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Author: Serge Tabachnikov
Publisher: American Mathematical Soc.
ISBN: 0821839195
Category : Mathematics
Languages : en
Pages : 192
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Book Description
Mathematical billiards describe the motion of a mass point in a domain with elastic reflections off the boundary or, equivalently, the behavior of rays of light in a domain with ideally reflecting boundary. From the point of view of differential geometry, the billiard flow is the geodesic flow on a manifold with boundary. This book is devoted to billiards in their relation with differential geometry, classical mechanics, and geometrical optics. Topics covered include variational principles of billiard motion, symplectic geometry of rays of light and integral geometry, existence and nonexistence of caustics, optical properties of conics and quadrics and completely integrable billiards, periodic billiard trajectories, polygonal billiards, mechanisms of chaos in billiard dynamics, and the lesser-known subject of dual (or outer) billiards. The book is based on an advanced undergraduate topics course. Minimum prerequisites are the standard material covered in the first two years of college mathematics (the entire calculus sequence, linear algebra). However, readers should show some mathematical maturity and rely on their mathematical common sense. A unique feature of the book is the coverage of many diverse topics related to billiards, for example, evolutes and involutes of plane curves, the four-vertex theorem, a mathematical theory of rainbows, distribution of first digits in various sequences, Morse theory, the Poincare recurrence theorem, Hilbert's fourth problem, Poncelet porism, and many others. There are approximately 100 illustrations. The book is suitable for advanced undergraduates, graduate students, and researchers interested in ergodic theory and geometry. This volume has been copublished with the Mathematics Advanced Study Semesters program at Penn State.
Author: Serge Tabachnikov
Publisher: American Mathematical Soc.
ISBN: 0821839195
Category : Mathematics
Languages : en
Pages : 192
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Book Description
Mathematical billiards describe the motion of a mass point in a domain with elastic reflections off the boundary or, equivalently, the behavior of rays of light in a domain with ideally reflecting boundary. From the point of view of differential geometry, the billiard flow is the geodesic flow on a manifold with boundary. This book is devoted to billiards in their relation with differential geometry, classical mechanics, and geometrical optics. Topics covered include variational principles of billiard motion, symplectic geometry of rays of light and integral geometry, existence and nonexistence of caustics, optical properties of conics and quadrics and completely integrable billiards, periodic billiard trajectories, polygonal billiards, mechanisms of chaos in billiard dynamics, and the lesser-known subject of dual (or outer) billiards. The book is based on an advanced undergraduate topics course. Minimum prerequisites are the standard material covered in the first two years of college mathematics (the entire calculus sequence, linear algebra). However, readers should show some mathematical maturity and rely on their mathematical common sense. A unique feature of the book is the coverage of many diverse topics related to billiards, for example, evolutes and involutes of plane curves, the four-vertex theorem, a mathematical theory of rainbows, distribution of first digits in various sequences, Morse theory, the Poincare recurrence theorem, Hilbert's fourth problem, Poncelet porism, and many others. There are approximately 100 illustrations. The book is suitable for advanced undergraduates, graduate students, and researchers interested in ergodic theory and geometry. This volume has been copublished with the Mathematics Advanced Study Semesters program at Penn State.
Author: Rozikov Utkir A
Publisher: World Scientific
ISBN: 9813276487
Category : Science
Languages : en
Pages : 224
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Book Description
A mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets. The ball moves and its trajectory is defined by the ball's initial position and its initial speed vector. The ball's reflections from the boundary of the table are assumed to have the property that the reflection and incidence angles are the same. This book comprehensively presents known results on the behavior of a trajectory of a billiard ball on a planar table (having one of the following forms: circle, ellipse, triangle, rectangle, polygon and some general convex domains). It provides a systematic review of the theory of dynamical systems, with a concise presentation of billiards in elementary mathematics and simple billiards related to geometry and physics.The description of these trajectories leads to the solution of various questions in mathematics and mechanics: problems related to liquid transfusion, lighting of mirror rooms, crushing of stones in a kidney, collisions of gas particles, etc. The analysis of billiard trajectories can involve methods of geometry, dynamical systems, and ergodic theory, as well as methods of theoretical physics and mechanics, which has applications in the fields of biology, mathematics, medicine, and physics.
