5R40. Lectures on Fluid Dynamics: A Particle Theorist’s View of Supersymmetric, Non-Abelian, Noncommutative Fluid Mechanics and d-Branes. CRM Series in Mathematical Physics. - R Jackiw (Center for Theor Phys, MIT, Cambridge MA 02139). Springer-Verlag, New York. 2002. 114 pp. ISBN 0-387-95422-8. \$49.95.

Reviewed by K Piechor (Inst of Fund Tech Res, Polish Acad of Sci, ul Swietokrzyska 21, Warsaw, 00-049, Poland).

This book by Jackiw differs significantly from what is traditionally understood as “lectures on fluid dynamics.” The author’s aim is to show that the apparatus, methods, language, etc, developed for physics of particles can be successfully applied to fluid mechanics. Roughly speaking, Jackiw shows how classical and very non-classical models of fluids can be derived from suitably constructed Lagrangian or Hamiltonian functionals, and that these models are just particular, specific cases of a general theory. As a result, some relations between seemingly different models, new invariants, and symmetries are discovered.

Chapter 1 is a brief introduction. In Chapter 2, the classical fluids, both irrotational and with nonvanishing vorticity, are discussed. To be able to give the canonical formulations for the latter case as well as that of magnetic fluids, the Clebsch parametrization is introduced and its use explained. Chapter 3 concerns specific models, both relativistic and nonrelativistic. As the first, Jackiw chooses so called Chaplygin gas for which the pressure is negative and proportional to the inverse of the density, and as the relativistic fluid, he takes the so called Born-Infeld model, which has the property to reduce to the Chaplygin gas in the nonrelativistic limit. In Chapter 4, it is shown that the Chaplygin gas as well as the Born-Infeld model follow from the Nanbu-Goto action for a “$d$-brane,” ie, a $d$-dimensional object in $d+1$-dimensional space. Two approaches to the problem are presented: the Chaplygin gas and the Born-Infeld model are derived either by a choice of a proper parametrization or by a hodograph transformation. In Chapter 5, it is shown that the $d$-brane theory is able to produce a fluid model with nonvanishing vorticity if one starts with a super $d$-brane. Then the resulting fluid model possesses supersymmetry. Chapter 6 deals with a one-dimensional case of the theories developed in the previous chapters. In this particular case, both the Chaplygin gas and the Born-Infeld model are completely integrable, therefore, many additional results concerning both models can be obtained. Chapter 7 concerns a non-Abelian fluid mechanics, and the final chapter, 8, is devoted to non-commutative fluid mechanics. The need for such theories follows mainly from magnetohydrodynamics. The monograph ends with solutions to problems, which are immersed in the book.

In this reviewer’s opinion, the beautiful monograph, Lectures on Fluid Dynamics: A Particle Theorist’s View of Supersymmetric, Non-Abelian, Noncommutative Fluid Mechanics and$d$-Branes will be more valuable for theoretical physicists and applied mathematicians than engineers. So, this reviewer does not see this book as a primary reading, but everyone interested in fundamentals and deep theoretical approach to fluid mechanics should become acquainted with it.