
Topological Methods in Hydrodynamics
Vladmir Arnold and Boris Khesin.
SpringerVerlag (1998), price $75.00, ISBN: 038794947.

Reviewer: R. Ghrist, University of Illinois UrbanaChampaign, 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 justreleased 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 wellknown 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 selfcontained. 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, ChernSimons 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.