Geometrical theory of dynamical systems and fluid flows.
Revised ed.

*(English)*Zbl 1201.37108
Advanced Series in Nonlinear Dynamics 23. Singapore: World Scientific (ISBN 978-981-4282-24-6/hbk). xxi, 421 p. (2010).

This is the second edition of an advanced introductory text on the geometric theory of dynamical systems, fluid flows and integrable systems published for the first time in [Advanced Series in Nonlinear Dynamics 23. Singapore: World Scientific. (2004; Zbl 1065.37001)]. In addition to efforts aimed at improving the overall readability of the book, the most significant changes in the new edition are as follows. The presentation of the material in Section 3.7 “Covariant derivative and parallel translation” has been improved to clarify the original idea of Arnold. Chapter 7 “Flows of an ideal fluid: variational formulation and gauge principle” has been completely rewritten to present a variational formulation of ideal fluid flows in the light of gauge theory. These changes affected also the beginning sections of Chapter 8. Finally, a new appendix I replaced the old one; a completely new appendix J entitled “Principle of gauge invariance” was added. With all the improvements made in the second edition, this nice guide to fluid mechanics and related subjects becomes an even more enjoyable reading.

Reviewer: Svitlana P. Rogovchenko (Umeå)

##### MSC:

37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |

35Q53 | KdV equations (Korteweg-de Vries equations) |

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

37Jxx | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |

76B47 | Vortex flows for incompressible inviscid fluids |

76Mxx | Basic methods in fluid mechanics |