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.
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?
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.
We can reason about whether things are necessary or sufficient.
To be sufficient as a theory of (n-body) gravity, a theory of gravity must also describe superfluidic gravity in order to describe gravitational dynamics within e.g. Bose-Einstein condensate superfluids.
Newtonian mechanics are insufficient to describe gravity in superfluids like Bose-Einstein condensates: Newtonian numerical methods do not predict superfluidic gravity effects with low error. Newtonian numerical methods probably cannot predict superfluidic gravity effects with low error. New
The three-body problem as commonly depicted is an abstract math problem wherein other physical fields are not modeled: electrostatic forces are not modeled in the three-body gravity problem as a classical mechanics problem (a Newtonian numerical methods problem).
Were there to be, say, solar wind pushing a three-body system in the vacuum of very cold space containing superfluidic Helium at very low pressure, we would then recognize the need to model solar wind and thus fluids in order to predict the relative positions of n masses with gravity in real physical space after time t.
Is that changing the goal posts? When do ideal 3 point attractor systems exist in isolation outside of abstact mathematics using expensive HPC time?
We observe steady flows in attractor systems which all eventually decay to states with less relativistic motion per Newtonian inertia and the second law of thermodynamics.
But entropy always increases,, so could there be actual 3-body (or n-body) perpetual motion machines in space? Well, there's e.g. solar pressure and superfluidity in space and we model that with fluids: with curl and viscosity, and vortices.
And it is unknown which fluidic solutions the posted numerical n-body solutions must correspond to (if a theory of gravity is suffcient, and any of such solutions are ever experimentally confirmed)
If Newtonian mechanics is not sufficient to model gravity in superfluid systems like Bose-Einstein condensates and Helium in the cold of space, then Newtonian mechanics cannot predict all n-body gravity solutions.
Other intersections between fluids and gravity? Planes, rockets, gliders; for these we must model fluids in order to predict "lift" and "thrust" counter to gravity (which is a weak force).
We often model fluids with Bernoulli's and Navier-Stokes.
Which axioms of relative motion model fluids and gravity in order to actually predict an experimental outcome given initial parameters?
Gravity is observed to be downward at 9.8ms/2 at sea level at many points on Earth.
The relation between gravitational force and distance is an inverse square relation.
Gravity decreases with the square of the distance from the greatest local mass centroid; . The relative gravitational force between objects at twice the distance is 1/(2*2)=1/4 the strength.
Electromagnetic signal power also decreases with the square of the distance. We're familiar with cross sections of EM field lines from e.g. experiments with metal shavings on (electro)magnetic field lines: while the force potential between two points is just one real complex scalar, there's an apparently deterministic unchanging field between two magnetic poles given metal shavings and magnets. But in real experiments, shortwave radio waves in and out and we say it's due to atomospheric disturbance and atmospheres are also fluidic.
Models of relativistic effects of gravity demonstrate the degree to which mass warps space. We like to start with quantized space (a regular 3d grid) and then add objects with mass and velocity; with tabula rasa as a closed system in isolation.
Typically we fail to model other fields due to specialization and lack of time for unified model search (because our solutions are internally consistent with the axioms chosen for simulation). But as with all of physics, real problems occur in real space and "there is yet no known way to subtract the effects of other fields that aren't modeled".
What the chosen predictive axioms fail to model with sufficient predictive error is what we should be concerned with over enumerating additional solutions given a known insufficient model of gravity and other fields. There are various theories of Quantum Gravity (QG), Quantum Field Theory (QFT), and alternative theories of non-quantum gravity. A sufficient theory of quantum gravity must describe n-body gravity within Bose-Einstein Condensates and also quantum levitation.
Newtonian mechanics (classical mechanics) does not explain quantum levitation or quantum locking; which is observably demonstrated in this video of Quantum Levitation of a (nitrogen-chilled) disc on a track formed into a mobius strip: https://www.youtube.com/watch?v=Vxror-fnOL4 and this video https://www.youtube.com/watch?v=f2Z8HyojgLQ. Note that the disc does not level around its mass centroid like maglev trains; the disc retains its locked position independent of gravity until the disc approaches thermal equilibrium with the track as it absorbs thermal entropy.
