## Reblog: Solitons and trains

Over at the Complex Projective 4-Space blog is a neat little problem: suppose you have a circular train track, and a couple trains of different length which roll at different speeds on the track, and interact by bouncing off one another and going the other way. Are their positions ever-changing, or do they, in time, come back to the way they were arranged when you first set them down, which is a kind of the recurrence problem mentioned in my bits about Arthur Christmas. The author, apgoucher, goes on to talk about vortex rings and solitons, and the ways they can interact. I think it’s worth some attention.

Suppose we have a circular track occupied by finitely many trains of various lengths travelling at the same speed. The trains collide elastically with each other. If the sum of the lengths of the trains is a rational multiple of the track length, then it can be proved that the trains will eventually return to their original configuration. Here’s an animated GIF I prepared earlier:

It’s actually quite fun to prove that the system is periodic (there is a very short elementary proof, but you have to think outside the box). I’ll leave this as an exercise to the reader, and possibly give a solution in about a week or so, when you’ve had sufficient time to think about it. If you find a proof, include it in the ‘comments’ section at the foot of this page.

Now for something completely different and not at all related: solitons. Solitons…

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