From Calculus to Chaos: An Introduction to Dynamics by David AchesonReview by: J. David LoganSIAM Review, Vol. 41, No. 1 (Mar., 1999), pp. 181-183Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2653183 .

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BOOK REVIEWS 181

that occur in applications like slurry trans- port, sand placement in the fractured oil and gas reservoirs, the removal of cuttings in drilling oil wells, fluidized beds, and coal combustors are controlled by inertia and elastic forces not present in theories that are presently popular.

The last two hundred pages of this work are on analytical and numerical techniques. In this part, one finds the analysis of vis- cometric flows, secondary flows, and un- steady flows that are used to characterize shear stresses, normal stresses, relaxation functions, etc., for different models of a vis- coelastic fluid. These are the usual topics in books on rheological fluid mechanics. Less usual is the comprehensive analysis of per- turbations and stability and bifurcation of simple flows presented here.

A fairly extended analysis of the types of partial differential equations (hyperbolic, elliptic, and parabolic) is presented in a sec- tion on qualitative dynamics. Topics taken up here are Hadamard instability, shear waves, change of type, and allied problems. Since I have contributed to these topics in my own work, I was gratified to see such an excellent account.

Computational rheological fluid dynam- ics is presently the most rapidly develop- ing area of rheological research. Nearly 80 pages have been devoted to this fast-moving subject; the account given is perhaps the most up-to-date available between two cov- ers. However, the subject is developing so fast that any account of it is likely to be outmoded in five years, or less.

Elsevier has done a nice job on this vol- ume: the printing is of high quality, and equations and figures are well presented.

This excellent volume will suit the needs of educated researchers in rheological fluid mechanics. It could be a well-used source of current theory and practice by graduate students and rheologists.

DANIEL D. JOSEPH University of Minnesota

From Calculus to Chaos: An Introduc- tion to Dynamics. By David Acheson. Oxford University Press, Oxford, UK, 1997. $65.00. ix+269 pp., hardback. ISBN 0-19-850257-5.

Expository writing in mathematics seems to have become a lost art. As a student in the 1960s, this reviewer remembers the plethora of popular science books, espe- cially in the areas of the new physics of the 20th century. But there were not many expository works in mathematics, and that trend has continued to the present day. Of course, physics has its Einsteins, Diracs, and so on, and it is easy to excite a general reader about relativity, determin- ism versus probability, cosmology, and H- bombs. Where is all the excitement in math- ematics? Certainly, one can argue that advances in mathematics are more diffi- cult to present to the general public; our language is more closely tied to symbols, and often the level of abstraction is at the top of the ladder. Further, there is natural human interest in twin paradoxes, big bangs, and nuclear war; but it is not clear what the public interest is in the al- gebraic geometric structure lying at the heart of the proof of Fermat's last the- orem, or in the underlying dynamics of problems in nonlinear science. Neverthe- less, mathematical advances unfold funda- mental truths in our intellectual existence, and they should be told. Of necessity, these accounts may have a strong technical com- ponent.

In a word or two, this book is a semitech- nical, semiexpository account of the ap- plications of calculus in differential equa- tions and dynamics, especially involving Newton's second law of motion. The au- thor's aim is to present, by means of sim- ple examples, some of the exciting re- sults and discoveries that have come out of the application of calculus and differen- tial equations since Newton's time, and to present important ideas without dwelling on the details. The applications include linear and nonlinear oscillations, plane- tary motion, least-action principles, fluid flow, instability and catastrophe (including ideas of bifurcation), and chaos. With re- spect to the latter, Chapter 11 presents the basic mathematical notions underly- ing chaotic behavior using the forced Duff- ing equation and the Lorenz equations. The exposition, which includes historical comments, photos of classical documents

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182 BOOK REVIEWS

and famous mathematicians, many dia- grams, and several equations, is about 180 pages; it is followed by a bibliography of about 70 entries, a brief tutorial on pro- gramming in QBasic, an appendix contain- ing QBasic programs for exploring dynam- ics, and solutions to the exercises, which occur at the end of each chapter. The only prerequisite listed is calculus, although readers would certainly benefit from some exposure to elementary physics (mechan- ics).

The exposition moves rapidly from a brief review of calculus and differential equations in the first few chapters to sophisticated ideas near the frontiers of research. Along the way there is an excellent, yet brief, introduction to using computers and nu- merical algorithms to self-explore some of the dynamics.

It is difficult to say who the specific audi- ence for this book is. The author hopes for a wide readership, including university stu- dents in mathematics and science, readers who are preparing for a university course, mathematics and science teachers in col- leges and universities, and general readers who "refuse to be put off by a few equa- tions." Regarding the latter, the man-or- woman-on-the-street should not be fooled; this is a mathematics book! As a text- book, it does not appear to fit into any of the courses in the standard university curriculum in the United States. When one reads the book, however, one thinks about the usual postcalculus differential equations course, or perhaps some of the recently developed undergraduate model- ing courses. This book could serve as a resource, or supplementary reading in such courses, especially in courses designed for honors students. Some undergraduate stu- dents will find this book exciting enough to fill their need to digest an expository book about mathematical advances in our age.