Newtonian numerical methods do not predict quantum levitation n-body solutions.
A sufficient theory of superfluid quantum gravity must predict for example quantum levitation n-body problems, Bose-Einstein Condensate n-body problems, what we call the Bernoulli effect, and might need to be compatible with GR: General Relativity.
Is downward gravity relevant to modeling the relative motions of objects in free-fall without wind resistance (in a zero-g plane, for example)? What about in microgravity? How does the gravitational mass centroid of the n-body system initially rooted at a zero-gravity Lagrangian point change the centroid of the Lagrangian point? Such dynamics are not modeled with closed-system numerical methods.
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
Ubarfgyl, V jnf rkcrpgvat n fgbel nep jurer uhznavgl qrsrngf gur gevfbynenaf naq ng yrnfg fheivirf, be orggre lrg orpbzrf n cbjreshy enpr va gurve bja evtug.
Rira vs jr qvq nyy qvr bhg, V jnfa'g rkcrpgvat vg gb or qhr gb bar jbzna'f vapbzcrgrapr.
BX, fb lrf, GUNG cneg qvq unccra orpnhfr bs bar jbzna'f vapbzcrgrapr. Ohg gur obbx raqf jvgu n pbzcyrgr qrfgehpgvba naq erovegu bs gur ragver havirefr! Gb zr gur cbvag jnf n erzvaqre gung ng gur raq bs vg nyy, nyy bs uhznavgl, gur jne jvgu Gevfbynevnaf, rira gur hafrra guerng bs gur nyy-cbjreshy nyvraf jnf hygvzngryl zrnavatyrff. Ba n havirefny fpnyr, vg raqf, naq zhfg erfgneg.
Senaxyl gb zr vg'f n ohzzre. V gel gb or na bcgvzvfgvp avuyvfg, ohg nf na ngurvfg vg'f n erzvaqre gb zr gung gur zbfg vzcnpgshy guvat V pbhyq rire qb sbe uhznavgl jvyy arire or rabhtu.
Vg'f bar bs gur ernfbaf jul V ernyyl ybir gur fpv sv pbaprcgf va gur frevrf, ohg pna'g oevat zlfrys gb er-ernq vg.
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 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.
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.
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.
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.
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.
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.
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
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).
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.
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.
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.
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.
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?
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.
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?
62 comments
[ 3.5 ms ] story [ 141 ms ] threadThe “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.
To be sufficient as a theory of (n-body) gravity, a theory of gravity must also describe superfluidic gravity in order to describe gravitational dynamics within e.g. Bose-Einstein condensate superfluids.
Newtonian mechanics are insufficient to describe gravity in superfluids like Bose-Einstein condensates: Newtonian numerical methods do not predict superfluidic gravity effects with low error. Newtonian numerical methods probably cannot predict superfluidic gravity effects with low error. New
The three-body problem as commonly depicted is an abstract math problem wherein other physical fields are not modeled: electrostatic forces are not modeled in the three-body gravity problem as a classical mechanics problem (a Newtonian numerical methods problem). Were there to be, say, solar wind pushing a three-body system in the vacuum of very cold space containing superfluidic Helium at very low pressure, we would then recognize the need to model solar wind and thus fluids in order to predict the relative positions of n masses with gravity in real physical space after time t.
Is that changing the goal posts? When do ideal 3 point attractor systems exist in isolation outside of abstact mathematics using expensive HPC time?
We observe steady flows in attractor systems which all eventually decay to states with less relativistic motion per Newtonian inertia and the second law of thermodynamics.
But entropy always increases,, so could there be actual 3-body (or n-body) perpetual motion machines in space? Well, there's e.g. solar pressure and superfluidity in space and we model that with fluids: with curl and viscosity, and vortices.
And it is unknown which fluidic solutions the posted numerical n-body solutions must correspond to (if a theory of gravity is suffcient, and any of such solutions are ever experimentally confirmed)
One application of n-body gravity problems: "Gravitational Machines" (2023) https://news.ycombinator.com/item?id=36266570
If Newtonian mechanics is not sufficient to model gravity in superfluid systems like Bose-Einstein condensates and Helium in the cold of space, then Newtonian mechanics cannot predict all n-body gravity solutions.
