Re: [math-fun] Physicists Discover 13 New Solutions To Three-Body Problem
The usual 2-body elliptical orbits are extremely robust, in the sense that any small (or even large) perturbations of any of the parameters (mass, energy, angular momentum, etc.) only make yet another ellipse with slightly different properties. Are there any 3-body configurations that show anything like this type of stability, other than a hierarchy like Sun, Earth, Moon ? The problem with many of these solutions is that the masses of the 3 objects must be identical. Even microscopic perturbations of the mass will screw up these solutions. This means that it is essentially impossible to find one of these solutions "in the wild". A truly stable solution will have to be able to tolerate deviations from mass (e.g., the impact of a meteor), deviations in energy (loss from tidal forces), deviations in angular momentum (more meteors or slight tugs from further away bodies), etc. Given the low cost of computation today, it might be simpler to do a brute force search with thousands of computers and millions of different initial configurations. If there are any somewhat stable configurations, surely one would pop up under such a brute force search. At 05:42 PM 3/9/2013, Allan Wechsler wrote:
I don't know whether I actually posted my conjecture a few years ago when the "hey-for-three" solution was discovered. I'm not sure how to rigorize the conjecture, but I am convinced that there is an at-least-countably-infinite family of such stable (using the word in the usual sense, pace Warren) configurations.
Basically, if you start off three bodies in a random way, then the system might spend some energy to eject one of the three, but if it doesn't, the system will eventually come very close to one of its previous configurations; if it does, then you can construct a stable solution by tiny perturbations.
Most of these stable modes are extremely shallow potential wells.
On Sat, Mar 9, 2013 at 5:25 PM, Cris Moore <moore@santafe.edu> wrote:
You may enjoy these orbits as well: http://arxiv.org/pdf/math/0511219v2.pdf
Cris
On Mar 9, 2013, at 1:23 PM, Ray Tayek wrote:
http://science.slashdot.org/story/13/03/09/1846253/physicists-discover-13-ne...
Posted by <http://unknownlamer.org/>Unknown Lamer on Saturday March 09,
@02:33PM
from the mystical-spheres dept. sciencehabit writes "It's the sort of abstract puzzle that keeps a scientist awake at night: Can you predict how three objects will orbit each other in a repeating pattern? In the 300 years since this '< http://en.wikipedia.org/wiki/Three-body_problem>three-body problem' was first recognized, just three families of solutions have been found. Now, < http://news.sciencemag.org/sciencenow/2013/03/physicists-discover-a-whopping.html>two physicists have discovered 13 new families. It's quite a feat in mathematical physics, and it could conceivably help astrophysicists understand new planetary systems." The <http://arxiv.org/abs/1303.0181>paper is available at arxiv.
--- co-chair http://ocjug.org/
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/
The Trojan configuration (3 bodies, equilateral triangle, some limits on mass ratios) is reasonably stable. Possibly also the banana variation, where the smallest body wanders in a banana shaped region around the middle body. There are several instances in the solar system of two lesser bodies orbiting a third, where the two bodies have a simple period ratio. I'm not sure if this qualifies as stable, since it may need some lossy tidal mechanism to persist. Rich --- Quoting Henry Baker <hbaker1@pipeline.com>:
The usual 2-body elliptical orbits are extremely robust, in the sense that any small (or even large) perturbations of any of the parameters (mass, energy, angular momentum, etc.) only make yet another ellipse with slightly different properties.
Are there any 3-body configurations that show anything like this type of stability, other than a hierarchy like Sun, Earth, Moon ?
The problem with many of these solutions is that the masses of the 3 objects must be identical. Even microscopic perturbations of the mass will screw up these solutions. This means that it is essentially impossible to find one of these solutions "in the wild".
A truly stable solution will have to be able to tolerate deviations from mass (e.g., the impact of a meteor), deviations in energy (loss from tidal forces), deviations in angular momentum (more meteors or slight tugs from further away bodies), etc.
Given the low cost of computation today, it might be simpler to do a brute force search with thousands of computers and millions of different initial configurations.
If there are any somewhat stable configurations, surely one would pop up under such a brute force search.
At 05:42 PM 3/9/2013, Allan Wechsler wrote:
I don't know whether I actually posted my conjecture a few years ago when the "hey-for-three" solution was discovered. I'm not sure how to rigorize the conjecture, but I am convinced that there is an at-least-countably-infinite family of such stable (using the word in the usual sense, pace Warren) configurations.
Basically, if you start off three bodies in a random way, then the system might spend some energy to eject one of the three, but if it doesn't, the system will eventually come very close to one of its previous configurations; if it does, then you can construct a stable solution by tiny perturbations.
Most of these stable modes are extremely shallow potential wells.
On Sat, Mar 9, 2013 at 5:25 PM, Cris Moore <moore@santafe.edu> wrote:
You may enjoy these orbits as well: http://arxiv.org/pdf/math/0511219v2.pdf
Cris
On Mar 9, 2013, at 1:23 PM, Ray Tayek wrote:
http://science.slashdot.org/story/13/03/09/1846253/physicists-discover-13-ne...
Posted by <http://unknownlamer.org/>Unknown Lamer on Saturday March 09,
@02:33PM
from the mystical-spheres dept. sciencehabit writes "It's the sort of abstract puzzle that keeps a scientist awake at night: Can you predict how three objects will orbit each other in a repeating pattern? In the 300 years since this '< http://en.wikipedia.org/wiki/Three-body_problem>three-body problem' was first recognized, just three families of solutions have been found. Now, < http://news.sciencemag.org/sciencenow/2013/03/physicists-discover-a-whopping.html>two physicists have discovered 13 new families. It's quite a feat in mathematical physics, and it could conceivably help astrophysicists understand new planetary systems." The <http://arxiv.org/abs/1303.0181>paper is available at arxiv.
--- co-chair http://ocjug.org/
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/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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 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)
-
Cris Moore -
Henry Baker -
rcs＠xmission.com