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## Physicists Discover 13 New Solutions To Three-Body Problem127

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 'three-body problem' was first recognized, just three families of solutions have been found. Now, 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 paper is available at arxiv.
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## Physicists Discover 13 New Solutions To Three-Body Problem

• #### Never thought it would be so hard to have a 3some (Score:5, Funny)

on Saturday March 09, 2013 @02:42PM (#43127045)
Though I'll admit it's entirely theoretical for me so far.
• #### Re: (Score:3, Funny)

by Anonymous Coward

You obviously have funding issues for your research. Adequate funding will resolve this research deficiency.

• #### Re: (Score:2)

Well There's the:
Cool Threesome
Uncool Threesome
Wait, is my girlfriend/boyfriend gay
Wait, is my girlfriend/boyfriend straight
Is he only asking for this threesome because he no longer likes me and wants to get it on with that girl from work/the gym?
We just woke up and he/she was there
The Siamese (shudder)
No, I'm just here to watch (we saw how that went in basic instinct)

And so on. That's just arranging a suitable partner, we haven't even touched on the physicality of it. I think I'll have to do some research

• #### having said that (Score:3)

on Saturday March 09, 2013 @02:45PM (#43127055)
would anyone care to explain how much accurate are the numerical analysis/numerical integration solutions ? ( which also apply to n-body problem, specific part of which is the 3 body problem ). Does the accuracy depend on how small is the dt we chose between each calculation ?
• #### Re:having said that (Score:4, Informative)

by Anonymous Coward on Saturday March 09, 2013 @02:51PM (#43127095)

would anyone care to explain how much accurate are the numerical analysis/numerical integration solutions ?

They are as accurate as you care to make them. The problem is that increased accuracy over a long period can
require an exponential increase in cost.

Does the accuracy depend on how small is the dt we chose between each calculation ?

Precisely.

• #### Re:having said that (Score:5, Informative)

on Saturday March 09, 2013 @02:55PM (#43127117)
so that actually means that for any dt, however small it is, given enough simulation time, there is a time point in the future after which the simulation is completely wrong ( for various values of "completely" )
• #### Re: (Score:2)

Depends on what you mean with completely wrong. There is a class of numerical algorithms called Symplectic Integrators, which make sure that energy is conserved. You can also choose algorithms with an adaptive stepsize, which means that the simulation should converge to within a given error tolerance. (The simulation can still suffer from e.g. accumulated floating point errors, though.)

The classic example of a simulation gone completely wrong, is the Flying ice cube [wikipedia.org] problem...
• #### Re: (Score:3)

would anyone care to explain how much accurate are the numerical analysis/numerical integration solutions ?

They are as accurate as you care to make them. The problem is that increased accuracy over a long period can require an exponential increase in cost.

Does the accuracy depend on how small is the dt we chose between each calculation ?

Precisely.

Well, for the same solver, it does. But the relative (and absolute) improvement realized by changing dt is quite dependent on what solver scheme you are using.

• #### Re: (Score:2)

what troubles me is the impossibility of a theoretical solution because it undermines my belief in a deterministic universe. There _must_ be some theoretical solution which can 100% accurately predict any future status of the n body problem!
• #### Re: (Score:1)

by Anonymous Coward

Your inability to predict something doesn't mean it isn't deterministic.

• #### Re: (Score:2)

IF the universe IS deterministic there should be a THEORETICAL solution to predict any future state of the universe provided that you a) know an initial state and b) know all the laws of the universe.
• #### Re: (Score:2)

Quantum mechanics rules out a deterministic universe from our point of view.
• #### Re: (Score:2)

i know, but quantum mechanics is not necessarily how the universe functions, it's just a probabilistic approximation just like the laws of thermodynamics. p.s. i think the universe cannot be non deterministic unless you believe in flying spaghetti monsters of pink unicorns
• #### Re: (Score:1)

p.s. i think the universe cannot be non deterministic unless you believe in flying spaghetti monsters of pink unicorns

The difference here being that unicorns are compatible with our understanding of the universe, and are thus a more reasonable thing to believe in based on the evidence than say, a flat earth, Biblical creation, or a deterministic universe. The latter three fly in the face of science, instead of simply being unsupported by evidence ala unicorns and spaghetti monsters.

