Mathematical Billiard

One of the most surprising properties of chaos is that it can appear in really simple systems. You've already seen the double-pendulum in the previous section. Another fascinating example are billiard tables.

• https://plus.maths.org/content/chaos-billiard-table
• https://en.wikipedia.org/wiki/Dynamical_billiards
• http://mathworld.wolfram.com/Billiards.html
• https://www.bristol.ac.uk/media-library/sites/pace/documents/corina-powerpoint%20standard.pdf
• http://mathcircle.wustl.edu/uploads/4/9/7/9/49791831/20160306-billiards-presentation.pdf
• http://people.maths.ox.ac.uk/tanner/Prospects2010/CUlcigraiTalk.pdf

Imagine you are trying to predict the path a ball on a billiard table will take in response to a push. The rules at play simple: the ball's acceleration is equal to the force applied divided by its mass (that's Newton's second law of motion) and when it hits a side, the angle of reflection is equal to the angle of incidence (strictly speaking you also need to include the effect of friction, but that's not too hard to do).

The trouble is that in your average pub setting you can't easily measure the exact amount of force applied to the ball, the exact angle with which it hits a side, and so on. As you make your calculations, this small initial uncertainty can snowball, so that pretty soon your prediction may become so uncertain as to be useless. That "sensitivity to ignorance" is the hallmark of mathematical chaos.

Two billiard tables next to each other, move one ball a small amount

Real Life Applications (semiconductors)

Elliptical Billiard

Move the yellow ball in the center, and see where it ends up after 100 collisions:

Chaotic Scattering

Gaspard–Rice system