**1R8. Introduction to Symmetry Analysis. - ** BJ Cantwell *(Sch of Eng, Stanford Univ, Stanford CA 94305). Cambridge UP, Cambridge, UK. 2002. 612 pp. Softcover, CD-Rom incl. ISBN 0-521-77740-2. $50.00. Also available in Hardcover ISBN 0-521-77183-8; $130.00.*

*Reviewed by TH Moulden (Dept of Aerospace Eng, Univ of Tennessee Space Inst, BH Goethert Pkwy, Tullahoma TN 37388-8897).*

The application of Lie group theory is far from ubiquitous in engineering courses. Mathematics has of course, long been familiar with the topic (for example see such standard texts as PJ Oliver (1986), *Applications of Lie Groups to Differential Equations*, Springer-Verlag or PE Hydon (2000), *Symmetry Methods for Differential Equations*, Cambridge University Press). As such the text under review is a very welcome addition to the engineering literature and one that should be widely adopted. The book is intended for beginning graduate students in engineering and the applied sciences. The author notes that most problems of interest to those fields are nonlinear problems. The study of nonlinear problems requires all the tools that are available in order to obtain insight into the structure and properties of such physical situations. Closed form solutions are usually not available for such equations.

The book under review starts with a short chapter on the history of group theory and its applications to the subject at hand. A more detailed history here is IM Yaglon (1988), *Felix Klein and Sophus Lie*, Birkhauser, quoted by Cantwell as well as the more recent book T Hawkins (2000), *Emergence of the Theory of Lie Groups*, Springer-Verlag.

The first formal chapter introduces symmetry and the basic symmetry groups of interest in the remainder of the text. Some of these symmetries are continuous (as in the properties of homogeneity and isotropy characterized by translation and orthogonal groups) and some discrete (as associated with specific geometric objects). Since an important physical phenomenon is the breaking of a continuous translational symmetry into a discrete symmetry, both types of symmetry must be discussed. One of the classical tools in any engineering study is dimensional analysis and Chapter 2 is devoted to its application in selected problems. It is here that insight into the physics of the problem under study becomes essential. The text does a good job, throughout, of comparing the group methods with this dimensional analysis (Section 10.3 being an example) and other standard techniques.

Chapter 3 moves onto review the classical background from the theory of differential equations (both ordinary and partial) needed for the rest of the text. Of particular importance for certain applications is the study of *critical points* (Section 3.9) in the phase space of a dynamical system. The application in Section 13.6 to the local flow structures in turbulence being a case in point (ad makes contact with such work as that of AE Perry and MS Chong (1987), *A Description of Eddying Motions and Flow Patterns Using Critical Point Concepts*, Annual Review Fluid Mechanics, Vol 19, pp 125-155, as well as Cantwell’s own work.

Chapter 4 treats classical dynamics mostly from the Hamiltonian viewpoint. This is mostly a review of classical material, along with examples, rather than a text on the subject. The cosmological two-body problem is treated in some detail.

One-parameter Lie groups are introduced in Chapter 5 while Chapter 6 applies this theory to first order ordinary differential equations. The Lie algebra of a Lie group (and its importance) is introduced in Chapter 5. Thus, in local kinematics the Lie algebra $SO3$ associated with the spin tensor maps (via the exponential function) to the Lie group $SO3;$ thus relating vorticity to fluid rotation. The reader may find that more background in Lie theory is required beyond what is provided by the text. Previous familiarity with algebra would be of utility, at this point, to provide background to some of the algebraic structures needed (the classic: IN Herstein (1975), *Topic in Algebra*, Wiley, would serve here and the recent text: A Baker (2002), *Matrix Groups*, Springer-Verlag, also provides some mathematical structure for special cases).

After an introduction of the notation for differential functions in Chapter 7, Chapter 8 treats ordinary differential equations of higher order from the Lie viewpoint while Chapter 9 extends this to partial differential equations. Higher order differential equations can be treated by means of successive reductions of order.

