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Book review

Author(s) Puta, Mircea
Title Hamiltonian Mechanical Systems and Geometric Quantization
Publisher Kluwer Academic Publishers
Year of publication 1993
Reviewed by Mihai Postolache

Hamiltonian Systems can be found at various places in physics, notably in dynamical systems theory, as well as in mechanics. This is a very specialized monograph which presents a coherent exposition of some important techniques, methods and results concerning the geometry of symplectic and Poisson manifolds. Also, some important applications of the treated results are given.

The book includes ten chapters, an introduction, a comprehensive bibliography, table of index, table of contents sufficiently detailed as well as a list of background notations used in the book. The first of these chapters deals with the symplectic geometry. The second, gives the basic elements concerning Hamiltonian mechanics. In the third chapter, some notions about Lie groups are presented. Chapter 4, deals with the Hamilton-Poisson mechanics and in Chapter 5 topics about the stability of Hamiltonian systems are investigated. The next two chapters deal with the geometric quantization. Finally can be found relations between geometric quantization and the Marsden-Weinstein reduction as well as with Poisson manifolds. Every chapter includes worked applications. The book is very well produced and is written in a very good English.

Professor Mircea Puta is a very well known Romanian scientist. His main area of research is dynamical systems, including Hamiltonian systems and links with the differential geometry.

Intended to attract readers interested both in theory and applications the book is written for more advanced students in mathematics, graduate students, research scientists as well as for applied mathematicians. Giving a detailed presentation of many applications, this book provides a interesting investigation of some important techniques of studying mechanical systems. Parts of this book will be useful for applied physicists. This book assumed prior acquaintance with advanced mathematical topics such as differential geometry, and higher algebra.

This research level material provides a very good introduction to the stated subject. The book being highly-specialized is designed mainly for the institutional market. Also the book would be a good investment for individuals seriously interested in the topic.