The 9th homework : problem 3.31 Billiard ball on diverse table

因吹斯汀

problem 3.31

Study the behavior for other types of tables. One interesting possibility is a square table with a circular interior wall located either in the center, or slightly off-center. Another possibility is an elliptical table.

● 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.

● background

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:

$$\frac{dx}{dt}=V_x\quad(1)\quad\quad\quad\frac{dy}{dt}=V_y\quad(2)$$
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 $(\frac{∂f}{∂x},\frac{∂f}{∂x})$, 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 $\overrightarrow{V_i}$ parallel and perpendicular to the wall. These are just:
$$\overrightarrow{V_{i⊥}}=(\overrightarrow{V_i}·\overrightarrow{n})\overrightarrow{n}\quad(3)\qquad \qquad \overrightarrow{V_{i∥}}=\overrightarrow{V_i}-\overrightarrow{V_{i⊥}}\quad(4)$$
Hence,after reflection from the wall, the velocity becomes:
$$\overrightarrow{V_{f⊥}}=-\overrightarrow{V_{i⊥}}\quad(5)$$
$$\overrightarrow{V_{f∥}}=\overrightarrow{V_{i∥}}\quad(6)$$

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● Content

Influence of diverse boundaries（我们来研究一下大力出奇迹）

●Square
Click to View the Code：方形边界.py


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
Click to View the Code：圆形边界.py

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 (y,$V_x$)

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

●Ellipse
Click to View the Code：椭圆边界（Start from origin）


Figure 5: This is the trajectory of the billiard ball in the case of elliptical boundary（Start from origin）. 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.

Click to View the Code：椭圆边界（Start from a focus）

Figure 6： This is the trajectory of the billiard ball in the case of ellipse boundary（Start from a focus）. 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 7: 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
The circle is in the center

Figure 8: This is the trajactory of the billiard ball, which shows a irregular curve.

Figure 9: 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.

偏离较大

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● Conclusion

In this situation, the trajectory is more complicated, the system is chaotic.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.

● Acknowledgments

Thanks to Chen Feng, Exercise_10: Chapter 3 problem 3.31: Billiard ball on diverse table