The middle section of the book (Chapters 1-13) applies the techniques developed in the early chapters of the book to problems in fluid mechanics. Chapter 10 addresses the classical boundary layer theory, Chapter 11, incompressible flows in general, Chapter 12 compressible in viscid flows, and Chapter 13 the turbulent shear flow. The boundary layer concept in fluid mechanics is about 100 years old now and Prandtl’s equations have been an essential feature of aerodynamics for much of that time. The advent of numerical solution techniques for the full Navier-Stokes equations has reduced this role, in a practical sense, but the conceptual ideas are still very relevant. The text first develops the classical Blasius equation and notes its invariance under the dilation group in order to extract a similarity solution. The other classical similarity solution the Falkner-Skan equation is also discussed.

Chapter 11 moves to the full incompressible flow Navier-Stokes equations. One of the early applications of Lie concepts to fluid mechanics was with these equations (RE Boisvert, WF Ames, and UN Srivastava (1983), *Group Properties and New Solutions of the Navier-Stokes Equations*, J Eng Math, Vol 17, pp 203-221). Ames has done much to promote symmetry analysis (acting, as he did, as editor for the translation of the important text LV Ovsiannikov (1982), *Group Analysis of Differential Equations*, Academic Press). The problems associated with unsteady jets flows are discussed in detail in this chapter.

A study of compressible flows form the subject of Chapter 12 with attention restricted to in viscid flows. This restriction is not significant for most of the problems discussed (spherical blast waves and similarity for airfoil flows, for example). The transonic flow similarity condition being an exception in the flow over thin airfoils since shock induced boundary layer separation (breaking the similarity) cannot be considered under this restriction.

Those with an interest in fluid turbulence will turn to the 50 pages of Chapter 13 to read about similarity rules in turbulent shear flows. Turbulence models are not discussed (but page 399 does make note of symmetry analysis as an *extremely useful tool* for the development of rational models of turbulence: no reference is given however). Section 13.6 contains the important analysis of the fine scale dissipation motions and their geometric structures. This is done by studying the invariants of the velocity gradient tensor. Not surprising is the finding that vortical structures play a dominant role. The important Galilean invariance constraint on turbulence models is not mentioned, however.

The remainder of the text (Chapters 14-16) is devoted to the Lie-Ba¨cklund transformations and their uses. These, as distinct from the Lie point transformations discussed in the earlier chapters, are contact transformations. Thus Chapter 14 introduces the transformations and their basic properties along with some examples; in particular the Blasius equation is discussed and compared with the earlier treatment of that equation in Chapter 8. Chapter 15 discusses various symmetries and conservation laws and the corresponding Lie-Ba¨cklund groups. Chapter 16 extends the discussion to include the non-local groups.

The text ends with three appendences dealing with background mathematical material with the first appendix devoted to basic calculus and the theory of contact. The need for a clear notation, to avoid confusion, is pointed out here. The second appendix deals with the invariance of contact conditions under Lie transformations while the third considers the Lie-Ba¨cklund transformations.

The important physical phenomena that lead to symmetry breaking (from, for example, continuous to discrete symmetry or via a Hopf bifurcation) are not a prominent feature of the text. This does limit the utility of the book as a reference while not harming its primary mission.

About the book in general: there are some meaningful exercises at the end of each chapter as well as a representative list. The book comes with a CD containing software for *Mathematica*®, which greatly assists the mechanics of finding Lie symmetry groups. Appendix 4 discusses this software. The text does warn, however, that this process can take much computer memory and a lot of processing time. It is best to approach this book with pen and paper in hand to work through the details and have the CD close by to use as appropriate. Very few typographical errors were noted. Other texts of interest for the basic Lie theory (but not the detailed applications in the text under review) include (in addition to texts mentioned above): GW Bluman and S Kumei (1989), *Transformations of Manifolds and Applications to Differential Equations*, Longman, while Cantwell makes frequent reference to NH Ibragimov (1994), *CRC Handbook of Lie Group Analysis of Differential Equations*, CRC Press.

*Introduction to Symmetry Analysis* is nicely presented by Cambridge University Press in its *Cambridge Texts in Applied Mechanics* series. It should find a wide audience in the engineering and applied sciences community and we wish it well.