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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

#71

Post by aaaaaaaaaaab »

zelphirkalt wrote: post #58
aaaaaaaaaaab wrote: post #54

Earlier quoted context omitted.

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 impossibl…

Indeed, there can be islands of stability in the phase space of chaotical systems.

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

#72

Post by Grustaf »

l332mn wrote: post #45
Grustaf wrote: post #29

Earlier quoted context omitted.

No that's not it. The two body problem has a closed, analytical solution, the three body one doesn't, so you need to simulate it. It's a fundamentally different approach.

That's not really the issue. The three body problem does have an analytical solution in the form of a power series, but the problem is that it converges so slowly to be of any practical use.

It is though, if you don’t have a closed form solution you need to use an iterative process to calculate the positions, meaning errors will accrue over time. For a closed form solution that wouldn’t be the case.

Thanks for mentioning the existence of an analytical solution at all though, I wasn't aware of that.

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

#73

Post by snowAbstraction »

The photo of Professor Hagai Perets (Left) and Ph.D. student Yonadav Barry Ginat seems to have them in their native environment: the university's Science and Math library. If you zoom you'll see the Math and Science topics listed on the card for the book shelves behind them.

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

#75

Post by user-the-name »

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.

That is not the issue. There are many problems that do not have analytical solutions, but can be approximated numerically to any precision you would like.

The issue with the three-body problem is that it is chaotic, meaning any error will eventually grow to take over the entire solution, making prediction impossible, even in theory. Every chaotic system lacks an analytical solution, but not every system without an analytical solution is chaotic.

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

#77

Post by ninkendo »

ordu wrote: post #14
thinkski wrote: post #13

When they say motions of three bodies are random and unpredictable, I assume they mean not able to be modeled with a closed form equation? Seems like the motions would still be entirely deterministic — could still predict the locations of the bodies computationally, given your computer computes faster than reality (at least for a reality with only 3 bodies), no?

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.

The motion isn’t deterministic if free will exists. Launching a rocket into space decreases earth’s rotation speed ever so slightly, which will have a small impact on the moon’s trajectory due to tidal interactions, and so on.

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

#78

Post by johncolanduoni »

Grustaf wrote: post #72
l332mn wrote: post #45

Earlier quoted context omitted.

That's not really the issue. The three body problem does have an analytical solution in the form of a power series, but the problem is that it converges so slowly to be of any practical use.

It is though, if you don’t have a closed form solution you need to use an iterative process to calculate the positions, meaning errors will accrue over time. For a closed form solution that wouldn’t be the case. Thanks for mentioning the existence of an analytical solution at all though, I wasn't aware of that.

There are iterative methods/systems that stabilize over time. For example, symplectic integrators on tame problems oscillate lightly around the true energy of the system over time. The issue here is the properties of the underlying problem, not the set of solution methods.

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

#79

Post by athrowaway3z »

ninkendo wrote: post #77
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.

The motion isn’t deterministic if free will exists. Launching a rocket into space decreases earth’s rotation speed ever so slightly, which will have a small impact on the moon’s trajectory due to tidal interactions, and so on.

Please define 'free will' before using it in a sentence about determinism.

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

#80

Post by mxxc »

ninkendo wrote: post #77
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.

The motion isn’t deterministic if free will exists. Launching a rocket into space decreases earth’s rotation speed ever so slightly, which will have a small impact on the moon’s trajectory due to tidal interactions, and so on.

that case is beyond scope of the three body problem
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