An Introduction to Mathematical Billiards

Utkir A Rozikov

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813276482
• Topic: Physical Sciences
• Subtopic: Chaotic Behavior in Systems

A mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets. The ball moves and its trajectory is defined by the ball's initial position and its initial speed vector. The ball's reflections from the boundary of the table are assumed to have the property that the reflection and incidence angles are the same. This book comprehensively presents known results on the behavior of a trajectory of a billiard ball on a planar table (having one of the following forms: circle, ellipse, triangle, rectangle, polygon and some general convex domains). It provides a systematic review of the theory of dynamical systems, with a concise presentation of billiards in elementary mathematics and simple billiards related to geometry and physics.

The description of these trajectories leads to the solution of various questions in mathematics and mechanics: problems related to liquid transfusion, lighting of mirror rooms, crushing of stones in a kidney, collisions of gas particles, etc. The analysis of billiard trajectories can involve methods of geometry, dynamical systems, and ergodic theory, as well as methods of theoretical physics and mechanics, which has applications in the fields of biology, mathematics, medicine, and physics.

Contents:

• Introduction
• Dynamical Systems and Mathematical Billiards
• Billiard in Elementary Mathematics
• Billiard and Geometry
• Billiard and Physics

Readership: Graduate students, young scientists and researchers interested in mathematical billiards and dynamical systems.
Key Features:

• The book features a very popular topic, therefore it is suitable for any researcher working in this field and students interested in dynamical systems
• Most material found in the book have not been published in English before
• The book contains results of many recent papers related to billiards
• In July of 2018, MathSciNet found more than 2420 entries for "billiards" in the entire database. The long list of literature devoted to billiards makes it difficult for a beginner to start reading the theory. Thus, this book will be useful to them

Chapter 1

Dynamical systems and mathematical billiards

In this chapter we briefly give the theory (without proofs) of discrete and continuous-time dynamical systems. This will be helpful for the reader, because a mathematical billiard which we want to discuss in this book is a particular case of dynamical systems. We define discrete and continuous-time dynamical systems, formulate the main problem in consideration of such a dynamical system and give a review of methods of solutions of the main problem. In the last section of this chapter we define mathematical billiards and formulate main problems related to them.

1.1Discrete-time dynamical systems

1.1.1Definitions and the main problem

What do we mean by a system? A system is a set of interacting or interdependent components, members or points.

For example, let S be the members of a family, i.e., S = {parents, childeren}, this set is a system, because there are some relations (interactions) between them. Consider now a set which is {chalk, coin, pen, key} this is a set but is not a system, because there is no interaction between these objects.

The state of a system is a collection of its properties, that are interesting to know.

For our example, the state of S (the family) can be considered as s = the number of boys, the number of schoolboys, salary of the father etc.

Measure of a system is the assignment of numbers to a state or a property of the system. It is a cornerstone of most natural sciences, technology, economics, and quantitative research in other social sciences.

For example, we can give the following distinct measures of the system S.

M_sport(S) = the number of boys in S.

M_school(S) = the number of schoolboys in S.

Let us assume that a thief robs the family, then the following measure is interesting to the thief:

M_thief(S) = Total sum of money in S.

So one can define different measures of the same system S.

A dynamical system is a system the state of which changes when time is increasing.

Again the system S (the above mentioned family) is a dynamical system because, states of S change when time increases: for example, the number of schoolboys, the salary of the father etc.

We note that, in case when a system contains infinitely many elements, then, investigation and measurement of such dynamical systems are very difficult and require powerful mathematical methods. You can just imagine how the system S will be complicated if there are infinitely many people in it. What will be M_thief(S) in the case where S is infinite? Is it still a finite number? If it is infinite, then the thief will die from a heart attack!

Thus a dynamical system is a rule that describes the evolution with time of a point in a given set. This rule might be specified by very different means like iterated maps (discrete-time), ordinary differential equations and partial differential equations (continuous-time) or cellular automata. There are some dynamical systems where instead of time parameters, one ...