Every once in a while it’s a good idea to remember that even the simplest-looking physical systems can have completely bonkers behavior. The pendulum is certainly one of those systems. It’s so simple that it’s a mainstay of freshman classes, for technical and non-technical majors alike, though even then we do have make an approximation that’s only valid for relatively small angles of the swing.
But the equation of motion that a pendulum obeys is pretty simple. String a few pendulums together and the equation of motion is still not too bad – all you have to do is a little somewhat tedious work in figuring out the expressions for the energy of each swinging arm and you’ve got the equation of motion. If you actually try to solve the equation of motion and calculate the paths that the pendulums will take, you’ll find that you can’t. Not in anything resembling a clean closed-form equation. Why? Just look:
I think that’s pretty amazing. Impossibly complicated chaotic behavior in the simple connection of three trivially solvable systems. It’s a nice reminder that even if physics does discover a Theory of Everything, that theory would not in and of itself close the book on the discipline. After all, the Lagrange equations of motion are a complete “theory of everything” as far as this particular system is concerned – but developing the consequences and implications of those laws is another story. The laws of nature have a richness that can’t be fully explored simply by knowing the basic statements of the laws.