Material

Buy a toy known as a pendulum man.

Mount it against a flat metal surface such as a refrigerator.

To Do and Notice

Start the pendulum moving. Try different speeds.

Notice that very slow, small motions of the array of pendulums are regular and predictable. Also high speed rotation is very predictable. However, over a wide range of speeds the motion of the pendulums is unpredictable.

Start the pendulums from the same initial position two times in a row. Notice that after just a few seconds the motion of the pendulums in the two trials is quite different.

What's Going On?

These pendulums exhibit the characteristic behavior of chaotic systems. Start the pendulums from exactly the same position twice in a row. After a few seconds the pendulums in these two different trials will be moving in totally different ways. The pendulums have been designed so that the slightest difference in the starting position grows exponentially in time.

Math Root

Exponential growth means that after some time, T, there will be a small difference, d, between the positions of the pendulums in the two trials. After the next interval of T, this difference will have doubled to 2d, after the next interval of T it will have quadrupled to 4T, and so on. This assures us that the smallest initial error in starting the pendulums during the two trials will result in completely unpredictable differences in their motions after a brief time.

So What?

After Isaac Newton discovered his laws of motion it seemed possible that the future motion of everything in the universe could be predicted from accurate measurements taken at one time. There seemed to be no free will. However, this simple set of pendulums with its chaotic exponential growth of unknowability combined with the absolute impossibility of starting the pendulum without slight differences in its position and speed guaranteed by Heisenberg's Uncertainty Principle of quantum mechanics, restores our faith in free will.

Scientific Explorations with Paul Doherty

© 2003

5 May 2003