Saturday, September 23, 2006

A Famous Instability

Take a hardcover book that you don't mind dropping on the floor and put a couple of rubber bands around it to keep it closed. Observe that there are three axes of symmetry of the book: longest, shortest, and intermediate. Toss the book in the air so that it spins about the shortest axis. That's the axis that passes from the front of the book to the back (unless you have picked a very strangely shaped book). Try to catch it, too, but notice that the book starts off spinning around this axis, and continues spinning around this axis. This is a stable spinning motion. Now toss the book in the air so that it spins around the longest axis, most probably the axis from the top of the book to the bottom. Stable. Now, one more time, spin about the middle axis. Whoops! It won't stay spinning about that axis, but tumbles all over the place. UNSTABLE about the middle axis.

Somewhere in my vast library I have a theorem about this, which I'll try to resurrect for another blog. But this time I just want to point out that I've reproduced this behavior in my little Mathematica laboratory of rigid-body motion. I'm not sure how to publish a Flash movie on this blog, or, in fact, anywhere else on the web, but I do have one. Here's a snapshot:

I've got three versions of this movie, one showing stable spinning about the x (red) axis, another showing stable spinning about the y (green) axis, and the third showing UNstable spinning about the z (blue) axis. All three of them use the same 4th-order Runge-Kutta integrator of the rigid-body equations of motion for free spinning, which I coded up by hand in Mathematica. I'll be explaining this in future installments.


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