## Physicists Discover 13 New Solutions To Three-Body Problem 127 127

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

## Re: (Score:3, Funny)

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)

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

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)

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completely wrong. There is a class of numerical algorithms calledSymplectic Integrators, which make sure that energy is conserved. You can also choose algorithms with anadaptive 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...

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Yes it is. We can't show which side of the sun the Earth will be on exactly 10,000,000 years from now (measured in standard seconds, etc.) And we can't show that it will still be orbiting the sun some number of years after that. The errors increase without limit, but slowly.

Note that some, but only a small part, of the error is due to unknowns, say astroids we haven't mapped the orbit of. Most of it is due to chaos. The calculations are EXTREMELY sensitive to initial conditions AND to errors in calcula

## Re: (Score:2)

We do have analytical solutions to the orbital problem,look up the parker sochacki solution to the picard iteration.

http://csma31.csm.jmu.edu/physics/rudmin/parkersochacki.htm [jmu.edu]

But there are limitations of how good our understanding of the initial position/velocity vectors are, so yes, we are also limited on the value of the results.

## Re: (Score:2)

I'm not sure that you can express the initial conditions as a finite series. If you can, then it looks as if you might be right. (Which doesn't say it would be possible to solve it in any feasible time on any feasible equipment.)

So, OK, MAYBE there is an analytic solution. But I doubt it, because I don't think the initial conditions can be satisfied. (OTOH, it's been decades since I even *looked* an a differential equation, so my opinions aren't worth weighing heavily.)

Even then, though, as we agree, t

## Re: (Score:2)

It's not as bad as you think, but the paper does discuss that towards the end. And yes, the author did use it elsewhere to show that the trojan asteroids cycled around their -- well, I'd actually call it their lagrangian spots on Jupiter's orbit.

## Re: (Score:1)

With a two body problem like the sun and the earth, we can accurately know exactly where the earth will be in 10,000,000 years. Because we will assign an exact number to its location and speed and exact numbers to the mass of the bodies and such. So the answer in relation to the article... is YES, we know exactly how to calculate 2 bodies in gravitational orbit. Of course, in reality, we dont know everything perfectly, so HiThere is correct in saying that in practice we cannot perdict that far out because t

## Re: (Score:2)

If there weren't anything but the earth and the sun, you would have a point. But you've left out not only the rest of the solar system, but even the moon. This doesn't work.

N.B.: You can't ignore both Jupiter and Saturn and have a prediction that's even good for a few decades.

## 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.

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

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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.

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## Re:having said that (Score:5, Insightful)

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 :)

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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)

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].

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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

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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.

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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

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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

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The Bell results only show that there are no

hiddenNon-local variables could never be proved to be impossible.localvariables.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_testsSo 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")

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The thing I find "odd" is that often (always?) the statistics of a chaotic system are extremely stab

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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

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

Are you sure?

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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.

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--

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

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Flip a coin

Heads or tails?

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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.

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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.

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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)

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)

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)

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

## Re: (Score:1)

No wonder you never get laid.

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

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)

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

"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.

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+1 Depressing Reality

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

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

Get him drunk before you ask him.

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## See the actual orbits (Score:5, Informative)

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)

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.

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

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

Right, it's impossible. In the real world, nothing lines up that perfectly. PROTIP: Don't be an asshole and others won't treat you like one.

GP's point is that three bodies DEFINE a plane. They're always coplanar. It's the fourth body that gets you in trouble.

## Re: (Score:1)

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

Right, it's impossible. In the real world, nothing lines up that perfectly. PROTIP: Don't be an asshole and others won't treat you like one.

GP's point is that three bodies DEFINE a plane. They're always coplanar. It's the fourth body that gets you in trouble.

Which is missing the point, since OP clearly meant that all the

velocity vectorslie in the same plane as the three bodies, so that theyremainin the same plane.## Re: (Score:2)

Except that you'd have to make an effort to think he meant just the positions of the objects rather than the velocity vectors that he obviously did.

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

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

stabilityof 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)

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

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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)

Yeah, I felt a bit stupid after posting that. This is one of those example where the problem has a complete and simple mathematical formulation. An analytical solution can be proved to be periodic. What these guys did was find numerical solutions. However, now that they have some ideas where to look, it should be easier to come up with analytical orbits.

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

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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)

Get cracking, math guys. Unt

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

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

RedattractsGreenand repelsBlue,GreenattractsBlueand repelsRed, andBlueattractsRedand repelsGreen? 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.

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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:

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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 Equations1982, 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, inAmerican 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

## Re:anonymous coward discovers new way to first pos (Score:5, Funny)

naked and petrified!

You mean the paleolithic version of the three body problem [s-nbcnews.com]?