[math-fun] Physicists Discover 13 New Solutions To Three-Body Problem
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/
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/
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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/
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/
_______________________________________________ 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/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
One nice fact is that if the force is 1/r^3, or more generally 1/r^a for a >= 3, then any braid at all corresponds to a valid trajectory of n bodies in the plane: you can get masses 1 and 3 to do-si-do 17 times counterclockwise, then 3 and 4 to twirl 6 times clockwise, and so on. The reason is that the action of any colliding path is infinite, so if you start off with a fictional trajectory that has the topology you want, you can relax it until it is actually a solution of the equations of motion, and it will keep the same topology throughout the relaxation. For 1/r^2 forces, on the other hand, some braids are allowed and others aren't. You can see some of these results here: http://tuvalu.santafe.edu/~moore/braids-prl.pdf - Cris On Mar 9, 2013, at 6:42 PM, 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/
_______________________________________________ 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/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ 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/
Cris's cool orbit movies have moved to: http://tuvalu.santafe.edu/~moore/gallery.html At 08:06 PM 3/9/2013, Cris Moore wrote:
One nice fact is that if the force is 1/r^3, or more generally 1/r^a for a >= 3, then any braid at all corresponds to a valid trajectory of n bodies in the plane: you can get masses 1 and 3 to do-si-do 17 times counterclockwise, then 3 and 4 to twirl 6 times clockwise, and so on.
The reason is that the action of any colliding path is infinite, so if you start off with a fictional trajectory that has the topology you want, you can relax it until it is actually a solution of the equations of motion, and it will keep the same topology throughout the relaxation.
For 1/r^2 forces, on the other hand, some braids are allowed and others aren't. You can see some of these results here: http://tuvalu.santafe.edu/~moore/braids-prl.pdf
- Cris
On Mar 9, 2013, at 6:42 PM, 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/
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/
Cris: If I understand you correctly, you posit the existence of periodic orbits which (therefore) have Fourier series, and then follow these constraints until everything matches ? At 08:06 PM 3/9/2013, Cris Moore wrote:
One nice fact is that if the force is 1/r^3, or more generally 1/r^a for a >= 3, then any braid at all corresponds to a valid trajectory of n bodies in the plane: you can get masses 1 and 3 to do-si-do 17 times counterclockwise, then 3 and 4 to twirl 6 times clockwise, and so on.
The reason is that the action of any colliding path is infinite, so if you start off with a fictional trajectory that has the topology you want, you can relax it until it is actually a solution of the equations of motion, and it will keep the same topology throughout the relaxation.
For 1/r^2 forces, on the other hand, some braids are allowed and others aren't. You can see some of these results here: http://tuvalu.santafe.edu/~moore/braids-prl.pdf
- Cris
In my 1993 paper, I naively discretized the orbit, and adjusted each point until the curvature of the path matches the acceleration it ought to feel. This is equivalent to gradient descent in the action. In the joint paper with Nauenberg, we used the first 10 or 20 Fourier coefficients, and did gradient descent in the action (evaluating the action by numerical integration). Of course, neither of these techniques proves rigorously that an orbit exists. That was done for the figure-8 by Chenciner and Montgomery in 2001, and for the Henon "criss-cross" orbit by Chen. Cris On Mar 10, 2013, at 10:59 AM, Henry Baker wrote:
Cris:
If I understand you correctly, you posit the existence of periodic orbits which (therefore) have Fourier series, and then follow these constraints until everything matches ?
At 08:06 PM 3/9/2013, Cris Moore wrote:
One nice fact is that if the force is 1/r^3, or more generally 1/r^a for a >= 3, then any braid at all corresponds to a valid trajectory of n bodies in the plane: you can get masses 1 and 3 to do-si-do 17 times counterclockwise, then 3 and 4 to twirl 6 times clockwise, and so on.
The reason is that the action of any colliding path is infinite, so if you start off with a fictional trajectory that has the topology you want, you can relax it until it is actually a solution of the equations of motion, and it will keep the same topology throughout the relaxation.
For 1/r^2 forces, on the other hand, some braids are allowed and others aren't. You can see some of these results here: http://tuvalu.santafe.edu/~moore/braids-prl.pdf
- Cris
_______________________________________________ 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 (4)
-
Allan Wechsler -
Cris Moore -
Henry Baker -
Ray Tayek