Considering that thousands were found, I'm guessing that they were unstable periodic orbits. These can still be used to characterize the system but they occur naturally with probability zero.
The three-body problem, as a concept, feels like a twist on the hammer-and-nail analogy: If analytic methods are the hammer, anything that doesn't resemble a nail is an anomaly (Framed as in this article, notoriously hard). Amusingly, this applies to most practical calculations (perhaps all when examined closely enough).
It reminds me of a YouTube course on general relativity where the instructor went from most basic elements and built up towards a complete description of the full math. I was able to follow along step-by-step at the time but there were a lot of quite complicated steps which I would have no ability to reproduce myself.
I recall a few of the steps made assumptions on our ability to calculate. I think, for example, they narrowed down the set of all vector spaces to just those spaces that were differentiable. I may be mis-remembering the precise detail but it was something along those lines, and this was just one of a few instances of this kind of "throw away cases that we are unable to calculate" along the path. In some cases the narrowing was justified but in a few the instructor admitted that the entire reason we were excluding possible sets of solutions was because they would otherwise make the next steps impossible.
We consider “cos 123” an exact solution even though to numerically calculate it requires a power series approximation. So 3 body problem is just as “exact” as that.
This isn't why. You can always use a convenient Taylor series to approximate cosine anything to arbitrary precision. For the three-body problem, small changes to initial conditions diverge into incalculably chaotic behavior.
> We consider “cos 123” an exact solution even though to numerically calculate it requires a power series approximation.
I think you're confusing what an exact solution is supposed to be with your own approximation of the exact solution. In your own example, cos(123) would represent a closed form solution to the problem. That solution doesn't cease to be exact if you decide to express it as a finite power series.
But couldn't one say the same about P vs NP? No polynomial algorithm for SAT being analogous to no analytic solution for 3-body?
It's not that closed form answers are required by the insistence of anyone, I just thought it's just of purely mathematical interest of what kinds of problems there are, like problems placed in P vs in NP.
You want to make course adjustments months or years in advance, which may involve things like gravity assists and slingshots that require very good predictions of where multiple different bodies will be a year from now. It can be thousands of times more expensive to course-correct "just in time", or even not possible at all.
As with most discussions of orbital mechanics, the best advice I can give if you want to learn more is: play Kerbal Space Program (1, not 2)
But predicting out real world chaotic orbits a few years in advance to the required precision is fairly trivial. You only really need high precision to line up a slingshot for the next slingshot not the slingshot after that. A last minute correction of 1 mile per hour * 1 month ~= 720 miles.
I've been getting the itch to go deeper into KSP after a scratching the surface of the first one (simple landing on Mun). I was considering buying 2. What's your review against playing KSP 2?
It is, because "close enough" requires a tolerably good model of multi-body orbital mechanics. Of course, you don't have to be able to model a three-body system perfectly because that's often impossible, but you've still got to be close enough for the cold gas thrusters to make up the difference.
It might be cool to see a competition with a problem like that
Is the fact that a fully expanded formula branches out exponentially with every step, since each future motion/velocity component of a body depends on all past components of all other bodies, which in turn depend on other components etc. relevant to that?
It is possible to write an n-body simulation program that only halts its step-by-step iteration if one body collides with another (specific) body or reaches a specific location.
Now, is it possible to tell analytically, without running the program, if it halts (at an arbitrary point in the future, or within a certain number of steps)? If so, the n-body/trajectory finding problem is solved, efficiently.
But (n>=3) n-body systems have been shown to be chaotic, depending sensitively on the initial state of the system, so this doesn't seem to be generally possible without brute force computation.
So what about this chaos/mathematical-logical-temporal relation/equations between state variables (body positions and velocities) make this halting problem (effectively/efficiently) unsolvable, and how does it relate to other computational systems where the halting problem applies, like turing machines?
But stable solutions are what the Trisolarans want to escape to, so while they're not how you become Trisolarans, they're definitely a way to be targeted by them.