• #### Re: (Score:2)

True, and I want to point out that flying spaghetti is totally consistent with string theory and a fully stochastic (and messy) universe.
• #### Re:having said that (Score:5, Insightful)

on Saturday March 09, 2013 @04:50PM (#43127719)

You misunderstand the laws of thermodynamics. They apply also at the quantum level, and deal mostly about the energy cost of transferring a bit of information. The trick being that the bit may or may not decay with some probability which depends on how much energy you put into preserving it. Where a "bit" is for example the excitation level of an electron.

The universe is truly nondeterministic. It really is a hugely complicated probability density function :)

• #### Re: (Score:2)

i tend to disagree. I don't think that subatomic particles have their own mind. every particle moves under the influence of all the known forces ( 5 - or maybe more ) of all subatomic particles in the universe and its position and momentum is certain despite the fact that it's impossible ( computationally ) for us to determine it. The laws for gases ( pressure etc. ) and quantum mechanics are just stochastic simplifications of the actual movements of the molecules ( or subatomic particles in the case of qua
• #### Re: (Score:3)

The universe is truly nondeterministic. It really is a hugely complicated probability density function :)

This is just an artifact of compression/optimization functions used to run the emulation.

• #### Re:having said that (Score:4, Informative)

<asmunder@@@stud...ntnu...no> on Saturday March 09, 2013 @06:09PM (#43128067)

It's not that a particle has a theoretical probability of being somewhere with some probable momentum, no it will be at a very real place at a very real time with a very actual momentum. It's just that practically it's so complicated to predict it, that the best way we have come up till now are quantum mechanics .

Nope, you're wrong. Here are the experimental [aps.org] evidence [aapt.org] which falsify [sciencemag.org] your hypothesis. Bonus: Zombie Feynman [xkcd.com].

• #### Re: (Score:3)

However weird the current accepted model is, and incompatible to what you want to believe in, if you really want to pursue science as a career or even as a hobby you need to understand that wanting things to be some way or feeling they should be some way are both hindrances to any scientist.

Science is the search for truth through logic and experiment, it accomplishes its goal mostly by ruling out the inconsistencies. Nobody can claim that the current statistical model is 100% correct, but what can be cla
• #### Re: (Score:2)

Well, not with absolute certainty. There's still the superdeterminism loophole. It's just that this is even weirder and less satisfying to many people than just dropping determinism, especially since, philosophically, it suggests that science is meaningless and anything we discover through the scientific method is coincidence that could change tomorrow, because literally every experimental result you've ever had is a part of a vast conspiracy of all the particles in the Universe.

• #### Re: (Score:2)

To say that superdeterminism is extremely implausible is to understate it. Furthermore as you said if that was so Science would be a pointless endeavor, and therefore by ignoring the possibility we don't really lose anything. We can also ignore the Flying Spaghetti Monster theory with similar results.
• #### Re: (Score:1)

Except that it was proven impossible for a local hidden variable theory (which is what you are suggesting) to reproduce the results of quantum mechanics. This result is called Bell's theorem [wikipedia.org]. This means that either the universe is non-deterministic or it is not completely local (i.e. there are effects which cannot be attributed to a local force). Either that or counterfactual definiteness [wikipedia.org] does not hold, which would essentially mean that the result of any experiment and the choice of measurement the experime

• #### Re: (Score:1)

Except that it has been proven impossible for a local hidden variable theory (which is what you are suggesting) to be able to replicate all of the results of quantum mechanics. This result is called Bell's theorem [wikipedia.org]. This essentially means that either the universe is non-deterministic or it is not completely local (i.e. there are effects not caused by local forces). Either that or counterfactual definiteness [wikipedia.org] does not hold (since Bell's theorem relies on it) due to the results of any experiment and the choice

