Exercise09-The Billiard Problem

熊毅恒 2014301020065

1.abstract

The billiard system can also be a chaotic system. This time I will discuss the problem of a ball moving without friction on a horizontal table. We imagine that there are walls at the edges of table that reflect the ball perfectly and that there is no frictional force between the ball and the table.

2.background

1.Between collisions

Except the collisions with the walls, the motion of the billiard is quite simple. Between every collisions the velocity is constant so we have ：

These equations can be solved using our usual Euler algorithm.

collisions

the components of $v_{i}$ parallel and perpendicular before collisions:

the velocity after reflection from the wall :

3.mainbody

The Lorenz model
it is easier to find the regularity to plot it in a phase space:

z versus x

Code
$\sigma$=10, b=8/3, r=25, the step of time is 0.0001.

When we plot it in phase space, The figure 1. gives some hints of an
underlying regularity.

while the time-dependent behavior is unpredictable, we can predict with certainty that the system will be found somewhere on the attractor surface in phase space.

z versus y when x=0

Code

figure 3.21

First, we consider a quite simple model-the motion of a billiard on a square table. The billiard started at point (0.2, 0).

Trajectory of a billiard on a square table

Code

figure 3.22
now let's change square to circle
Circular stadium

Code
However, if we cut the table along the x axis, and pull the two semicircular halves apart (along y), a distance $2\alpha$ r, the trajectory will be definitely not symmetric.

Stadium billiard \alpha=0.01

Code

figure 3.23 3.24
phase-space plots for the trajectories of $\alpha=0, 0.001$, 0.01, 0.1
Stadium Billiard: \alpha=0, 0.001, 0.01, 0.1

Code

problem 3.30 :
Here, the initial separation of the billiards was 0.00001.
Separation

Code

4.conlusion

Ths chaotic system will present when the boundary is composed by some shapes (different or same). If the boundary is a simple regular shape, the chaotic situation cannot be observed.