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.
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