• #### Re: (Score:2)

If you believe in free will, you have to admit the possibility that the Universe isn't deterministic. It might not be possible to prove that any posited set of laws is ultimate, so that question might remain unsettled for a long time.
• #### Re: (Score:3)

The key there is 'from our viewpoint'

The Bell results only show that there are no hidden local variables. Non-local variables could never be proved to be impossible.
For all we know all quantum events are determined by a single 128-bit LFSR.
• #### Re: (Score:1)

Mm, but Bells result only show that there are no hidden local variables assuming function can be written in form blahttp://en.wikipedia.org/wiki/Local_hidden_variable_theory#Local_hidden_variables_and_the_Bell_tests

So strictly speaking it doesn't prove that no hidden local variables theory is possible. (even though hidden local vars functions that could be chosen to match quantum results do not "feel natural")

• #### Re: (Score:3)

The clockwork metaphor of the universe fell apart about 100yrs ago. The universe is random at a fundamental level but even if it were deterministic one of the laws in your point (b) is that most systems in nature are mathematically chaotic [wikipedia.org], no mater how well you measure the starting conditions it can NEVER be accurate enough to reliably predict the behavior of the system past a certain point in the future.

The thing I find "odd" is that often (always?) the statistics of a chaotic system are extremely stab
• #### Re: (Score:2)

How can something be possibly random at a fundamental level ? it would go against the law of conservation of motion. In my opinion there is no randomness at all. It's just that every particle in the universe is affected by every other particle ( nomater how small those forces may be ), thus the particles' movement seem random to us.

the practical difficulty in computing does not mean that there is a chaotic or random factor. It's just means the factors that affect the particular phenomenon so many, that i
• #### Re: (Score:2)

the practical difficulty in computing does not mean that there is a chaotic or random factor.

Are you sure?

• #### Re: (Score:2)

How can something be possibly random at a fundamental level?

Because nothing has a fixed position or speed. Everything is at a fundamental level a probability wave, that only collapses to a fixed position or a fixed speed when interacting with other probability waves.

• #### Re: (Score:3)

Chaotic systems have attractors. Chaotic systems will be mostly stable around the attractors, it's the details (where around the attractor they are) that vary.
• #### Re: (Score:1)

by Anonymous Coward

--
And did you exchange a walk on part in the war for a lead role in a cage? - Pink Floyd.

"Meek and obedient you follow the leader down well-trodden corridors into the valley of steel" - Ditto

• #### Re: (Score:2)

Sorry, but this is not the way it works. You have problems such that you can prove there exists an optimal algorithm to solve them, and simultaneously prove you cannot actually write it.

Or for cases such as this, there may not be a finite number of solutions. In fact, there may not be a countable infinity of solutions. At which point, since the axiom of choice may not be true (your choice!) it may be that you may not be able to pick all the solutions which are true and exist, nor even write them as families

• #### Re: (Score:2)

i'm not sure i understood your last point on whether it could be not computable. If it is deterministic and we had all the initial data and the laws we could "replay" the whole thing out. Of course we would have to have a massive supercomputer outside this universe, so as not to affect the prediction.
• #### Re: (Score:1)

Flip a coin

• #### Re: (Score:2, Informative)

by Anonymous Coward

It really just means no closed form solution... falls under advanced algebra. Interesting results, boring class.

• #### Re: (Score:2)

Of course there's a theoretical solution and you can give it as a power series [wikipedia.org]. The three-body problem just can't be solved via first integrals, and the power series is pretty much useless for practical purposes as it converges too slowly.

• #### Re: (Score:2)

what troubles me is the impossibility of a theoretical solution because it undermines my belief in a deterministic universe.

As they say: The universe doesn't care what you believe.

