Chapter 3 Section 3.7 The Biliard Problem

Background

A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed. Billiard dynamical systems are Hamiltonian idealizations of the game of billiards, but where the region contained by the boundary can have shapes other than rectangular and even be multidimensional. Dynamical billiards may also be studied on non-Euclidean geometries; indeed, the very first studies of billiards established their ergodic motion on surfaces of constant negative curvature. The study of billiards which are kept out of a region, rather than being kept in a region, is known as outer billiard theory.
　　　　　　　　　　　　　　　　　　　　　　　 　　　　---- wikipedia

Circular Stadium

对桌面为正圆形的情况，小球的运动是十分规则的
如图：


在经历足够长的时间后小球轨迹会覆盖除去中心圆部分的整个桌面

Stadium billiard

当桌面情况与跑道类似时，小球的运动轨迹会随着$\alpha$的增大而逐渐出现混沌效应。


上图为$\alpha = 0.1$时的轨迹图，title有错误


但是，由上两图可以看出，当$\alpha$继续增大时，球场模型会逐渐接近矩形模型，此时小球轨迹会逐渐表现出规律性。可以预见，当$\alpha = \infty$时，混沌会再次消失。

Phase space

在相空间研究Stadium Billiard
由下图给出: