Texts on Mathematical Billiards

Question

I want to study the theory of mathematical billiards, and was looking for a text for self-study. If you have a text in mind that is more general i.e. on dynamical systems as a whole but still contains some material on billiards, that would be great too. I'm an undergraduate with background in intermediate analysis, basic algebra, linear algebra, basic differential equations and applied complex analysis.

Answer 1

I think these are some books you might find interesting, regarding mathematical billiards at a relatively introductory level

• An Introduction to Mathematical Billiards, by Utkir A. Rozikov;
• Chaotic Billiards, by Nikolai Chernov;
• Geometry and billiards, by Sergei Tabachnikov.

Different approaches and fascinating mathematics can also be found in

• Invariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities, by Anatole Katok et al.

Hope these prove useful.

Answer 2

There are several very nice discussions of billiards in A Panaramic View of Riemannian Geometry, by Marcel Berger, ranging from elementary (first variational formula) to advanced (billiards on a Riemannian manifold). The book is replete with references as well (1310 of them to be precise), so it can lead you to other references on specific points of interest. Relevant sections include 1.2.3, 1.4.3, 1.9.3, 8.1, 10.1.

Billiards come up several times in Arnold's classic, Mathematical Methods of Classical Mechanics, most often in the context of toral billiard tables. This book is significantly more advanced, but a masterpiece. Relevant sections include Appendix 1, 8, 10, 16 (Note that these are not your normal appendices. The appendices are almost half the book.)

Finally, an interesting place that billiards come up is in Quantum Computation and Quantum Information, by Nielsen and Chuang. This is the standard reference for quantum computing. In section 3.2.5 they construct a billiard ball computer capable of universal computation. The discussion is elementary and easy to follow.