Talk:Quasiperiodic motion

Unclear

@Charles Matthews: I don't understand what you wrote here:

A quasiperiodic function on the real line is the type of function (continuous, say) obtained from a function on T, by means of a curve 

R → T

which is linear (when lifted from T to its covering Euclidean space), by composition.

Can you explain? Please ping me. Eric Kvaalen (talk) 08:48, 16 August 2024 (UTC)

@Eric Kvaalen: T as a torus is a quotient E/L where E is a Euclidean space and L is a lattice in it. What is meant is that we think about a function from the real line R, that is a linear mapping from R into E, then composed with the quotient map to E/L = T. Geometrically the real line is wrapped around the torus. Unless the image in E goes through lattice points, the wrapping in T does not return to the origin in T. Now we take some (continuous) function F on T, and think of it as a function on R. This is a quasi-periodic function - if the wrapping does not return to the origin, it is not exactly periodic, but since the wrapping comes close to the origin, it nearly assumes the same values again. HTH. Charles Matthews (talk) 09:03, 16 August 2024 (UTC)

@Charles Matthews: Yeah, that helps. Of course, the resulting function can be periodic even if the first function doesn't go through lattice points – I mean, the function y=0.2+x doesn't go through lattice points placed at integral combinations of x and y, yet the function on the torus is periodic.

Now I'm not sure I understand "It is therefore oscillating, with a finite number of underlying frequencies." Are those terms (oscillating and underlying) well-defined terms? Obviously one could have a function such as

$f(t)=g(t,at+b)$

where $g(x,y)$ is a double Fourier series:

$g(x,y)=\Sigma _{j}\Sigma _{k}C_{jk}\sin(2\pi jx)\sin(2\pi ky)$

Then there are an infinite number of frequencies:

$f(t)=\Sigma _{j}\Sigma _{k}C_{jk}\sin(2\pi jt)\sin(2\pi kat+2\pi kb)$

Eric Kvaalen (talk) 11:49, 16 August 2024 (UTC)

@Eric Kvaalen: I have started to rewrite the article, in the hope of clarifying the points you raise. Charles Matthews (talk) 10:06, 17 August 2024 (UTC)

@Charles Matthews: I've had a look at it. I find the mention of one-parameter subgroups confusing. What group are they subgroups of, in this case? Eric Kvaalen (talk) 11:34, 18 August 2024 (UTC)

@Eric Kvaalen: One-parameter subgroup is a standard concept for Lie groups, and the torus T is a Lie group. The previous explanation said just the same thing. Perhaps the point to mention here is that a one-parameter subgroup of G is not strictly a subgroup, because it is a homomorphism from the real line into G. The image in T might be a circle group, or it might wind around the torus. In the latter case the image is dense in some sub-torus. Charles Matthews (talk) 11:50, 18 August 2024 (UTC)

@Charles Matthews: Obviously the torus can be given group structure, but what for? I think you're making the article overly complicated, bringing in notions that are not really necessary or helpful. Now it's compactifying, and direction cosines. We should concentrate on making the article easy to understand, while supplying the pertinent information. Eric Kvaalen (talk)

@Eric Kvaalen: Well, I'm making an effort to develop the article in an encyclopedic way now; having given it about half an hour of my time 19 years ago. There is now a link to linear flow on the torus, which is closer to an orthodox mathematical physics discussion of the dynamics. You might find that easier to read, with explicit differential equations and formulae. I would argue, actually, that I have been developing a kinematics point of view, in that the differential equations giving rise to the motion are really not necessary to understanding how straight lines bend around a torus. This is somewhat a matter of taste: there is some geometry involved, and some number theory in the condition on the frequencies.

