Send it over to Trisolaris!
Wasn't talking to them the whole problem in the first place?
yep I was about to say, first the UFOs and now this, turn the radio beacons off already
Scientists find an effective solution for the three-body problem
61–70 of 264 posts
Re: Scientists find an effective solution for the three-body problem
#62Earlier quoted context omitted.
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.
Conversely, there are also extremely simple closed-form recurrence relations that exhibit chaotic behavior, e.g. the logistic map.
Re: Scientists find an effective solution for the three-body problem
#63Re: Scientists find an effective solution for the three-body problem
#64Earlier 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.
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.
What you're describing is not randomness, it's independence.
It's hard to define randomness. I think non-determinism is better than your definition.
Re: Scientists find an effective solution for the three-body problem
#65Earlier 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…
Not to answer your question, but you may be interested to know that chaotic systems can often be effectively controlled by small perturbations: https://en.wikipedia.org/wiki/Control_of_chaos
Re: Scientists find an effective solution for the three-body problem
#66Earlier 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…
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Re: Scientists find an effective solution for the three-body problem
#67Earlier quoted context omitted.
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?
Not really because they're already so chaotic you couldn't be sure what divergences from simulation were inherent and which were due to external perturbation.
Re: Scientists find an effective solution for the three-body problem
#68Earlier quoted context omitted.
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.
Conversely, there are also extremely simple closed-form recurrence relations that exhibit chaotic behavior, e.g. the logistic map.
True but a recurrence relationship is not the same as a closed form solution. The differential equation for a three body problem is also very simple.
Re: Scientists find an effective solution for the three-body problem
#69Re: Scientists find an effective solution for the three-body problem
#70Earlier quoted context omitted.
Conversely, there are also extremely simple closed-form recurrence relations that exhibit chaotic behavior, e.g. the logistic map.
True but a recurrence relationship is not the same as a closed form solution. The differential equation for a three body problem is also very simple.
Of course. My point was just that you can evaluate a recurrence relation exactly (i.e. with zero numerical error) and still get chaotic behavior. OP’s mistaken point was that the three body problem’s chaos arises solely from numerical error during simulation, which is untrue.