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April 20[edit]

Xia's five-body configuration[edit]

One of my favorite webcomics, Saturday Morning Breakfast Cereal, once made a joke about Xia's five-body configuration (comic here). I went looking for what it was talking about and found our article on the Painlevé conjecture, to which I added a redirect-with-possibilities. As an aside, the article could use some TLC; the most glaring problem is that it introduces variables without saying what they represent.

Anyway I was trying to figure out where this infinite energy was supposed to be coming from. My best guess so far is that the five bodies are idealized as point masses, which means that the gravitational energy released as you let two of them approach one another grows without bound. This of course would make them black holes in our actual universe, so that energy wouldn't be available, but in the universe of the comic, I guess this wouldn't be a problem (I've never thought very deeply about what happens to general relativity as c approaches ∞, so I'm not sure about that). But in any case there's no new energy appearing that wasn't there before, so the comic's claim that "the universe collapses" seems wrong.

No, really, I actually do have a sense of humor. I just want to know if I've understood this correctly. Does anyone have a different understanding? --Trovatore (talk) 19:50, 20 April 2024 (UTC)[reply]

O/T That article rather overeggs the pudding. The various proofs are for the 2d case ie planar. Perhaps it is obvious that if it works in 2d it'll work in 3d. Greglocock (talk) 21:22, 22 April 2024 (UTC)[reply]

I'm not too sure what it means to "overegg the pudding" but I was not interpreting the configuration as planar; if that's correct then I've misunderstood the drawing. Can you point me to why you say that? --Trovatore (talk) 21:43, 22 April 2024 (UTC)[reply]

It means exaggerating the utility of . The repeated use of the word planar is what I was getting at.Greglocock (talk) 05:45, 23 April 2024 (UTC)[reply]

The last section of the AMS article by Saari and Xia has this: "While we now know that noncollision singularities exist, several mysteries remain. Any partial listing has to include whether n = 5 is the cut-off for this surprising behavior, or whether the four-body problem can propel particles to infinity in a finite time. Can, for instance, Anosov’s suggestion be carried out? Are there planar examples with small n values?" This implies IMO that Xia's construction is non-planar. I think the sketch of the construction also implies this: the orbits of the pair of point masses m₁ and m₂ are said to be parallel to the x-y plane and highly elliptical, while m₃ moves along the z-axis. The orbits of the pair m₄ and m₅ are also orthogonal to the z-axis, with their major axes shown at an angle to those of m₁ and m₂ in the accompanying figure. --Lambiam 12:52, 23 April 2024 (UTC)[reply]

Energy borrowed from the potential energy shed by point masses approaching each other closely but not, in the finite time, "arbitrarily closely", can only be finite. To reach infinity their distance has to become less than any positive number, which means it is zero. Doesn't that qualify as a collision? What is worse, in Jinxin Xue's 4-body solution all four bodies are said to escape to infinity in a finite time. Do they scoot off in four different directions? --Lambiam 13:18, 23 April 2024 (UTC)[reply]

I believe the idea is this: as you approach the time of the singularity, the separation between some pairs of masses approaches zero, but the positions of those masses diverge to infinity. If you consider a collision to be when two masses have identical, finite coordinates, then that never happens in this scenario. --Amble (talk) 22:59, 23 April 2024 (UTC)[reply]

How hard would it be to construct it from satelitees? Zarnivop (talk) 22:41, 25 April 2024 (UTC)[reply]

You could put satellites in the right pattern at some starting time, but they wouldn’t actually behave as predicted by Xia’s model. That’s because the model requires perfect point masses in a Newtonian universe affected only by one another’s gravitational pull. Real satellites have other forces acting on them, have negligible influence from each other’s gravity, and are not point masses. And of course, real satellites aren’t in a Newtonian universe. —Amble (talk) 02:25, 26 April 2024 (UTC)[reply]