Ikeguchi Laboratories has posted one of the most fantastic "physics in action" videos I've seen in a long time: [embed]http://youtu.be/JWToUATLGzs[/embed] The concept is simple -- 32 metronomes on a table, all set to the same tempo, but started at slightly different times. But here's the fun bit -- although they begin "out of phase", after about 2 minutes, they all lock onto the same phase and synchronize! (Well, almost all -- there's a rebel on the far right that takes an extra minute to get with the program). So what's going on? The key is that the metronomes are not on a solid table, but instead are on a slightly flexible platform hanging from a string. Thus, as a metronome's pendulum rod changes direction, it imparts a small force to the platform, which leads to small motions in the platform. The moving platform then gives small nudges back to the metronomes. These forces will tend to push the other metronomes to speed up or slow down to match the timing of the original metronome, bringing the metronomes "in phase". Now the really fun bit (for me at least), was watching exactly how this played out in practice. If you watch the video closely, you'll see that the synchronization does not happen all at once, nor does it happen randomly. Instead, the synchronization tends to take place first in pairs, with adjacent metronomes locking onto the same phase. This behavior makes a lot of sense, because the strongest forces on a metronome will initially be from its nearest neighbors, at least until enough metronomes are in phase that you start getting a large scale coherent swaying of the platform (which starts to happen about a minute in, becoming increasingly strong during the next minute). The pairs are also more likely to be oriented along the rows (side-by-side), reflecting the direction in which the metronomes cause the platform to move. The other phenomena you can notice is that adjacent pairs will frequently spend quite a bit of time "180 degrees out of phase" (i.e., showing totally opposite behavior from the neighboring pair -- going "tick" exactly when the other goes "tock"). If you watch one of these sets, after happily going along for a while, the equilibrium will shift, and the pairs start to change their tempo. The relative phases will shift, and will gradually drag the pair that's 180 degrees out of phase back in line with the rest of them, before reverting back to the natural frequency of the metronome. This behavior is probably clearest in the "rebel on the right", which is in a quasi-stable equilibrium, and spends an extra minute beating a syncopated tempo. So, lots of interesting stuff to see in a remarkably simple set-up!
(Ed: In comments, Andy Rundquist linked to a post of his analyzing an earlier metronome video, along with bonus Mathematica code, along with a link to a much older publication about modeling the system.)
(Ed: Aaaaand now Paul Gribble (in comments) has just checked relevant Python code into Github. Y'all are nuts. My kind of nuts, but nuts.)