We don't have enough information to know whether it's deterministic or not. Whatever the case, it is what it is. And if it is deterministic, that still doesn't necessarily imply that predicting the future is actually computationally feasible.

• #### Re: (Score:2)

i know that the question is still open and i'm not sure myself. However I cannot "see" how it can be non deterministic. Stochastic theories are always approximations when it's too difficult to compute each element. The fact that we have built a stochastic theory which gives accurate results more or less on a "whole" level ( of the phenomenon we are studying ) does not mean that this stochastic theory reflects reality. It's just a tool to get practical results ( just like feynman's on the back of the hand ca
• #### Re: (Score:2)

Actually, even if it is deterministic there can be problems with no solutions, such as the Halting Problem. Predicting the future in all cases is impossible, even in a fully deterministic universe.
• #### Re: (Score:1)

In the real world, there are only n body problems where n is a very very large number.
• #### Re: (Score:2)

Actua\ly, not always. Accuracy is often dependent on getting convergence at all (existance and uniqueness), and then on not getting an infinitely slow convergence (iirc, the mcLauren/Taylor solution to the ATAN function is an example.)

After that, you are limited in a very real way by computing power. Thus, any time you can eliminate whole swathes of calculation by refining your model -- or coming up with an exact solution -- it's always a big plus.

• #### Re: (Score:2)

But don't forget that pretty much any numerical analysis will take place on a computer with a limited ability to represent floating point numbers. There will be a diminishing point of returns when decreasing dt when the increased precision from the smaller dt is eaten up by the increased errors in the floating point numbers.

One of my favorite descriptions of this problem comes from RW Hamming's book, "Numerical Methods for Scientists and Engineers": http://books.google.com/books/about/Numerical_Methods_fo [google.com]

• #### Re:having said that (Score:5, Informative)

on Saturday March 09, 2013 @03:59PM (#43127439) Journal

I don't think they did it that way, rathe, they are using the computer to help them predict repeating lissajous patterns (for want of a better term) on their transformed sphere-space.

That then relates back to a specific repeating orbit in 3-space.

This is rather interesting, in that it is quite similar (methinks) to the knot classification problem.

But looking at the lissajous figures, it doesn't really seem to me that there are fourteen new classes, unless the lagrange solutions -- which are all a single class -- were counted as five.

But it's no less impressive, what they have done. They have started to transform from physicists to mathematicians.

• #### Re: (Score:3)

Pretty much all integrators used for celestial mechanics have variable dt. The reason is that there are long periods where almost nothing happens, and then you come very close to a star (or two of the 3 bodies come very close together) where you have very rapid changes of velocity and you need very small dt. Because most of the newly found solutions include such close encounters, their accuracy may be questionable.

• #### Oh, you're talking about THAT three-body problem. (Score:5, Funny)

on Saturday March 09, 2013 @02:48PM (#43127079)

The one that *usually* keeps scientists awake at night is, "how can I get my girlfriend and her cute roommate into bed at the same time?"

• #### Re:Oh, you're talking about THAT three-body proble (Score:5, Funny)

by Anonymous Coward on Saturday March 09, 2013 @02:55PM (#43127119)

I think just getting the girlfriend into bed (or having one, for that matter) is sufficient of a problem for most scientists.

• #### Re:Oh, you're talking about THAT three-body proble (Score:5, Funny)

on Saturday March 09, 2013 @03:40PM (#43127351)

I think just getting the girlfriend into bed (or having one, for that matter) is sufficient of a problem for most scientists.

Well, at least they've already solved in for a spherical girlfriend in vacuum.

• #### Re: (Score:2)

Or for that matter, keeping the girlfriend out of the bed if you're called Sheldon.
• #### Re:Oh, you're talking about THAT three-body proble (Score:5, Informative)

on Saturday March 09, 2013 @03:09PM (#43127197)

"how can I get my girlfriend and her cute roommate into bed at the same time?"

