Geometry and Billiards

Serge Tabachnikov - Collection Student Mathematical Library

180 pages, parution le 20/10/2005

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Résumé

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 Poincaré 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.

This volume is copublished with the Mathematics Advanced Studies Seminars.

Readership : Advanced undergraduates, graduate students, and research mathematicians interested in ergodic theory and geometry.

L'auteur - Serge Tabachnikov

Serge Tabachnikov, Penn State, University Park, PA

Sommaire

Motivation: Mechanics and optics
Billiard in the circle and the square
Billiard ball map and integral geometry
Billiards inside conics and quadrics
Existence and non-existence of caustics
Periodic trajectories
Billiards in polygons
Chaotic billiards
Dual billiards
Bibliography

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Caractéristiques techniques

	PAPIER
Éditeur(s)	American Mathematical Society (AMS)
Auteur(s)	Serge Tabachnikov
Collection	Student Mathematical Library
Parution	20/10/2005
Nb. de pages	180
Format	14 x 21,5
Couverture	Broché
Poids	235g
Intérieur	Noir et Blanc
EAN13	9780821839195
ISBN13	978-0-8218-3919-5