Chaos Theory and the Pendulum

Date: 12/31/2001 at 04:50:45
From: Glenn Polii
Subject: Chaos theory

Greetings!

I'm in the last grade in high school. I've read your explanations 
about chaos theory. I've been thinking of such things for a few years, 
since before I knew that other people have been thinking about such 
things too and call them "chaos theory." How is chaos theory related 
to physics?

Date: 12/31/2001 at 09:23:08
From: Doctor Tom
Subject: Re: Chaos theory

Hi Glenn,

This is a big site, and I'm not exactly sure what article you read 
here about chaos theory, but I can surely give you an example from 
physics. (It's actually a math problem, but the math is used to solve 
a physics problem.)

A single pendulum is a rod hanging from a pivot that can swing back 
and forth. A double pendulum has one pendulum swinging from the tip of 
another pendulum. A triple pendulum hangs a third pendulum from the 
tip of the second, etc.

It is quite easy to write down the equations of motion for any such 
system - single, double, triple, ..., pendulum systems.

Both the single and double pendulum problems are well-behaved, but the 
triple (and quadruple, ...) pendulum problems are chaotic.

Even the single pendulum problem cannot be solved analytically. (The
usual "exact" solution you see is not exact - it assumes that the
motion is very small and that for small angles of motion sin(x) = x.)

But all the systems can be solved numerically. In other words, you can 
write down the differential equations of motions of the rods in the 
pendulums and take tiny time-steps with a computer problem to figure 
out what the system will look like 1/100 (or 1/1000 or 1/10000) of a 
second later. So if you know the initial configuration, and want to 
see what it looks like 10 seconds later, you can get a good idea by 
taking 1000 steps of 1/100 seconds. If that's not accurate enough for 
you, you can do 10000 steps of 1/1000 seconds, etc.

Suppose you try to model a real system of pendulums. You need to
measure the rods, weigh the rods, etc., and there will be some tiny 
error in measurement. Thus your initial conditions are not exact; they 
are only as accurate as your measuring devices. Suppose you've got 
really good measuring devices, and make an error of only 1/1000000 
grams in mass or 1/1000000 centimeters in length.

Obviously, the values your computer program will calculate are in 
error if you use your measured values instead of the calculated 
values. (I will assume that the computer is infinitely accurate - that 
there are no numerical errors in its calculations. We only assume that 
there is a measurement error made in the initial numbers and that the 
computer calculates perfectly, but beginning with imperfect numbers.)

In the single and double pendulum case, imagine that you start two
computers calculating with the "true" measurements and with your
measurements that are in error. In the non-chaotic case, if you 
compare the solutions that the two programs give, they will stay close 
together. In other words, a small error at the start will result in a 
small error after a long calculation.

But with a triple pendulum (or any chaotic system of equations), after 
a short time, the predicted positions of the pendulum parts will be 
completely different. How different depends on the accuracy of your 
measurements, but it is not too long.

Suppose that the calculation goes wildly bad after 100 seconds, so you 
think, "Maybe I can get better results with more accurate initial 
measurements, or by making more time-steps between positions." So you 
measure everything 100 times as accurately, and you make your time 
steps 100 times smaller. The two calculations will agree for a longer 
time, but perhaps now it will be 110 seconds before the system 
calculations are wildly different. So for 100 times more work in 
measurement, you only get a 10% improvement in accuracy.

This continues, and since it is really impossible to measure things
to, say, one part in a million, any attempts to simulate the motion
of a triple pendulum for a long time are doomed to failure.

That's the basic idea of a chaotic system - it's usually an iterated 
system (where each calculation depends on the previous one), and where 
a tiny difference in the initial conditions leads relatively rapidly 
to wildly different results in the calculations. In other words, even 
with perfect calculations and a huge amount of computer time and more 
accurate measurements, you can't predict what the system will look 
like after a relatively small amount of time.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/