Try turning the lights off and leaving the room.

• #### Re: (Score:2)

+1 Depressing Reality

• #### Re: (Score:2)

And they wont accept numerical solution: http://xkcd.com/613/ [xkcd.com]

• #### Re: (Score:2)

"how can I get my girlfriend and her cute roommate into bed at the same time?"

Get him drunk before you ask him.

• #### Re: (Score:1)

Sorry, how did the scientist get a girlfriend? You lost me there.
• #### See the actual orbits (Score:5, Informative)

by Anonymous Coward on Saturday March 09, 2013 @02:50PM (#43127083)

The orbit gallery [ipb.ac.rs]

Click on an orbit and look at the "real space" diagram to see the actual paths of the planets.

• #### Re: (Score:1)

This is essentially 'Experimental Mathematics' at its best - the conclusions are (I assume) valid, but no theoretical framework is provided to show that there are no other solutions. I think their work is very important, but it lacks mathematical elegance; it may be that we will never find a practical and elegant mathematical theory to cover this - I hope I am wrong!

I know of no proof that determines if the number of solutions (disregarding symmetries and topological invariant transformations ,,,) is finit

• #### Very special cases (Score:5, Informative)

on Saturday March 09, 2013 @02:53PM (#43127107)

While the results are interesting, it looks like the 13 new solutions all involve 3 equal mass bodies with total zero angular momentum and coplanar. Of course, all the periodic solutions are probably special cases of some sort.

• #### Re: (Score:3)

I wonder what stability these solutions have. I.e., whether they are more like L1/L2 points (small deviations amplified over time) or more like L4/L5 points (small deviations lead to loops around the center).
• #### Re: (Score:2)

I believe that stability is exactly what this team will be investigating next.
• #### Re: (Score:3)

PROTIP: It isn't a "very special case" to get 3 coplanar bodies.

• #### Re:Very special cases (Score:5, Informative)

on Saturday March 09, 2013 @05:02PM (#43127787)

While the results are interesting, it looks like the 13 new solutions all involve 3 equal mass bodies with total zero angular momentum and coplanar. Of course, all the periodic solutions are probably special cases of some sort.

From the point of view of "conceivably help astrophysicists understand new planetary systems" (TFA claim): the zero angular momentum doesn't bother me that much: it'd be a planetary system that rotates in time. The coplanar... mmmhh... maybe an acceptable approximation. It is the mass equality that one doesn't see too often.

• #### Re: (Score:2)

The coplanar... mmmhh... maybe an acceptable approximation.

I'd say that depends on the stability of those systems. It's not just about point solutions in the parameter space, for astronomers, it's more about stable regions, like the L4/L5 Lagrangian solution. You simply won't hit a point solution with real objects, be it the mass or coplanarity, it doesn't matter.

• #### Re: (Score:2)

The coplanar... mmmhh... maybe an acceptable approximation.

I'd say that depends on the stability of those systems. It's not just about point solutions in the parameter space, for astronomers, it's more about stable regions, like the L4/L5 Lagrangian solution. You simply won't hit a point solution with real objects, be it the mass or coplanarity, it doesn't matter.

You reckon?

1. when you speak stability of the system, what reference of duration you think of? Because, look, I'm pretty sure the Solar System is mathematically unstable in the absolute sense, however the changes in the planet orbits are so minute on millennial scale that we can consider it "pretty stable" even if, hundreds of millions of years the changes would be notable (my point: unless catastrophically unstable to exist, a real astronomical star system does not impose/require stability in the absolute

• #### um (Score:1)

by Anonymous Coward

The paper is four pages. These could hardly be considered "solutions", there are no proofs at all.

• #### Re: (Score:2)

True, but now when you know where to look, it is far easier to retrofit proper theory. The presence of the solutions could trivially be verified in any (really well-written) independent numerical simulation.
• #### Re: (Score:1)

There are no proofs in physics, only experiments. Experiments are difficult in this case, so these solutions were found with numerical simulation. Additional simulations by other physicist (and for this problem, there will be many) will show whether these are proper solutions or caused by the authors' mistakes.

