Geometrical Theory of Dynamical Systems and Fluid Flows

By T. Kambe
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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.

 

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