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Scientists find an effective solution for the three-body problem

phys.technion.ac.il

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Re: Scientists find an effective solution for the three-body problem

#51

Post by db48x »

ko27 wrote: post #41

How good are we at predicting three body movement today with our computers? This is the question I could never find an answer too. Can we do it real time, or few years into the future? Can we do it accurately, to an arbitrary precision? Or is it always fuzzy with statistical outcomes?

We can do it accurately, to any precision you care to pay for. Since n-body gravitation is a chaotic system, getting more precise predictions requires more precise measurements of the current state of the solar system. When it’s not possible to measure things more precisely, we instead run many simulations with small random perturbations in the current state, then classify the simulations to get probabilities.

Re: Scientists find an effective solution for the three-body problem

#52

Post by im3w1l »

kragen wrote: post #11
guerrilla wrote: post #10

Earlier quoted context omitted.

You sure about that? The fourth body, being so small, can't really effect the motion of the other three; however if you have the evolution of the first three then you can determine the motion of the fourth.

Because three-body systems are chaotic, the fourth body can affect the motion of the other three, however small it is.

Does this mean that you could you use a three body system as a measurement device?

Re: Scientists find an effective solution for the three-body problem

#53

Post by Y_Y »

ko27 wrote: post #41

How good are we at predicting three body movement today with our computers? This is the question I could never find an answer too. Can we do it real time, or few years into the future? Can we do it accurately, to an arbitrary precision? Or is it always fuzzy with statistical outcomes?

One way to deal with these kinds of issues is to express your initial conditions as intervals (or even distributions) in the sense that you include the a range of possible values, rather than the most probable one (which is normally implicitly done). So if you measure the earth to be (6±1)e24 kg, then you work with something that looks like [5,7]e24 kg i.e. the segment of the number line corresponding to the possible physical realities that led to your measurement. You'll get a range of different outcomes in the end, and their relative probabilities given your priors. You can do this exactly for some systems, but usually you'll do some monte carlo and hope it's valid. This is similar to classical (linear) error propagation where you carry around an "uncertainty", but chaotic systems don't generally allow you to make the assumption of narrow Gaussians used there.

Re: Scientists find an effective solution for the three-body problem

#54

Post by aaaaaaaaaaab »

ko27 wrote: post #41

How good are we at predicting three body movement today with our computers? This is the question I could never find an answer too. Can we do it real time, or few years into the future? Can we do it accurately, to an arbitrary precision? Or is it always fuzzy with statistical outcomes?

It is a chaotic system. Arbitrary small deviations in the initial conditions will result in completely different outcomes. So your simulation will eventually diverge from reality as you cannot measure the initial conditions exactly.

Re: Scientists find an effective solution for the three-body problem

#55

Post by bottled_poe »

Grustaf wrote: post #28
Aeolun wrote: post #19

Earlier quoted context omitted.

Huh? If this is the case, wouldn’t a 2 body problem also be practically impossible to calculate? I mean, you can certainly still predict to a certain (probably high) level of accuracy, but ultimately that motion is also influenced by factors outside your model.

No, the fundamental difference is that the two-body problem can be solved analytically, you can write down a formula. For three bodies and up you only have numerical solutions, simulations, and they will break down over time.

I think this is accurate and I don’t understand why you are being downvoted.

Re: Scientists find an effective solution for the three-body problem

#56

Post by aaaaaaaaaaab »

Aeolun wrote: post #19
ordu wrote: post #14

Earlier quoted context omitted.

They would be deterministic. But they are unpredictable. Minuscule fluctuations (coming from influence of some much smaller bodies, that are not in the model, or approximations in calculations) can lead to dramatic differences in the outcome. So theoretically speaking, they are deterministic, but practically they are unpredictable.

Huh? If this is the case, wouldn’t a 2 body problem also be practically impossible to calculate? I mean, you can certainly still predict to a certain (probably high) level of accuracy, but ultimately that motion is also influenced by factors outside your model.

No. A small error in the initial conditions of a 2-body system produces a small error in the result. In case of a 3-body system, a small error will result in drastically different outcomes. The phase space of a 2-body system is nice and smooth, but a 3-body system’s is more like a fractal.

Re: Scientists find an effective solution for the three-body problem

#58

Post by zelphirkalt »

aaaaaaaaaaab wrote: post #54
ko27 wrote: post #41

How good are we at predicting three body movement today with our computers? This is the question I could never find an answer too. Can we do it real time, or few years into the future? Can we do it accurately, to an arbitrary precision? Or is it always fuzzy with statistical outcomes?

It is a chaotic system. Arbitrary small deviations in the initial conditions will result in completely different outcomes. So your simulation will eventually diverge from reality as you cannot measure the initial conditions exactly.

Nitpick + a little more thought: Isn't it more correct to say, that initially slightly different conditions might (instead of "will") result in a very much different outcome? Does chaotic mean, that two states which only differ a little must result in vastly different outcomes? I wonder whether there could be states, which are very similar and some condition drives them to converge again. Or is such a thing impossible?

Re: Scientists find an effective solution for the three-body problem

#59

Post by phreeza »

Grustaf wrote: post #28
Aeolun wrote: post #19

Earlier quoted context omitted.

Huh? If this is the case, wouldn’t a 2 body problem also be practically impossible to calculate? I mean, you can certainly still predict to a certain (probably high) level of accuracy, but ultimately that motion is also influenced by factors outside your model.

No, the fundamental difference is that the two-body problem can be solved analytically, you can write down a formula. For three bodies and up you only have numerical solutions, simulations, and they will break down over time.

The problem is that a three-body system is inherently unstable/chaotic. So even if you run a numerical simulation with the same granularity for a two-body and a three-body system, the three-body simulation will degrade much faster than the two-body system. This is unrelated to the fact that there is a closed form solution, there are many stable, non-chaotic dynamical systems that don't have a closed form solution.
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