As sibling above points out, people will probably try to find analytical solutions that match these.

• #### Re: (Score:2)

There is a class of problems named NP. Have you heard about them?

• #### Re: (Score:2)

Yes, but not all problems are Np. For example, the Parker-sockaki has allready passed existence and uniqueness, but AFAIK, NP is still out there. it would be nice to know that it was NP, but right now it only might be.

• #### Re: (Score:2)

Solving differential equations (like this one) is NP.

• #### At least a beginning (Score:2)

I've been told by software simulation vendors that no way can their stuff - even if it was running on every supercomputer on the planet for years, could solve the 10e19 body problem I have simulating a fusor's emergent behavior. The math guys have let us down here in science-ville. And if you can't even really do it feedforward for 3 bodies that only attract (vs ions, electrons, charge exchange, and neutrals) I don't have any hope of it being done for my field in my lifetime.

Get cracking, math guys. Unt

• #### See groundbreaking work by Klaatu et al. (Score:1)

Wasn't this solved in 1951 as shown in that documentary "The Day The Earth Stood Still"?

LAst line for those who don't get the joke [wsu.edu]
• #### This is of course ...the (Score:2)

the... a solution to the three body problem under a universal unidirectional inverse square law -- still the simplest case of the three body problem which one can analyse.

What if the force is dependent not on mass, which cannot be negative, but on electric charge, which can be? What about a hypothetical coloured force (like the stuff out of quantum chromodynamics) in which Red attracts Green and repels Blue, Green attracts Blue and repels Red, and Blue attracts Red and repels Green? What if there is a fou

• #### Stability of the solutions (Score:2)

The authors do not check the stability of the found peridioc orbits, which is a necessary condition for expecting such orbits in nature. When stable nearby orbits diverge typically linearly in time and stay similar to the periodic solution (like the planets in the solar system stay close to elliptic orbits), while when unstable the divergence is exponential and quickly the 3 bodies are widely separated.

• #### Re: (Score:2)

From the article:

The next step for the Belgrade physicists is to see how many of their new solutions are stable and will stay on track if perturbed a little. If some of the solutions are stable, then they might even be glimpsed in real life.

The authors are, in fact, smarter then you. They are well aware of this issue. RTFA. You just make yourself look stupid when you post the obvious.

My sig says it all:

• #### Re: (Score:2)

Did I say they were not aware? To me the authors look amateurish because checking stability is rather trivial once you know how to integrate ordinary differential equations.

The other aspect that should be said is that in such hamiltonian systems periodic orbits are dense in phase space, but most of them are unstable. So actually one should expect an infinity of such orbits, most of them are not interesting for practical applications. So the minimum the authors could have checked stability before publish

• #### Did the authors forget some known solution work? (Score:2)

It's strange how they don't mention the solutions of the 3-body problem explored in the 19th century by G W Hill: see e.g. "Hill's Lunar Equations and the Three-Body Problem": K R Meyer, D S Schmidt, Jnl of Differential Equations 1982, 44, 263-272 https://math.uc.edu/~meyer/jde82.pdf [uc.edu]. Part of his work was one of the first things published in the American Journal of Mathematics, (G W Hill, in American Journal of Mathematics, Vol. 1, No. 1 (1878), pp. 5-26).

• #### 3 body lagrange problem (Score:1)

3 bodies can remain in static orbit according to lagrange: https://en.wikipedia.org/wiki/Lagrangian_point [wikipedia.org]

Some of these orbit locations are "attractors", meaning that bodies close to these points will tend to "fall in" to the orbital points and remain stable. These solutions generally have 2 large bodies and one much smaller body. What I always wondered if it was possible for 2 black holes to orbit one another. If so, then they should have these lagrange orbital points where other objects would fall in and

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