Differential Equations: Theory and Applications

634 Pages·2010·8.98 MB·English

by  David Betounes (auth.)

Differential Equations: Theory
and Applications
Second Edition
David Betounes
Differential Equations: Theory
HHand Applications
Second Edition
David Betounes
Department of Physics, Astronomy
and Geosciences
Valdosta State University
1500 N. Patterson Street
Valdosta, GA 31698
USA
Additional material to this book can be downloaded from http://extra.springer.com
ISBN 978-1-4419-1162-9 e-ISBN 978-1-4419-1163-6
DOI 10.1007/978-1-4419-1163-6
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2009938714
Mathematics Subject Classification (2000): 34A34, 34A30, 34A12, 34D05, 34D20, 70E15, 70H05, 70H06, 70H15
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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. His-
torically, 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 mathe-
matics, thematerialisorganized sothatthebookcanalsobeusedinawider
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 in-
structor’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 to-
gether the traditional theoretical material on differential equations and the
new, exciting techniques afforded by computer algebra systems (CAS), like
Maple, Mathematica, or Matlab. The electronic material for mastering and
enjoying the computer component of this book is on Springer’s website.
Ways to Use the Book
The book is designed for use in a one- or two-semester course (preferably
two). The core material, which can be covered in a single course, consists of
Chapters 1, 2, 3, 4, and 5 (maybe Chapter 6, depending on what you think
is basic). The other chapters consist of applications of the core material
to integrable systems (Chapter 7), Newtonian mechanics (Chapters 8), and
Hamiltonian systems (Chapter 9). These applications can be covered in a
sequelto afirstcourseon thecore material, or, dependingon thedepthwith
whichthecorematerialistreated,partsoftheseapplications(likeChapter9)
can besqueezed into the firstcourse. Thereis also much additional material
v
vi PREFACE
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
electronic material serves as an extensive 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
differentialequations(DEs)beforetheybeginstudyingthetheoryinChapter
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 con-
cept of the flow generated by the vector field associated with such systems.
Additional material on Lipshitzmaps and Gronwall’s inequality is presented
in Appendix B.
Chapter 4 presents the basic theory for linear systems of differential
equations, and this material, given its heavy dependence on concepts from
linearalgebra, cantake awhiletocover. Somemightarguethatthismaterial
oughttobeinChapter1,becausethetheoryforlinearsystemsisthesimplest
and most detailed. However, I have found, over the years developing this
PREFACE vii
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.
Chapter5describesthelinearizationtechniqueforanalyzingthebehavior
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, purposeof 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,
linearlyequivalent—andhelpsclarifythebasisoftheLinearizationTheorem.
Chapter 6 covers a number of results connected with the stability of sys-
tems of differential equations. The standard result for the stability of fixed
points for linear systems in terms of the eigenvalues of their coefficient ma-
trices leads, via the Linearization Theorem, to the corresponding result for
nonlinear systems. The basic theorem on Liapunov stability, determined by
aLiapunovfunction,isdiscussedandshowntobemostusefulinthechapters
on mechanics and Hamiltonian systems. A brief introduction to the stabil-
ity of periodic solutions, characteristic multipliers, and the Poincar´e map is
also provided as an illustration of the analogies and differences between the
stability techniques for fixed points and closed integral curves (cycles).
Chapter7isabriefintroductiontothetopicofintegrablesystems,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
simple, geometrically oriented, and are particularly suited to study with
computer graphics.
Chapter 8 begins the application of the theory to the topic of Newtonian
mechanics and, together with Chapter 9, can serve as a short course on
mechanics and dynamical systems. A large part of Chapter 8 deals with
rigid-body motion, which serves to illustrate a number of the concepts stud-
ied for linear systems and integrable systems.
Chapter 9 comprises the elementary theory of Hamiltonian systems and
includes proofs of Arnold’s Theorem, the Transport Theorem, Liouville’s
Theorem, and the Poincar´e Recurrence Theorem. This chapter is indepen-
dent of Chapter 8, but certainly can serve as an important complement to
that chapter. Because of the independence there is a certain amount of rep-
etition 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.
viii PREFACE
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
eitherinthetextorintheappendices(withtheexceptionoftheInverse
Function Theorem, Taylor’s Theorem, and the change of variables for-
mula). Minor results are often proved or the proofs are assigned as
exercises. Even if the book is used in an applied way, without an em-
phasis on proofs, students may at some later point in their careers
become more interested in, and in fact need, the additional under-
standing 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 natureof the solutions of differential equations. Most of these
have been done with Corel Draw and Maple. However, the text has a
numberofhand-drawnfigures,whicharereproducedsoastoappearas
blackboard sketches. These are meant to convey the belief that many
aspects of visualization are still easiest and best done by hand.
• Electronic Component: The electronic material on Springer’s web-
site is provided to complement and supplement the material in the
text. It is a major resource for the book. Many of the pertinent ex-
amples in the text that use Maple are on the website, in the form of
Maple worksheets, along with extensions and additional commentary
on these examples. These can be beneficial to students in working re-
lated computer exercises and can greatly reduce the amount of time
spent on these exercises.
An important part of the electronic component of the book 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 electronic component of the book. This can serve as material
for a short course in itself.
PREFACE ix
The electronic component also contains many worksheets that are tu-
torial 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) implementat-
ing 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 electronic material is organized by chapters, corresponding to the
chapters in the text. You can access all the Maple worksheets con-
stituting 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 electronic material.
x PREFACE
Preface to the 2nd Edition
In this the 2nd Edition of the book, all the chapters have been revised
and enhanced, and in particular, extensive additional examples, exercises,
and commentary have been added to Chapter 4 (Linear Systems), Chapter
7 (Stability Theory), and Chapter 9 (Hamiltonian Systems). The electronic
material (obtained from Springer’s website) has been revised and extended
too, with all files now compatible with any version of Maple from Maple 5
to Maple 12.