Author: Nikolai Chernov
Publisher: American Mathematical Society
ISBN: 1470474425
Category : Mathematics
Languages : en
Pages : 330
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Book Description
This book covers one of the most exciting but most difficult topics in the modern theory of dynamical systems: chaotic billiards. In physics, billiard models describe various mechanical processes, molecular dynamics, and optical phenomena. The theory of chaotic billiards has made remarkable progress in the past thirty-five years, but it remains notoriously difficult for the beginner, with main results scattered in hardly accessible research articles. This is the first and so far only book that covers all the fundamental facts about chaotic billiards in a complete and systematic manner. The book contains all the necessary definitions, full proofs of all the main theorems, and many examples and illustrations that help the reader to understand the material. Hundreds of carefully designed exercises allow the reader not only to become familiar with chaotic billiards but to master the subject. The book addresses graduate students and young researchers in physics and mathematics. Prerequisites include standard graduate courses in measure theory, probability, Riemannian geometry, topology, and complex analysis. Some of this material is summarized in the appendices to the book.
Author: Vladimir Dragović
Publisher: Springer Science & Business Media
ISBN: 3034800150
Category : Mathematics
Languages : en
Pages : 293
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Book Description
The goal of the book is to present, in a complete and comprehensive way, areas of current research interlacing around the Poncelet porism: dynamics of integrable billiards, algebraic geometry of hyperelliptic Jacobians, and classical projective geometry of pencils of quadrics. The most important results and ideas, classical as well as modern, connected to the Poncelet theorem are presented, together with a historical overview analyzing the classical ideas and their natural generalizations. Special attention is paid to the realization of the Griffiths and Harris programme about Poncelet-type problems and addition theorems. This programme, formulated three decades ago, is aimed to understanding the higher-dimensional analogues of Poncelet problems and the realization of the synthetic approach of higher genus addition theorems.
Author: Utkir A. Rozikov
Publisher:
ISBN: 9789813276475
Category :
Languages : en
Pages : 223
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Book Description
Author: Richard Evan Schwartz
Publisher: American Mathematical Soc.
ISBN: 0821853686
Category : Mathematics
Languages : en
Pages : 330
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Book Description
The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.
Author: A.V. Bolsinov
Publisher: CRC Press
ISBN: 0203643429
Category : Mathematics
Languages : en
Pages : 752
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Book Description
Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularites, and topological invariants. The authors,
Author: Serge Tabachnikov
Publisher: SMF
ISBN: 9782856290309
Category : Billiards
Languages : en
Pages : 142
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Book Description
Author: Stephanie Alexander
Publisher: Springer
ISBN: 3030053121
Category : Mathematics
Languages : en
Pages : 88
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Book Description
Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.
Author: Tim Ander
Publisher: CRB Publishing
ISBN: 8827502343
Category : Sports & Recreation
Languages : en
Pages :
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Book Description
Take Your Pool Skills to the Next Level and Win Big! Inside How to Play Pool, you’ll discover the rules for many popular variations of the game: Eight-Ball Nine-Ball One-Pocket and Snooker With this book, you can strengthen your pool game with the right posture, physics, and geometry. You’ll learn to execute many different types of shots, such as straight, angled, and spin shots. For example, you’ll learn to combine top/back with left/right spin and get all kinds of impressive results! How to Play Pool explains how you can use your cunning to plan ahead and out-strategize your opponents. You’ll find out why to use just the right amount of force to avoid reflections and “own” pockets. By targeting clumps of balls, you can set yourself up for a great endgame layout. If you pay close attention to the cue ball’s trajectory after it hits the target ball, you’ll set yourself up for shot after easy shot. With these simple and powerful pool-playing tips and techniques, you’ll dominate the table – and the competition! You’ll even learn how to pull off a variety of crowd-pleasing trick shots: Pocketing the Eight-Ball on the Break Jumping Over Obstacles Sinking the 4-in-a-Line Shot Don’t wait – Take the plunge and become a pool shark today with How to Play Pool! It’s fast and easy to order – just scroll up and click the BUY NOW WITH ONE CLICK button on the right-hand side of your screen.
