Thinking About Ordinary Differential Equations by Robert E. O'Malley,Review by: J. David LoganSIAM Review, Vol. 40, No. 1 (Mar., 1998), pp. 163-164Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2653016 .

Accessed: 12/06/2014 17:03

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Review.

http://www.jstor.org

This content downloaded from 62.122.76.48 on Thu, 12 Jun 2014 17:03:10 PMAll use subject to JSTOR Terms and Conditions

BOOK REVIEWS 163

many others, have not really aged. One thus expects a new book on the subject to live up both by its content and its expo- sition to the standards of these references. I regret to say that the present book falls short of this expectation. I should hasten to repeat that the book is very readable and to add that it does provide a quick glance the asymptotic approaches mentioned above. However, the narrow focus makes the book unsuitable for an undergraduate learning about planar dynamical systems and need- ing a broader view, while its insufficient depth does not recommend it to a special- ist. The exposition is focused, in places too narrowly, on the formalism: for instance, in the derivation of perturbation expansions for periodic solutions of the systems

y +y+EY2 =0 and y"+y=E(1-y2)y/'

it is not mentioned whether or why the formal expressions actually represent (ex- isting) periodic solutions; the underlying geometry of the problem is underempha- sized. Of course, this flaw in exposition could easily be corrected by a few extra sentences, but it is still serious as stands. In the book with an applied flavor it would be important to give more emphasis to the physical significance of the results derived for some of the systems modeled in the beginning of the book.

The subject of planar autonomous dy- namical systems is one of the most ex- tensively developed and best understood areas of differential equations. The great attraction of this subject lies in the fruitful interaction of different subjects: physical questions are translated into differential equations, which are then studied using analysis, algebra, and topology, with the products of these more abstract methods reinterpreted for the original physical sys- tems. Unfortunately the present book does not convey this sense of excitement arising from the marriage of different disciplines.

REFERENCES

[1] A. A. VITT, A. A. ANDRONOV, AND S. E. KHAIKIN, Theory of Oscillators, Dover, New York, 1996.

[2] N. N. BOGOLIUBOV AND Y. A. MITROPOLSKI, Asymtotic Methods in the Theory of Non-Linear Oscillations, Gordon and Breach, New York, 1961.

[3] E. A. CODDINGTON AND N. LEVINSON, Theory of Ordinary Differential Equa- tions, McGraw-Hill, New York, 1955.

MARK LEVI Rensselaer Polytechnic Institute

Thinking About Ordinary Differential Equations. By Robert E. O'Malley, Jr. Cambridge University Press, Cambridge, UK, 1997. $24.95. x+247 pp., paperback. ISBN 0-521-55742-9.

This well-written book is designed as a textbook for advanced undergraduates in mathematics, science, and engineering who have had some previous experience in dif- ferential equations, either in a standard, elementary, postcalculus course or perhaps in the calculus course itself. The perspec- tive is on solution methods and techniques rather than on specific applications or on mathematical proofs of existence, unique- ness, and stability.

In Chapter 1 there is a review of some of the basic first-order equations: separa- ble, exact, homogeneous, linear, and so on. The author also examines special second- order equations whose solution can be found by substitution and quadrature. Chapter 2 contains a standard treatment of second- order linear equations along with the usual applications to oscillators and circuits. It presents the annihilator method for solv- ing nonhomogeneous equations. Chapter 3 focuses on power series solutions and dis- cusses singular points and the method of Frobenius. It also contains a brief pre- sentation of Bessel functions and Lengen- dre polynomials. Chapter 4 discusses linear systems, the fundamental matrix, eAt, and so forth. Here the reader will require a knowledge of elementary matrix theory. In Chapter 5 there is a brief discussion of phase plane phenomena and stability. Fi- nally, perturbation methods are presented in Chapter 6: algebraic equations, pertur- bation expansions, and boundary layers.

This content downloaded from 62.122.76.48 on Thu, 12 Jun 2014 17:03:10 PMAll use subject to JSTOR Terms and Conditions

164 BOOK REVIEWS

As one can observe from the chapter contents, the material is quite classical and standard. There is no new territory opened here. The approach is analytical and re- quires good calculus skills on the part of the reader. There are essentially no nu- merics in the text. There is only a brief mention of Sturm-Liouville problems. The publisher advertises on the cover that the text offers an "applied perspective." How- ever, the reviewer would quarrel with that claim; if "applied" means developing and understanding differential equation models in the pure and applied sciences, then the book does not meet that definition. It is applied in the sense that analytical cal- culations are emphasized and the exposi- tion is discussion oriented, like in a science book, rather than wrapped in a definition- theorem-proof format.

Nevertheless, the book is attractive and well written. It stresses an analytical ap- proach which can be of tremendous value to students who have neglected their calcula- tion skills in favor of calculator or computer algebra manipulations. The book also has a good size-about 240 pages, just enough for one semester. In this latter sense, one should not expect that every topic will be covered in detail. Generally, within the con- straint of an analytical-based course, the author has been circumspect in his choice of topics.

Another nice feature of this text is the large number of exercises at the end of each chapter (usually numbering over forty). Brief solutions in outline form are included for about twenty percent of them. The dif- ficulty level varies; some are routine and others require more careful thought. The exercises are outstanding and have a strong potential to reinforce the reader's analytical skills.

In summary, this brief text is an excel- lent choice for a one-semester, traditional course in differential equations at the un- dergraduate level whose goal is to develop analytical technique-rather than focus on applications (modeling) or theory (proving theorems). One might even be tempted to try the text in a post-calculus differential equations course for honors students. An instructor would be quite content with stu- dents' knowledge of differential equations

if they were to master the topics in this well-written and well-conceived text.

J. DAVID LOGAN University of Nebraska

Global Aspects of Classical Integrable Systems. By Richard H. Cushman and Larry M. Bates. Birkhaiuser-Verlag, Basel, Switzerland, 1997. $54.95. xv+435 pp., hardcover. ISBN 3-7643-5485-2.

This text is a study of five very classical Hamiltonian systems using modern tools, notably differential topology, Lie group ac- tions, and the language of geometric me- chanics. The systems are the harmonic os- cillator, the Kepler problem, the free rigid body, the spherical pendulum, and the La- grange top. Their general solutions have been known for centuries. Why restudy them now? The answer is contained in the word "global" from the book's title. The 20th century development of Poincar6's analysis situs into algebraic and differen- tial topology, together with the other tools listed above, give us the language for de- scribing how individual solutions fit to- gether globally to fill phase space.

If one is going to choose five integrable systems among the plethora of integrable Hamiltonian systems, then the authors made a wise choice. Their five systems carry enormous historical weight. They also serve as a good vehicle for the au- thors' global agenda, and for developing the various mathematical tools which they employ. Moreover, each of the five systems can be looked at in several different ways and arises in different contexts. The au- thors explore many of these. This adds to the richness of the book. For example, the authors show explicitly how the equa- tions of motion for a free rigid body can be looked at as geodesic equations on the three-dimensional rotation group, and how after regularization Kepler's problem (for negative energies) becomes the problem of finding geodesics on the round three-sphere.

Their five systems are examples of inte- grable systems. This fact, and the global point of view, is the glue which holds these five examples together into a book. In- tegrable systems are special types of the

This content downloaded from 62.122.76.48 on Thu, 12 Jun 2014 17:03:10 PMAll use subject to JSTOR Terms and Conditions