Pendulum
This project displays damped Lissajou curves, Lissajou curves whose amplitude decreases exponentially. They are the trajectories of Blackburn pendulums, pendulums oscillating in two dimensions, but with different frequencies along two directions.
Check Paul Wainwright's website for beautiful long exposure photographs revealing the trajectories of a Blackburg pendulum (and check his video about how he made them, including a giant pendulum in a barn!).
The frequency along the horizontal direction is labeled a, and the frequency along the vertical direction is labeled b. The trajectories are plotted so that the thickness of the curve is inversely proportional to the speed of the pendulum.
Integer frequencies
The following pictures show the curves for a and b running from 2 to 10: (a, b ) = (2, 2), (2, 3), (2, 4) ... (3, 2), (3, 3), ...
Note that when the two frequencies are equal, the curve collapses to a line segment. Also, the (p, q) and (q, p) curves are mirror of each other along the diagonal.
Irrational frequencies
Now let's keep the same values for a but take b to be an irrational number, for instance n + pi/10, where n = 2, ..., 10. We now get more chaotic looking curves, although how chaotic it is depends a lot of the particular value of a and b, with some curve looking still very regular.
