fixed Points.............. 1.5 Reduction to 1st-Order, Autonomous. 1.6 Summary................ v 1 1 14 23 26 27 32 2 Techniques, Concepts, and Examples 2.1 Euler's Numerical Method.. 2.1.1 The Geometric View. 2.1.2 The Analytical View 2.2 Gradient Vector Fields... 2.3 Fixed Points and Stability. 2.4 Limit Cycles.... 2.5 The Two-Body Problem.. 2.5.1 Jacobi Coordinates. 57 2.5.2 The Central Force Problem 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 33 34 34 36 39 45 51 55 59 Xl

xii Contents 3.5 The Flow for Autonomous Systems 3.6 Summary.............. 4 One-Dimensional Systems 4.1 Autonomous, One-Dimensional Systems 4.1.1 Construction of the Flow for 1-D, Autonomous Systems.. 4.2 Separable Differential Equations.. 4.3 Integrable Differential Equations.. 4.4 Homogeneous Differential Equations 4.5 Linear and Bernoulli Differential Equations 4.6 Summary.. 5 Linear Systems 5.1 Existence and Uniqueness for Linear Systems 5.2 The Fundamental Matrix and the Flow... 5.3 Homogeneous, Constant Coefficient Systems. 5.4 The Geometry of the Integral Curves. 5.4.1 Real Eigenvalues... 5.4.2 Complex Eigenvalues.. 5.5 Canonical Systems....... 5.5.1 Diagonalizable Matrices 5.5.2 Complex Diagonalizable Matrices. 5.5.3 The Nondiagonalizable Case: Jordan Forms 5.6 Summary....................... 6 Linearization and Transformation 6.1 Linearization.... 6.2 Transforming Systems of DEs.... 6.2.1 The Spherical Coordinate Transformation 6.2.2 Some Results on Differentiable Equivalence 6.3 The Linearization and Flow Box Theorems 7 Stability Theory 7.1 Stability of Fixed Points.... 7.2 Linear Stability of Fixed Points........... 7.2.1 Computation of the Matrix Exponential for Jordan Forms. 7.3 Nonlinear Stability. 7.4 Liapunov Functions. 104 112 115 116 123 128 135 147 151 155 157 162 165 174 180 182 192 211 214 217 219 227 231 231 247 253 257 266 275 276 279 280 290 292

Contents 7.5 Stability of Periodic Solutions. 8 Integrable Systems 8.1 First Integrals (Constants of the Motion) 8.2 Integrable Systems in the Plane...... 8.3 Integrable Systems in 3-D.... 8.4 Integrable Systems in Higher Dimensions 9 Newtonian Mechanics 9.1 TheN-Body Problem 9.1.1 Fixed Points. 9.1.2 Initial Conditions. 9.1.3 Conservation Laws 9.1.4 Stability of Conservative Systems. 9.2 Euler's Method and then-body Problem 9.2.1 Discrete Conservation Laws.. 9.3 The Central Force Problem Revisited. 9.3.1 Effective Potentials... 9.3.2 Qualitative Analysis.... 9.3.3 Linearization and Stability 9.3.4 Circular Orbits... 9.3.5 Analytical Solution..... 9.4 Rigid-Body Motions.... 9.4.1 The Rigid-Body Differential Equations. 9.4.2 9.4.3 9.4.4 9.4.5 Kinetic Energy and Moments of Inertia The Degenerate Case........... Euler's Equation............. The General Solution of Euler's Equation 10 Motion on a Submanifold 10.1 Motion on a Stationary Submanifold 10.1.1 Motion Constrained to a Curve 10.1.2 Motion Constrained to a Surface 10.2 Geometry of Submanifolds. 10.3 Conservation of Energy.. 10.4 Fixed Points and Stability. 10.5 Motion on a Given Curve. 10.6 Motion on a Given Surface 10.6.1 Surfaces of Revolution 10.6.2 Visualization of Motion on a Given Surface xiii 303 323 324 329 334 348 361 362 365 366 366 374 384 392 401 404 405 409 410 412 424 432 438 446 447 451 463 464 471 476 484 493 495 502 513 520 526

xiv Contents 10.7 Motion Constrained to a Moving Submanifold. 11 Hamiltonian Systems 11.1 1-Dimensional Hamiltonian Systems 11.1.1 Conservation of Energy... 11.2 Conservation Laws and Poisson Brackets. 11.3 Lie Brackets and Arnold's Theorem. 11.3.1 Arnold's Theorem 11.4 Liouville's Theorem A Elementary Analysis A.1 Multivariable Calculus.... A.2 The Chain Rule................. A.3 The Inverse and Implicit Function Theorems A.4 Taylor's Theorem and The Hessian A.5 The Change of Variables Formula. B Lipschitz Maps and Linearization B.1 Norms.... B.2 Lipschitz Functions.... B.3 The Contraction Mapping Principle B.4 The Linearization Theorem..... C Linear Algebra C.1 Vector Spaces and Direct Sums C.2 Bilinear Forms........ C.3 Inner Product Spaces... C.4 The Principal Axes Theorem C.5 Generalized Eigenspaces... C.6 Matrix Analysis........ C.6.1 Power Series with Matrix Coefficients D CD-ROM Contents Bibliography Index.... 531 541 544 547 551 565 567 582 589 589 595 596 602 606 607 608 609 613 619 633 633 636 638 642 645 656 662 665 669 675