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 are stable localised waves which propagate at a constant speed, and occur as the solutions to certain partial differential equations (e.g. the Sine-Gordon equation). When two solitons collide, they temporarily interact and pass through each other, eventually (in the limit as the elapsed time tends to infinity) recovering as if there were no interaction. Indeed, this could be given as a somewhat convincing answer to the question ‘what happens if an unstoppable force meets an immovable object?’.
It has been proposed that soliton solutions to the Sine-Gordon equation may be used to propagate impulses of information at sub-neuronal scales in cellular structures called microtubules. More information is available in D. D. Georgiev’s paper on the subject. More elaborate soliton interactions are possible, such as this collision between an antikink and a standing breather:
A much more artificial example of this type of reaction is Dave Greene’s Heisenburp device in Conway’s Game of Life, where a glider (moving localisation) collides with a complicated arrangement of machinery, which sends a fast neuronal impulse along a diagonal wire (much more quickly than the glider could have travelled in vacuo) to a receiver arrangement, which subsequently replaces the glider in exactly the same position and phase it would have occupied had the machinery not been present.
As is visible above, the device resembles a myelinated neurone, both in layout and function. Unlike actual neurones, however, this arrangement is incredibly fragile (and can be broken, for example, by sending two gliders in quick succession, or a glider along a different lane). Indeed, configurations in Conway’s Game of Life, at least on small scales, are the antithesis of solitons — the smallest perturbation can cause chaotic devastation.
Solitons also occur in both fundamental particle physics and in fluids, as explained in this video of David Tong having a bath:
These vortex ring solitons also support more interesting interactions. It is possible to project a faster vortex ring through the centre of a slower one, causing them to ‘leapfrog’ over each other. I seem to recall this being the subject of a talk at the Trinity Mathematical Society, but cannot remember the title of the talk. Nevertheless, a quick search found a YouTube video showing a simulation of vortex rings exhibiting this type of behaviour:
Train proof sketch (ROT13’d):
Vs jr erqhpr rnpu genva’f yratgu gb mreb, fb gung vg vf n fvatyr cbvag, juvyr fvzhygnarbhfyl fuevaxvat gur frtzrag bs genpx gung vg bpphcvrf, gur gbgny genpx yratgu fuevaxf gb n engvbany zhygvcyr Y’ bs vgf sbezre fvmr Y, naq gur genvaf orpbzr rynfgvpnyyl-pbyyvqvat cbvagf. Gur frtzragf bs habpphcvrq genpx rkuvovg qlanzvpf gung ner pbzcyrgryl hapunatrq, rkcnaqvat juvyr fgngvbanel, pbagenpgvat juvyr fgngvbanel, be fgnlvat n svkrq fvmr naq zbivat pybpxjvfr be pbhagrepybpxjvfr ng gur genva fcrrq f whfg nf gurl qvq orsber gur fuevax-bcrengvba.
Ohg abj gur genvaf ner cbvagyvxr, naq jr pna ercynpr rynfgvp pbyyvfvbaf jvgu *cnffvat guebhtu rnpu bgure* jvgubhg punatvat gur qlanzvpf bs gur *frtzragf bs habpphcvrq genpx*. Jvgu gung, gur cbvagf zhfg arprffnevyl erghea gb gurve vavgvny cbfvgvbaf nsgre n gvzr g rdhny gb gur (fuehaxra) genpx yratgu qvivqrq ol gur genva fcrrq: g = Y’/f.
Ng gur raq bs bar bs gurfr plpyrf, gur fuehaxra genvaf ner onpx *frgjvfr* ng gurve bevtvany cbfvgvbaf, ohg gur ercynprzrag bs rynfgvp pbyyvfvbaf jvgu cnff-guebhtu orunivbe jvyy unir vaqhprq n cbffvoyl abagevivny crezhgngvba C ba gurz. Ohg nsgre fbzr vagrtre senpgvba bs a! plpyrf, jurer a vf gur ahzore bs genvaf, gur genvaf ner onpx *cbvagjvfr* ng gurve bevtvany cbfvgvbaf, nf gur beqre bs gur crezhgngvba C nf n zrzore bs gur flzzrgevp tebhc F_a zhfg qvivqr a!.
Va cnegvphyne, ng gvzr a!Y’/f, gur *bevtvany*, hafuehaxra flfgrz jvyy unir nyfb erghearq gb vgf vavgvny pbasvthengvba *eryngvir gb fbzr cnegvphyne genva raqcbvag*. Ng gur fnzr gvzr, gur flfgrz nf n jubyr unf n arg irybpvgl nebhaq gur genpx. Gur ahzore bs genvaf geniryvat pybpxjvfr vf vainevnag: rirel pbyyvfvba syvcf bar pybpxjvfr genva gb pbhagrepybpxjvfr naq bar pbhagrepybpxjvfr genva gb pybpxjvfr. (Be ybbx ng gur fuehaxra, cnff-guebhtu cbvag-genvaf, juvpu arire punatr qverpgvba, gb frr gung gur ahzore geniryvat rnpu jnl vf pbafgnag). Pbafrdhragyl, gur flfgrz nf n jubyr unf n arg irybpvgl gung vf rffragvnyyl pbafgnag. Gung irybpvgl vf gur ahzore bs pybpxjvfr genvaf zvahf gur ahzore bs pbhagrepybpxjvfr genvaf qvivqrq ol a naq gvzrf f pybpxjvfr, fb vf n engvbany senpgvba bs f, fnl nf/o. Gur flfgrz nf n jubyr pna gurersber bayl ercrng nsgre crevbqf bs gvzr gung ner zhygvcyrf bs oY/nf, naq *jvyy* ercrng nsgre nal crevbq bs gvzr gung vf na vagrtre zhygvcyr bs obgu a!Y’/f naq oY/nf, va juvpu gurer’f n pbzzba snpgbe bs 1/f naq nyy bs gur bgure dhnagvgvrf ohg sbe Y’ naq Y ner vagrtref. Gur arprffnel naq fhssvpvrag pevgrevba sbe guvf vf guhf gung Y’ naq Y unir na vagrtre pbzzba zhygvcyr, juvpu vf gehr tvira gur ulcbgurfvf gung gur fhz bs gur genva yratguf, Y – Y’, vf n engvbany senpgvba bs Y.
Well done. That is precisely the solution I had in mind, although you can replace n! with n since only cyclic permutations are possible.
Ah of course, since they won’t ever get shuffled around.
apgoucher provided an animated GIF which loops. Therefore, it is periodic.
That only exhibits one example with periodic behavior, insufficient to prove the general case.
I’m pretty sure this was simply a joke-proof 😛
So am I. 🙂
I think that the TMS talk you’re thinking of was Dr. David Acheson’s “What’s the Problem with Maths?” in which he described, along with lots of other things, his research on vortex leapfrogging ( http://iopscience.iop.org/0143-0807/21/3/310 ).
Also, skip to 3:45 in this video http://www.youtube.com/watch?v=mHyTOcfF99o for vortex leapfrogging actually happening in real life.
Thank you! Yes, that was indeed the TMS talk of which I was thinking.
Reblogged this on nebusresearch and commented:
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? 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.
But the converse is not true… A system that is periodic does not have to have the sum of the lengths of the trains a rational multiple of the track length
Yes, that is correct, with the obvious counter-example being a single train on a track of incommensurate length.
The most trivial cases are zero, one or two trains, when the rationality does not matter… but there are also other examples with more trains where the number of trains is even. I think for odd number of trains there is no periodic system with irrational ratio.
oh I take it back… there are also systems with odd trains and irrational ratio that are periodic of course
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