Chapter 3 Problem 3.31 The Billard Problem

计算物理

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Name:陈锋
Student Number: 2013301020145
April 28,2016

Abstract

The billiard system can also be a chaotic system. In this problem, I will try to solve the trajectories of different kinds of tables and show the phase diagram to see whether the system is chaotic or not.

Introduction

In this problem, I will not consider the effect of friction, which means the billiard ball will move without friction on a perfect billiard table.
Except for the collisions with the walls, the motion of the billiard is quite simple. Between collisions the velocity is constant so we have:

$$\begin{equation} \frac{dx}{dt} = v_x \\ \frac{dy}{dt} = v_y \end{equation}$$
where $v_x$ and $v_y$ change only through collisions with the walls. These equations can be solved using our Euler algorithm.
For a circle or an ellipse, the direction of the normal vector is $(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})$, which means (x,y) for circle and $(\frac{2x}{a^2}, \frac{2y}{b^2}) for ellipse$.
It is then useful to calculate the components of $\vec{v}_i$ parallel and perpendicular to the wall. These are just:
$$\begin{equation} \vec{v}_{i,\bot} = (\vec{v}_i\cdot\hat{n})\hat{n} \\ \vec{v}_{i,\|} = \vec{v}_{i} - \vec{v}_{i,\bot} \end{equation}$$
After reflection, the velocity becomes:
$$\begin{equation} \vec{v}_{f,\bot} = -\vec{v}_{i,\bot} \\ \vec{v}_{f,\|} = \vec{v}_{i,\|} \end{equation}$$

Influence of diverse boundaries

Square

Figure 1: This is the trajectory of the billiard ball in the case of square boundary. Actually if the ratio of $v_x$ and $v_y$ is rational, the square cannot be fully filled, for the quantity of irrational number is larger.

Circle

This is not a chaotic case.

Figure 2: This is the trajectory of the billiard ball in the case of circular boundary. It shows that the billiard ball cannot reach the center part, which is determined by the position of tha ball and direction of initial. velocity.

Figure 3: This is the phase diagram of (x,$v_x$) and (x,$v_y$)

Figure 4: This is Poincare section by scattering the points only when the billiard ball crosses the y = 0 axis.

Ellipse

This is also not a chaotic case.

Figure 5: This is the trajectory of the billiard ball in the case of elliptical boundary. It shows that the billiard ball cannot reach the left and the right part, which is determined by the position of tha ball and direction of initial.

Figure 4: This is Poincare section by scattering the points only when the billiard ball crosses the y = 0 axis, which is a ellipse-liked curve.

Square with an inner circle

Both of these are chaotic cases.

The circle is in the center

Figure 7: This is the trajactory of the billiard ball, which shows a irregular curve.

Figure 8: This is Poincare section by scattering the points only when the billiard ball crosses the y = 0 axis.

The circle is slightly off-center

Which is nearly the same as the former case.

VPython

Circle:

Ellipse:

Conclusion

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.

Programme Code

-Chapter 3 Problem 3.31