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Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts
Valelriĭ V. Kozlov and Dmitriĭ V. Treshchëv
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Translations of Mathematical Monographs
1991; 171 pp; hardcover
Volume: 89
ISBN-10: 0-8218-4550-0
ISBN-13: 978-0-8218-4550-9
List Price: US$84
Member Price: US$67.20
Order Code: MMONO/89
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Starting with the work of G. D. Birkhoff, billiards have been a popular research topic drawing on such areas as ergodic theory, Morse theory, and KAM theory. Billiard systems are also remarkable in that they arise naturally in a number of important problems of mechanics and physics.

This book is devoted to mathematical aspects of the theory of dynamical systems of billiard type. Focusing on the genetic approach, the authors strive to clarify the genesis of the basic ideas and concepts of the theory of dynamical systems with impact interactions and also to demonstrate that these methods are natural and effective. Recent limit theorems, which justify various mathematical models of impact theory, are key features. Questions of existence and stability of periodic trajectories of elastic billiards occupy a special place in the book, and considerable attention is devoted to integrable billiards. A brief survey is given of work on billiards with ergodic behavior. Each chapter ends with a list of problems.

Reviews

"Written by well-known mathematicians who have contributed a great deal to the field. The exposition ... is very thorough."

-- Mathematical Intellingencer

Table of Contents

  • Introduction: Elements of impact theory
  • The genetic method in the dynamics of systems with one-sided constraints
  • Periodic trajectories of the Birkhoff billiard
  • The Hill equation
  • Integrable problems
  • Nonintegrable billiards
  • Appendix I. Systems with elastic reflections and KAM theory
  • Appendix II. On the connection of dynamic and geometric properties of periodic trajectories
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