The clarification of what is meant by a solution is in the last sentence of the article:
The “solutions” were instances when these three bodies found a way to maintain an orbit around one another.
avidiax
This is my question. Does this mean that the solution orbits are stable equilibria, i.e. small errors or perturbations return to a stable orbit?
scythe
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.
westurner
Are all of the identified solutions (also) fluid attractor systems; and - if correct - shouldn't a theory of superfluid quantum gravity predict all of the n-body gravity solutions already discovered with non-fluidic numerical solutions?
uoaei
I don't see a reason to think any of them have relevance to superfluid physics. The three-body problem is always cast in terms of Newtonian physics.
As far as I know, no one has ever analyzed these systems even in settings of special or general relativity, so it's mostly a toy problem divorced from our current understanding of reality anyway. If such a solution is unstable, it probably couldn't be constructed in reality due to relativistic effects because it requires overly simplistic physics.
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).
zoogeny
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.
cynicalkane
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.
rewmie
> 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.
calf
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.
JKCalhoun
> Understanding the extremely subtle movements among three orbital bodies is important for space travel...
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.
feoren
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)
Retric
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.
MaxRegret
You're right, the real problem with predicting trajectories through those "keyholes" [1] isn't solving the dynamics equations: our numerical integrators are more than good enough. The problem is we can't measure the initial states with enough precision.
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?
zardo
Two still has bugs that effect basic gameplay. E.g. the game being in the same state when you reload as when you saved.
feoren
The reviews have been so universally negative on KSP 2 that I haven't bought it yet. Anything I'd say would be hearsay; you can go read the reviews yourself. My impression is it's standard enshittfication: some asshole MBA saw a company that wasn't as shitty as it possibly could be yet, so they bought it, fired/drove out the original team, and made a cheap-ass knockoff 2nd game hoping the laurels of the first would drive sales long enough for them to make some quick bucks before buzzing off to parasitize another company. The exact details may be different but it's basically the same story happening over and over and over and over and over in our dystopian cancerous capitalism. We can't have nice things anymore.
taneq
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?
mock-possum
Why should there be a relation between the n-body problem and the halting problem?
7373737373
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?
Do you want Trisolarans? Because that's how you get Trisolarans
schwartzworld
Trisolarans lived in a 4 body system. Also that book is hot garbage.
captainclam
As evidenced by lots of people enjoying it.
schwartzworld
Have you read it? The plot is tropey at best, and the characters' motivations bizarre. Whatever interesting ideas the author has are overshadowed by his lack of storytelling ability.
captainclam
I have, and I totally agree with your critiques, but I still enjoyed it. Do you have any suggestions for really good science fiction?
schwartzworld
It's a big category.
In the last year, two of my favorite reads were Children Of Time and it's sequels (Adrian Tchaikovsky) and Embassytown (China Mieville). I also really enjoyed reading the Lilith's Brood series (Octavia Butler). All three of these present a more nuanced look at humanity's interaction with alien intelligence
hn92726819
Did they? I thought it was 3 suns and a planet and the planet had a negligible impact on the suns orbit.
andrewflnr
Stable solutions are explicitly not how you get Trisolarans.
taneq
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.
gadders
Good books but the protagonist in the 3rd book was INFURIATING.
burnte
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.
MenhirMike
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.
moribvndvs
All of the books follow this pattern where 2/3rds or more is split between really bad and unlikeable characters meandering and waxing philosophically for whole chapters at a time–putting the plot in limbo–and terse, emotionless, transactional conversations sprinkled in to break up the log jam with boring exposition. It was absolutely infuriating, and I originally quit in the middle of book 2 because I could not stand another second of Luo, until my reading buddies at work hounded me to finish. They would all climax with a sudden scramble to wrap everything up in an interesting way, but the problem became worse with each subsequent book. I really don’t get how this series took off.
MenhirMike
> I really don’t get how this series took off.
My guess: The novelty of Science Fiction from a Chinese Author, which isn't commonly translated to English, and a legitimately interesting SciFi concept. If you read the plot summary on Wikipedia, it actually sounds amazing, it's just that the books themselves don't deliver on the premise nearly as well.