There is one group of individuals who could greatly benefit from read- ing this book, namely, those without an applications-oriented background who teach elementary differential equations courses in our colleges and universities. It is not unusual for faculty whose training has been in pure mathematics to teach

such courses, and many of these individ- uals have not studied even the most rudi- mentary principles of physics. Acheson's book could give them a deeper perspective into the interdependence between math- ematics and physics. Further, the book has several interesting examples that would make exciting classroom demonstrations and projects.

Regarding general readers, it is a pity this little book is so expensive. At $65 most undergraduates or general readers will not purchase it at their local bookstore; in- structors will be reluctant to require the book as a text or a supplement. Putting the book in a paperback format at an af- fordable price would have been a better idea.

One interesting and novel feature of the book is the presentation of computer pro- grams designed to let the reader explore dynamics on his or her own. The fourth chapter (17 pages) on computer solutions introduces the Euler method, the im- proved Euler method, and the Runge-Kutta method for numerically solving differential equations. (The examples used to illustrate these methods are extremely well chosen.) Then, in the appendices, 10 program list- ings are provided to allow the reader to explore some of the examples in the text. These programs can be easily typed in line- by-line, or they can be downloaded from the author's website. Some will question the author's choice of QBasic for the pro- grams, rather than, for example, a com- puter algebra package like Maple or Math- ematica. But the author argues that so- phisticated software packages mask what is really going on, and using a noncompiled programming language like QBasic allows the user to keep track of what the com- puter is doing as well as take the satis- faction that comes from solving the prob- lem. This reviewer typed in a few of the programs on a PC with Windows 95, and he was rewarded by solution curves and nice color graphics. Even the most inex- perienced computer user can easily partic- ipate in the dynamics of the three-body problem, planetary motion, a double pen- dulum, and linear and nonlinear oscilla- tors that become chaotic. A word of warn- ing: sometimes the programs crash be- cause the magnitudes of the variables go

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BOOK REVIEWS 183

out of bounds. The user should insert stopping criteria to prevent these occur- rences.

In summary, Acheson's book is an at- tractive and clear presentation of the evolu- tion of ideas in nonlinear dynamics. Math- ematicians, physicists, and engineers from all levels will find it rewarding and an en- tertaining exposition of a semitechnical na- ture.

J. DAVID LOGAN University of Nebraska-Lincoln

Algebraic Methods in Quantum Chem- istry and Physics. By Francisco M. Fernandez and Eduardo A. Castro. CRC Press, Boca Ra- ton, FL, 1996. $110.00. vii+269 pp., hardback. ISBN 0-8493-8292-0.

This is a book on applications of Lie al- gebras and operator methods in theoret- ical physics and chemistry, and like any other book on subjects as vast as these, it has a point of view that guided the selec- tion of topics. The authors' point of view is to omit the exposure of basic notions and theorems in the theory of Lie groups and Lie algebras, assuming that the reader is familiar with the main ideas and con- cepts relevant to those fields, and, instead, to provide some useful mathematical tools needed for derivation of the theoretical re- sults in a most economic and elegant way. In particular, the Campbell-Baker-Hausdorff formula, disentangling of the exponential operators, and Liouville transformation re- ceive such a treatment. At the same time, this book can serve as a complement to the standard quantum-mechanical course, because most of the fundamental notions, like observables, representations, eigenvec- tors, eigenvalues, matrix elements, selection rules, coherent states, equations of motion, approximation and perturbation methods, and the Schr6dinger, Heisenberg, and inter- mediate pictures in quantum mechanics are introduced, explained, and exemplified. It should be stressed that quantum mechan- ics per se is a well-established discipline, and the actual question that arises is: How should one solve the concrete problem?

Let us also point out that quantum- mechanical problems capable of exact

solution are not numerous and are tra- ditionally solved in the textbooks on the subject mainly by means of Schrodinger's wave equation. On the contrary, the opera- tor methods are usually applied only in a few instances, like harmonic oscillator and an- gular momentum problems. The present book shows that a large number of one- and some three-dimensional models are solvable by algebraic, representation-independent methods relying only on the standard com- mutation relations. In the more realistic situations, it is shown how one can enlarge the above approach by making use of var- ious combinations of ladder or shift opera- tors methods, of the virial and hypervirial theorems, and sometimes of computer cal- culations in order to find the appropriate solution. Applications of all these methods to the calculation of eigenvalues, matrix el- ements, and wave functions are discussed in detail and, in many cases, up to nu- merical evaluations. The concrete physical problems that receive such treatment in- clude the computation of the vibrational overlap integrals (Franck-Condon factors), some exactly solvable models with spheri- cal symmetry, and the vibration-rotational spectrum of diatomic molecules, just to mention a few. The more mathematically inclined reader will be pleased to find a clear exposition (besides the above-mentioned Campbell-Baker-Hausdorff formula) of the Magnus theorem, the Fokker-Planck equa- tion, the Wigner-Kirkwood expansion, and the Euler-MacLaurin and Poisson summa- tion formulae.

In short, this book is a nice introduction to the use of operator methods in quan- tum mechanics and chemistry and can also serve as a reference source because of the numerous problems solved in it. Without any doubt, it is quite suitable for use by students in intermediate quantum mechan- ics courses and also by more advanced post- graduate students and researchers who wish to see how efficient the algebraic methods are in solving concrete quantum-mechanical problems. The experienced reader can also consult some of the relevant competitive books on the subject (and not cited in this book), which are listed below in order of their appearance.

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