Edit: While we're discussing the Remembrance of Earth's Past series, there are few books I have more mixed feelings about. On the one hand, it had many fascinating ideas. While trying to avoid spoilers, the dark forest theorem is far too plausible, the Swordholder gambit is well done, the first encounter with the Trisolaran teardrop probe was really well done, the 2D weapon was legitimately terrifying, the curvature drives' effect on spacetime was a nice twist, I could go on. On the other hand there were a lot of things that just didn't gel. The apathy of the humans after the end of the Deterrence Era, a lot of human reactions to events (could just be cultural though?) and why does the sun have a crust?! Argh.
I couldn't make it through book 2. Book 1 set up some interesting ideas without exploring it too much, and book 2 completely moved on without exploring things, and it was slow, boring, and annoying. I read the wikipedia summary and realized I was much better off forgetting the entire trilogy.
Luo also wasn't THAT much better, mainly because he's an author self-insert that got to get his perfect mail order bride and then proceeded to play his role without any fail or mistakes once he has to step up. But the whole idea of the Dark Forest concept was IMHO legitimately good science fiction. But yeah, Cheng is a complete and utter moron. Maybe that makes her human, but I stopped reading Death's End halfway through and looked up the remaining plot summary on Wikipedia. I don't know if the author had a real plan while writing the story, because the series begins really strong and then just runs out of good ideas.
Readit News
The “solutions” were instances when these three bodies found a way to maintain an orbit around one another.
I recall a few of the steps made assumptions on our ability to calculate. I think, for example, they narrowed down the set of all vector spaces to just those spaces that were differentiable. I may be mis-remembering the precise detail but it was something along those lines, and this was just one of a few instances of this kind of "throw away cases that we are unable to calculate" along the path. In some cases the narrowing was justified but in a few the instructor admitted that the entire reason we were excluding possible sets of solutions was because they would otherwise make the next steps impossible.
I think you're confusing what an exact solution is supposed to be with your own approximation of the exact solution. In your own example, cos(123) would represent a closed form solution to the problem. That solution doesn't cease to be exact if you decide to express it as a finite power series.
It's not that closed form answers are required by the insistence of anyone, I just thought it's just of purely mathematical interest of what kinds of problems there are, like problems placed in P vs in NP.
Is it? I assumed "close enough" was good enough as long as you have realt-time feedback (where am I really?) and a means to make course adjustments.
As with most discussions of orbital mechanics, the best advice I can give if you want to learn more is: play Kerbal Space Program (1, not 2)
https://arxiv.org/abs/2308.16159
What I was looking for was some visualizations of the stable orbits, and the paper has some in the "results" section.
Also, the difficulty of finding a delta-v/time efficient trajectory between two locations in such a system.
https://space.stackexchange.com/q/64392/38733
It might be cool to see a competition with a problem like that
Is the fact that a fully expanded formula branches out exponentially with every step, since each future motion/velocity component of a body depends on all past components of all other bodies, which in turn depend on other components etc. relevant to that?
Now, is it possible to tell analytically, without running the program, if it halts (at an arbitrary point in the future, or within a certain number of steps)? If so, the n-body/trajectory finding problem is solved, efficiently.
But (n>=3) n-body systems have been shown to be chaotic, depending sensitively on the initial state of the system, so this doesn't seem to be generally possible without brute force computation.
So what about this chaos/mathematical-logical-temporal relation/equations between state variables (body positions and velocities) make this halting problem (effectively/efficiently) unsolvable, and how does it relate to other computational systems where the halting problem applies, like turing machines?
Edit: https://cs.stackexchange.com/questions/43181/is-the-unsolvab...
Edit: While we're discussing the Remembrance of Earth's Past series, there are few books I have more mixed feelings about. On the one hand, it had many fascinating ideas. While trying to avoid spoilers, the dark forest theorem is far too plausible, the Swordholder gambit is well done, the first encounter with the Trisolaran teardrop probe was really well done, the 2D weapon was legitimately terrifying, the curvature drives' effect on spacetime was a nice twist, I could go on. On the other hand there were a lot of things that just didn't gel. The apathy of the humans after the end of the Deterrence Era, a lot of human reactions to events (could just be cultural though?) and why does the sun have a crust?! Argh.
Apparently Netflix are due to do an english version of it too.
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