Geometry and Billiards
About this Title
Serge Tabachnikov, Penn State, University Park, PA
Publication: The Student Mathematical Library
Publication Year 2005: Volume 30
ISBNs: 978-0-8218-3919-5 (print); 978-1-4704-2141-0 (online)
MathSciNet review: MR2168892
MSC: Primary 51-02; Secondary 37-01, 37D50, 37J10, 70H05, 82C05
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.
The 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 (but contains more material than can be realistically taught in one semester). Although the minimum prerequisites include only the standard material usually covered in the first two years of college (the entire calculus sequence, linear algebra), readers should show some mathematical maturity and strongly rely on their mathematical common sense. As a reward, they will be taken to the forefront of current research.
A special feature of the book is a substantial number of digressions covering diverse topics related to billiards: evolutes and involutes of plane curves, the $4$-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.
Advanced undergraduates, graduate students, and research mathematicians interested in ergodic theory and geometry.
Table of Contents
- Chapter 1. Motivation: Mechanics and optics
- Chapter 2. Billiard in the circle and the square
- Chapter 3. Billiard ball map and integral geometry
- Chapter 4. Billiards inside conics and quadrics
- Chapter 5. Existence and non-existence of caustics
- Chapter 6. Periodic trajectories
- Chapter 7. Billiards in polygons
- Chapter 8. Chaotic billiards
- Chapter 9. Dual billiards