|
Geometrical Theory of Dynamical Systems and Fluid Flows
T. Kambe,
World Scientific (2004), 416 pp., price EU 88.-
ISBN: 9812388060.
|
Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.
|
Level: intermediate.
This text serves as an introduction to the large body of geometric
methods in dynamical systems on manifolds, with a particular
emphasis upon applications to fluid dynamics and integrable systems.
The first half of the text consists of a primer on manifolds, Lie
theory, and Riemannian geometry. This portion of the book is
structured so as to present the minimal amount of mathematical
formalities necessary to present the introduction to geometric fluid
dynamics in the second half of the text.
As a tutorial on geometric fluids, this book is a gentle and
informative introduction. It covers the geometric prerequisites
quickly enough to allow the reader to get to the applications before
fatigue and frustration conspire to quash enthusiasm. It is
therefore recommended for the non-mathematician who wants to learn
or appreciate geometric fluid dynamics. Some mathematicians will be
put off by the occasional inaccuracy in the definitions (e.g., the
definition of a tangent bundle). Most readers, however, will find
the geometric tools to be presented in a reasonable and mercifully
imprecise fashion.
The second half of the text covers a broad range of topics in
geometric fluids and integrable systems. These applications are in
harmony with tools surveyed, and are themselves of great interest:
Euler equations, vortex filament equations, KdV, and Sine-Gordon. As
compared to the text by Arnold and Khesin, the mathematics here is
less precise but more carefully and sequentially developed. There
are also fewer connections to open problems in this text as compares
with that of Arnold-Khesin. Overall, however, it is a good text for
students or researchers interested in geometric fluids.