The article has had references added, and is 150% longer now, so I don't think I have been wasting my time. I tend to write prose here, rather than rely on formulae: the "direction cosines" paraphrase the treatment in the Encyclopedia of Mathematics. That's coordinate geometry at an elementary level. The fact that a torus is compact is rather fundamental in understanding what "quasiperiodic" means. A hypertext encyclopedia can use hyperlinks as an expository mechanism. Charles Matthews (talk) 04:03, 19 August 2024 (UTC)

@Charles Matthews: I learned about quasiperiodic motion decades ago in connexion with Kolmogorov–Arnold–Moser theory, and recently I was thinking about quasiperiodic motion in relation to the moon, and I looked at our article on KAM theory to refresh my memory. It talks about "quasiperiodic orbits" but didn't say what exactly that means, so I clicked on the link and read what was here last week. It didn't give much explanation, and there was that sentence about quasiperiodic functions (which is actually a different subject) that I didn't understand, and I asked you. Now you have added a lot to the article, but it still doesn't give a nice, clear definition of a quasiperiodic orbit, and if someone comes to this article to find out what they are, he will be confronted with lots of terms and concepts that are not necessary for the understanding of the concept. You could put all that stuff later in the article if you really think it's relevant, but we need something concise but more complete at the beginning. Eric Kvaalen (talk) 17:00, 19 August 2024 (UTC)

@Eric Kvaalen: To be precise then, the KAM article has the wikitext [[Quasiperiodic motion|quasiperiodic]] [[orbit (dynamics)|orbit]]. I could certainly add something about the "quasiperiodic orbit" idea. To understand KAM theory, you certainly need the idea of a torus in phase space, as a manifold; that by good choice of coordinates on it you can think of it as a standard torus like the unit cube with faces identified; and that in the setting the "orbit" is one of the linear flow lines we have been discussing, mapped into the torus. Then the orbit terminology would make more sense. I have also added an image, because the geometrical intuition is more helpful here than formulae. To be honest, while this article clearly needs more work, the editor who put "quasiperiodic orbit" into the KAM article could be asked to unpack that terminology also. (Not going to happen, IP number from 2006, so a talk page message for that article could help.) The book Quasi-Periodic Motions in Families of Dynamical Systems: Order amidst Chaos is now used as a reference in this article, and reference [5] goes to https://books.google.co.uk/books?id=UGdqCQAAQBAJ&pg=PA2, which as you can see uses the Lie group language. To sum up, now that you have explained the actual expository issue that brought you here, much of the early (pre-2007) writing in articles here needs reconsideration, and I have recently been involved with unitarian trick and rigged Hilbert space also. Please don't treat this situation as a blame game. I'd like to make the point that encyclopedia articles are not written in textbook style. I see that orbit (dynamics) talks about curves in phase space, and the point you first brought up is how to explain how the real line is mapped into a torus as a curve. It is quite reasonable and concise to say that (starting the orbit at 0 in the torus) it is a one-parameter subgroup; it is also reasonable but verbose to explain it the way I first did. I would see the problem lying in the use of two adjacent wikilinks. That is anyway against the Manual of Style, and ought to be fixed in the KAM article. Charles Matthews (talk) 05:06, 20 August 2024 (UTC)

@Charles Matthews: By the way, I don't understand the sentence about what David Ruelle says, and you have put a link to Google Books, but it tells me that I can't read that page! What's the point of a link that doesn't work? Same thing for reference #2. Eric Kvaalen (talk) 10:39, 20 August 2024 (UTC)

@Eric Kvaalen: I only add the URL for Google Books links in cases where I can read the page, here in the United Kingdom. What you can read on Google Books may vary from country to country, but I am not able to give you details of that. Charles Matthews (talk) 03:04, 21 August 2024 (UTC)

@Charles Matthews: So what does David Ruelle mean? I don't understand at all. If there is a finite number of frequencies, then why doesn't it make sense to ask what they are? And as I said here back on August 16, there are an infinite number of frequencies. Eric Kvaalen (talk) 10:30, 21 August 2024 (UTC)

@Eric Kvaalen: He is making the point that it doesn't make sense here to talk about a fixed set of fundamental frequencies. That is unlike ordinary Fourier series. The same point is made in Remark 1 of reference [5] where it is said that the frequency vector is determined only up to an nxn invertible matrix tranformation. There is no distinguished basis.

I am always willing to discuss content. But my conclusion here is that quasiperiodic orbit should redirect to orbit (dynamics), and an explanation should appear in that article, using the orbit concept. Charles Matthews (talk) 15:50, 22 August 2024 (UTC)

Thanks. I'm able to read a few of the pages in that book. They use the term "internal frequency". I'm gonna see whether I can improve the article. Eric Kvaalen (talk) 11:04, 23 August 2024 (UTC)