Topological Methods in Hydrodynamics Vladmir Arnold and Boris Khesin. Springer-Verlag (1998), price $75.00, ISBN: 038794947.
Reviewer: R. Ghrist, University of Illinois Urbana-Champaign, USA.

Countless students and researchers in dynamical systems have profited from the clarity and insight imparted from the books of V. I. Arnold. The text under review, written in collaboration with B. Khesin, is not a just-released text (1998); nor has it (apparently) been read as widely as it deserves to be. This is however a monogram well worth the investment in time and energy.

The title of this text evokes that of MMCM (Mathematical Methods in Classical Mechanics). Here, the proposed methods are of a more topological flavor, and the proposed applications are to fluid dynamics and magnetohydrodynamics. Just as MMCM was instrumental in providing a convincing argument that symplectic geometry and differential forms are the right languages for classical mechanics, the volume under review gives a broad argument for the injection of contemporary topological and geometric perspectives in the study of fluid flows and related systems.

The volume begins with the Lie group/algebra approach to the Euler equations. The well-known results of Arnold give a natural algebraic link between the Euler equations for a rigid body and the Euler equations for the motion of a fluid. Throughout this section (and the remainder of the book), the thread of Hamiltonian dynamics runs.

Subsequent chapters change emphases from dynamics to topology to geometry. These chapters cover, among other things, steady Euler fluid flows, vortex motion, magnetic fields, vorticity and helicity, energy relaxation, hydrodynamic stability, and dynamos. The last chapter may be of particular interest to the reader coming from a dynamical systems perspective: this chapter covers dynamical systems which are of a hydrodynamical flavor. Material covered includes the KdV equation, NLS and the Hasimoto transformation, and more.

This book is not what one would call introductory. The authors assume a working knowledge of differential forms, Hamiltonian and symplectic dynamics, Lie groups, and some algebraic topology. The text is also not one to be read linearly; indeed, the authors make it clear that the chapters were written to be somewhat independent and self-contained. This has the unfortunate consequence that the reader will attempt to learn mathematical fluid dynamics by reading from the beginning of the text. This is not recommended. The first clear treatment of the Euler equations for an inviscid fluid appears in the more readable Chapter 2, the previous chapter having its fluid dynamics content buried in and among rarified discussions of coadjoint orbits, etc.

One of the best potential features of the book is its ability to inform the reader about contemporary ideas in topology and geometry within the context of very grounded applications. It provides both motive and means for learning about topics such as asymptotic linking, Chern-Simons theory, symplectic topology, and the Virasoro algebra.

Despite the high learning curve for this volume, it is well worth the investment. Several of the results in the book are previously unpublished. Furthermore, the text is teeming with interesting open problems and directions for further research.