We often model fluids with Bernoulli's and Navier-Stokes.
Which axioms of relative motion model fluids and gravity in order to actually predict an experimental outcome given initial parameters?
Gravity is observed to be downward at 9.8ms/2 at sea level at many points on Earth.
The relation between gravitational force and distance is an inverse square relation.
Gravity decreases with the square of the distance from the greatest local mass centroid; . The relative gravitational force between objects at twice the distance is 1/(2*2)=1/4 the strength.
Electromagnetic signal power also decreases with the square of the distance. We're familiar with cross sections of EM field lines from e.g. experiments with metal shavings on (electro)magnetic field lines: while the force potential between two points is just one real complex scalar, there's an apparently deterministic unchanging field between two magnetic poles given metal shavings and magnets. But in real experiments, shortwave radio waves in and out and we say it's due to atomospheric disturbance and atmospheres are also fluidic.
Models of relativistic effects of gravity demonstrate the degree to which mass warps space. We like to start with quantized space (a regular 3d grid) and then add objects with mass and velocity; with tabula rasa as a closed system in isolation.
Typically we fail to model other fields due to specialization and lack of time for unified model search (because our solutions are internally consistent with the axioms chosen for simulation). But as with all of physics, real problems occur in real space and "there is yet no known way to subtract the effects of other fields that aren't modeled".
What the chosen predictive axioms fail to model with sufficient predictive error is what we should be concerned with over enumerating additional solutions given a known insufficient model of gravity and other fields. There are various theories of Quantum Gravity (QG), Quantum Field Theory (QFT), and alternative theories of non-quantum gravity. A sufficient theory of quantum gravity must describe n-body gravity within Bose-Einstein Condensates and also quantum levitation.
Newtonian mechanics (classical mechanics) does not explain quantum levitation or quantum locking; which is observably demonstrated in this video of Quantum Levitation of a (nitrogen-chilled) disc on a track formed into a mobius strip: https://www.youtube.com/watch?v=Vxror-fnOL4 and this video https://www.youtube.com/watch?v=f2Z8HyojgLQ. Note that the disc does not level around its mass centroid like maglev trains; the disc retains its locked position independent of gravity until the disc approaches thermal equilibrium with the track as it absorbs thermal entropy.
Newtonian numerical methods do not predict quantum levitation n-body solutions.
A sufficient theory of superfluid quantum gravity must predict for example quantum levitation n-body problems, Bose-Einstein Condensate n-body problems, what we call the Bernoulli effect, and might need to be compatible with GR: General Relativity.
Is downward gravity relevant to modeling the relative motions of objects in free-fall without wind resistance (in a zero-g plane, for example)? What about in microgravity? How does the gravitational mass centroid of the n-body system initially rooted at a zero-gravity Lagrangian point change the centroid of the Lagrangian point? Such dynamics are not modeled with closed-system numerical methods.
Gur yrnq srznyr punenpgre va gur 3eq obbx znqr rirel vapbeerpg qrpvfvba vg jnf cbffvoyr gb znxr, naq ol ure vapbzcrgrapr qbbzrq uhznavgl.
Rira vs jr qvq nyy qvr bhg, V jnfa'g rkcrpgvat vg gb or qhr gb bar jbzna'f vapbzcrgrapr.
Senaxyl gb zr vg'f n ohzzre. V gel gb or na bcgvzvfgvp avuyvfg, ohg nf na ngurvfg vg'f n erzvaqre gb zr gung gur zbfg vzcnpgshy guvat V pbhyq rire qb sbe uhznavgl jvyy arire or rabhtu.
Vg'f bar bs gur ernfbaf jul V ernyyl ybir gur fpv sv pbaprcgf va gur frevrf, ohg pna'g oevat zlfrys gb er-ernq vg.
The first has a bit too much lead-up. The third is a bit too philosophical.
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.
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.
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.
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
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.
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.
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...