Apparently Netflix are due to do an english version of it too.
anonporridge
Sadly being produced by D&D. After their massive Game of Thrones season 8 fuck up, I'm not holding my breath.
Then again, maybe it'll be fine because all the source material is finished and they don't have to fail at writing the end of the story on their own.
ndsipa_pomu
I didn't realise that.
I was already somewhat sceptical as I heard that the story is being moved to the U.S. which seemed odd, but it was apparently signed off by Liu Cixin.
I'm still a bit in shock as to how bad GoT season 8 was. So many plotlines and character development were just shredded, burnt and then thrown away. And that battle looked impressive on-screen (if you like watching a black screen with little points of light) but just made absolutely no sense tactically.
irrational
If only there was someone else who could have written the ending of the GoT for them.
gadders
Still waiting...
coremoff
the first 4 seasons, where they had complete source, were really good; fell off a cliff afterwards obviously - they should do OK with a complete trilogy, though personally I don't understand the interest in those books beyond the premise.
hooverd
Eh, as long as they have source material they're okay.
tanvach
For me the second book was way worse.
torstenvl
The second book was by far the best.
The first has a bit too much lead-up. The third is a bit too philosophical.
tene80i
How so?
gadders
I don't want to give any spoilers so I've converted my comment into Trisolaran (ROT13):
Gur yrnq srznyr punenpgre va gur 3eq obbx znqr rirel vapbeerpg qrpvfvba vg jnf cbffvoyr gb znxr, naq ol ure vapbzcrgrapr qbbzrq uhznavgl.
deanCommie
Ohg gung'f gur cbvag? Gung'f yvgrenyyl gur cbvag bs gur obbx. Ubj pna lbh or znq ng gur punenpgre/cebgntbavfg? Lbh qba'g unir gb or unccl gung uhznavgl vf qbbzrq, lbh pbhyq/fubhyq srry natel nobhg vg. Ohg vg'f n gurzr guebhtubhg gur fgbel. Synjrq uhznaf if Vapbzcerurafvoyr Havirefr-fpnyrq guerngf naq gvzrfpnyr
The “solutions” were instances when these three bodies found a way to maintain an orbit around one another.
As far as I know, no one has ever analyzed these systems even in settings of special or general relativity, so it's mostly a toy problem divorced from our current understanding of reality anyway. If such a solution is unstable, it probably couldn't be constructed in reality due to relativistic effects because it requires overly simplistic physics.
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)
[1] https://en.wikipedia.org/wiki/Gravitational_keyhole
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...
In the last year, two of my favorite reads were Children Of Time and it's sequels (Adrian Tchaikovsky) and Embassytown (China Mieville). I also really enjoyed reading the Lilith's Brood series (Octavia Butler). All three of these present a more nuanced look at humanity's interaction with alien intelligence
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.
My guess: The novelty of Science Fiction from a Chinese Author, which isn't commonly translated to English, and a legitimately interesting SciFi concept. If you read the plot summary on Wikipedia, it actually sounds amazing, it's just that the books themselves don't deliver on the premise nearly as well.
Apparently Netflix are due to do an english version of it too.
Then again, maybe it'll be fine because all the source material is finished and they don't have to fail at writing the end of the story on their own.
I was already somewhat sceptical as I heard that the story is being moved to the U.S. which seemed odd, but it was apparently signed off by Liu Cixin.
I'm still a bit in shock as to how bad GoT season 8 was. So many plotlines and character development were just shredded, burnt and then thrown away. And that battle looked impressive on-screen (if you like watching a black screen with little points of light) but just made absolutely no sense tactically.
The first has a bit too much lead-up. The third is a bit too philosophical.
Gur yrnq srznyr punenpgre va gur 3eq obbx znqr rirel vapbeerpg qrpvfvba vg jnf cbffvoyr gb znxr, naq ol ure vapbzcrgrapr qbbzrq uhznavgl.