Re: [math-fun] Physicists Discover 13 New Solutions To Three-Body Problem
CMoore>The figure-8 orbit is _slightly_ stable, in the sense that small perturbations of the initial positions and momenta, or small changes in the masses away from them being identical, result in smooth oscillations, phase changes, or rotations rather than escape. The same seems to be true of the "criss-cross" orbit, which was discovered by Henon; see some numerical evidence of this in http://arxiv.org/pdf/math/0511219.pdf . Still, you would have to be incredibly lucky to be in the basin of attraction of either orbit. Cris On Mar 9, 2013, at 9:57 PM, Henry Baker wrote: Are there any 3-body configurations that show anything like this type of stability, other than a hierarchy like Sun, Earth, Moon ? Cristopher Moore Professor, Santa Fe Institute ------ Since there's a (small) continuum of possible perturbations, is there a continuum of possible oscillation periods, mostly incommensurable with the main orbits? More generally, are there concrete examples of "immortal", aperiodic three body systems with comparable masses that never eject one? --rwg
This is a nice question: Richard Montgomery or Alain Chenciner might know the answer. Certainly there are 3-body orbits with different masses that are locally stable. But since the system is Hamiltonian, that just means that the eigenvalues of the Jacobian are on the unit circle, so that perturbations don't grow to first order. When all the nonlinearities are taken into account, I don't know whether there is an open set (or even a dense set) around these orbits that stay in that region forever. Cris On Mar 10, 2013, at 3:02 PM, Bill Gosper wrote:
Since there's a (small) continuum of possible perturbations, is there a continuum of possible oscillation periods, mostly incommensurable with the main orbits? More generally, are there concrete examples of "immortal", aperiodic three body systems with comparable masses that never eject one? --rwg
Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
Let's just look at your Figure Eight. (Have you succeeded in publishing a paper where it is Fig. 8?) Are those perturbations which don't eject somebody periodic? Do all three bodies follow the same (slightly bent?) orbit, or is the orbit actually blurred? Does it tolerate small nonplanar perturbations? Also, can you get enough decimal places of the aspect ratio of the unperturbed orbit to try an inverse symbolic calculator? Neil B. showed me a picture captioned with a claim it was a quasiperiodic three-body solution, but I can't reconstruct its url. But here's a paper, QUASI-PERIODIC SOLUTIONS OF THE PLANE THREE-BODY PROBLEM NEAR EULER'S ORBITS<http://link.springer.com/article/10.1007%2FBF01230666?LI=true#page-1> that seems to affirm aperiodic immortality, which is what I was after. The idea that such a simple, bounded system can produce endless novelty is a little hard to swallow. --rwg CM> This is a nice question: Richard Montgomery or Alain Chenciner might know the answer. Certainly there are 3-body orbits with different masses that are locally stable. But since the system is Hamiltonian, that just means that the eigenvalues of the Jacobian are on the unit circle, so that perturbations don't grow to first order. When all the nonlinearities are taken into account, I don't know whether there is an open set (or even a dense set) around these orbits that stay in that region forever. Cris On Mar 10, 2013, at 3:02 PM, Bill Gosper wrote: Since there's a (small) continuum of possible perturbations, is there a continuum of possible oscillation periods, mostly incommensurable with the main orbits? More generally, are there concrete examples of "immortal", aperiodic three body systems with comparable masses that never eject one? --rwg Cristopher Moore Professor, Santa Fe Institute
Let's just look at your Figure Eight. (Have you succeeded in publishing a paper where it is Fig. 8?) Are those perturbations which don't eject somebody periodic? Do all three bodies follow the same (slightly bent?) orbit, or is the orbit actually blurred?
If you change the masses, initial positions, and momenta slightly, you still get an approximate figure-8, sometimes with an overall precession. We think this can last forever, although I think we only know that to first order. A larger perturbation causes this precession to become more irregular, until a large enough perturbation makes the whole thing fall apart.
Does it tolerate small nonplanar perturbations?
Yes, those too.
Also, can you get enough decimal places of the aspect ratio of the unperturbed orbit to try an inverse symbolic calculator?
Good question. My own method only gets a few digits (and the fact that using the resulting initial conditions stays in an 8 for many revolutions is good evidence of stability in itself). But Carles Simo has much more precise results, with 10-20 digits of accuracy.
Neil B. showed me a picture captioned with a claim it was a quasiperiodic three-body solution, but I can't reconstruct its url. But here's a paper, QUASI-PERIODIC SOLUTIONS OF THE PLANE THREE-BODY PROBLEM NEAR EULER'S ORBITS<http://link.springer.com/article/10.1007%2FBF01230666?LI=true#page-1> that seems to affirm aperiodic immortality, which is what I was after.
Thanks for finding that paper! It mentions that Arnol'd proved the existence of some quasiperiodic orbits, presumably with the theory of KAM tori.
The idea that such a simple, bounded system can produce endless novelty is a little hard to swallow.
Well, it's highly chaotic, so as Henry says it pulls information up from the low-order bits. And chaotic systems often have very rich spectra of periodic orbits; this is true even for the logistic map in one dimension. Cris
Does the approximate figure-8 behave like the original one (up to homeomorphism of a neighborhood of its orbit in phase space)? If so, can it be that it belongs to some kind of hyperbolic set with at least local structural stability? --Dan On 2013-03-15, at 5:41 PM, Cris Moore wrote:
If you change the masses, initial positions, and momenta slightly, you still get an approximate figure-8, sometimes with an overall precession. We think this can last forever, although I think we only know that to first order. A larger perturbation causes this precession to become more irregular, until a large enough perturbation makes the whole thing fall apart.
On Mar 15, 2013, at 10:42 PM, Dan Asimov wrote:
Does the approximate figure-8 behave like the original one (up to homeomorphism of a neighborhood of its orbit in phase space)?
I guess a homeomorphism here means that the topology remains the same... that is, if you take (R^2)^3 and remove the points where two masses collide, the resulting space has a fundamental group, and a periodic orbit is an element in that group. (By removing a few degrees of freedom this space looks like a sphere with three holes.) When you perturb it, though, there is often an overall rotation, so you have to go into a rotating frame to make the topology the same again. (It's the difference between a 3-straid braid, and a 3-strand braid with some overall twists to it.)
If so, can it be that it belongs to some kind of hyperbolic set with at least local structural stability?
I _think_ so. But like I said, I don't thing anything is known rigorously beyond the first-order stability. Cris
--Dan
On 2013-03-15, at 5:41 PM, Cris Moore wrote:
If you change the masses, initial positions, and momenta slightly, you still get an approximate figure-8, sometimes with an overall precession. We think this can last forever, although I think we only know that to first order. A larger perturbation causes this precession to become more irregular, until a large enough perturbation makes the whole thing fall apart.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
participants (3)
-
Bill Gosper -
Cris Moore -
Dan Asimov