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