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?
Scientists find an effective solution for the three-body problem
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Re: Scientists find an effective solution for the three-body problem
#42Earlier 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.
For a two-body problem the discrepancies between your model and reality increase gradually with time, but it's still possible to predict eclipses decades in advance with a precise albeit imperfect measurement of initial conditions. With a three-body problem any slight shift causes a wildly different trajectory, bearing no resemblance to the original so your measurements of the initial condition have to be perfect.
arent eclipses three body problems?
Re: Scientists find an effective solution for the three-body problem
#43How 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?
The system is chaotic so there is a strong dependence on the initial conditions. I suppose that if you don't know these precisely, then at some point even the best computer simulation can't help you much.
Re: Scientists find an effective solution for the three-body problem
#44When 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.
Random doesnt mean non-deterministic anyway.
X is random with respect to Y, if knowing Y makes no difference to your predicting that X.
QM systems are indeterminate, they are random in the above sense /because/ they are indeterminate. But that isnt what random means.
Re: Scientists find an effective solution for the three-body problem
#45Earlier quoted context omitted.
For a two-body problem the discrepancies between your model and reality increase gradually with time, but it's still possible to predict eclipses decades in advance with a precise albeit imperfect measurement of initial conditions. With a three-body problem any slight shift causes a wildly different trajectory, bearing no resemblance to the original so your measurements of the initial condition have to be perfect.
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.
Re: Scientists find an effective solution for the three-body problem
#46Earlier 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.
The question is what happens with a small perturbation and how does it "grow". For a two body problem, you nudge one of the bodies and it is forever off by a small amount, but your predictions into infinity require only a small adjustment to compensate. For a three body problem you nudge one of the bodies and only for a very short time do your previous predictions stay true, the change amplifies until nothing you tho…
> a common occurrence is one of the bodies being ejected.
So the problem eventually solves itself?
Re: Scientists find an effective solution for the three-body problem
#47Earlier quoted context omitted.
For a two-body problem the discrepancies between your model and reality increase gradually with time, but it's still possible to predict eclipses decades in advance with a precise albeit imperfect measurement of initial conditions. With a three-body problem any slight shift causes a wildly different trajectory, bearing no resemblance to the original so your measurements of the initial condition have to be perfect.
arent eclipses three body problems?
For all practical purposes eclipses are two two-body problems ... Sun vs Earth-Moon pair, and Earth vs Moon. The scale of the Sun is so vastly larger and distances so great from sun to earth there's no need to do 3-body.
If the three had roughly the same mass would be different story.
Re: Scientists find an effective solution for the three-body problem
#48Please saw off the head of Mr. Cankut Orakcal. You can stop the coronavirus pandemic, the next 9/11 or financial crisis. Decapitate on sight as needed.
Re: Scientists find an effective solution for the three-body problem
#49Earlier quoted context omitted.
The question is what happens with a small perturbation and how does it "grow". For a two body problem, you nudge one of the bodies and it is forever off by a small amount, but your predictions into infinity require only a small adjustment to compensate. For a three body problem you nudge one of the bodies and only for a very short time do your previous predictions stay true, the change amplifies until nothing you tho…
> a common occurrence is one of the bodies being ejected. So the problem eventually solves itself?
Well, not if the ejected body contains you.
Re: Scientists find an effective solution for the three-body problem
#50When 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?
The initial conditions are usually known up to a small epsilon in practice. I guess this initial error grows exponentially with time, hence "unpredictability".
This exponential increase in error is described by the Lyapunov exponent of the system: https://en.wikipedia.org/wiki/Lyapunov_exponent