An Interactive Introduction to Mathematical Billiards
One of my areas of research is billiard dynamics, using both analytic approaches and numerical simulation. This is designed to be an interactive
introduction for students or anyone who is interested. While it turns out that such dynamical systems are broadly applicable, as the name suggests,
they may be visualized as the paths of billiard balls as they move until they collide with a wall and then bounce off. For our model, we assume no energy
is lost, so they never slow and stop. While our table might resemble a rectangular billiard table (without pockets), more generally we consider tables of any shape you can imagine.
Choices for the simulation:
- Billiard table: Choose from a variety of shapes
- Mass distribution: We assume our ball is very small relative to the table size and we only draw the path, but the mass distribution will influence the collisions at walls. For example, if all the mass is at a point, spin does not matter and the collisions will be "angle in equals angle out." On the other hand, if the balls have even (uniform) mass or all the mass is at the rim (hollow) more interesting collisions occur.
- Number of collisions: Determines how long to run the simulation.
- Angle: The red arrow gives the starting direction of the billiard. This angle can be changed. Note that it is measured from the inward normal, so 0 degrees would be straight out.
- Spin: We can put English on the balls! The spin will change at collisions for Uniform and Hollow settings.
One of the general questions we are interested in is when is there regularity and structure and when is there seeming randomness and choatic behavior? Feel free to play around and investigate informally, or consider some of the specific questions given below.
- Can you find settings which appear to fill the entire table in an orderly way?
- Can you find settings which appear to fill the entire table is a random or chaotic way?
- Can you find an example of "invariance", that is, an example where the trajectories will not
fill the billiard table but be restricted to certain portions?
- A trajectory is "periodic" if it returns to its starting position and velocity after a finite number of collisions and then repeats.
Can you choose settings to get a periodic trajectory?
- Can you find a table and a mass distribution such that ALL of the trajectories are periodic?