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An atlas of periodic solutions to the three-body problem
- khalic 17d agoI'm going to spend so much time on this website, very good work
- RugnirViking 17d agovery ai but also pretty cool. wheres the data source? could I find my own periodic solution?
- porphyra 17d agoIt has a list of data sources here: https://www.threebodyorbits.com/about https://www.threebodyorbits.com/about This whole thing does have a strong "GPT-6 Astra look" with the big text on the left style. But it is indeed very cool.
- deskamess 17d agoI guess I misunderstood or mis-scoped the problem. Does the 3-body problem state 'the general case' has no solution, but that does not preclude some configurations from having a solution?
- incognito124 17d agoThat's exactly the case
- isolli 17d agoIt's also not computable, as in chaotic. Small differences in initial positions will lead to unpredictably large differences in trajectory (with small and large having specific meanings to match the formal definition of a chaotic system).
- layer8 17d agoThe three-body problem only states the problem to solve, it doesn’t itself state anything about the existence or non-existence of solutions. It has been proven that there is no general closed-form solution. And there are obvious solutions for trivial special cases, such as three equal masses in an equilateral triangle rotating around each other. Further reading: https://en.wikipedia.org/wiki/Three-body_problem#Solutions https://en.wikipedia.org/wiki/Three-body_problem#Solutions
- Sharlin 17d agoThere is no closed-form solution for finding the roots of >4th degree polynomials in general, but that doesn’t preclude many families of >4th degree polynomials from having closed-form solutions. As a trivial example, x^5 - 1. The exact same thing with the three-body problem.
- mr_mitm 17d agoI'm pretty sure there is always a unique solution to the equations of motions (safe for some pathological edge cases perhaps). Classical mechanics is deterministic, after all. But for more than two bodies, there is in general no solution in closed form, and it's often chaotic, so not even computeable for arbitrary time frames. The "about" info states that all of these are computed numerically.
- jrflo 17d ago[dead]
- dreamcompiler 17d agoYes. It's kind of like the halting problem: You cannot write a general computer program that will analyze the source code of any random other computer program and tell you if it will halt. You can write a program that will analyze the code of a few specific other programs and tell you if they will halt. You just can't do it in general. The 3-body problem is like that. Except it's much harder to find stable 3-body problems than computer programs that are predictable.
- IAmBroom 17d ago> Except it's much harder to find stable 3-body problems than computer programs that are predictable. Proving that statement is true might be harder than either of the other two issues.
- btilly 17d agoThe general case always has a solution. At least until the point where two of the three bodies meet (which is a singularity). We can approximate that solution numerically. The problem is that the solutions very strongly tend to be chaotic. Meaning that small differences in initial conditions, tend to grow exponentially with time. Which means that if you measure everything to 3 digits of precision, in finite time it will stop looking like the actual solution. Every additional digit of precision adds a similar finite time to how long the approximation is good for. So when finally found, say, the 1953 BC conjunction described in https://en.wikipedia.org/wiki/Conjunction_%28astronomy%29?#Notable_conjunctions https://en.wikipedia.org/wiki/Conjunction_%28astronomy%29?#N... - that was a very good stress test for our estimated planetary data. Because surprisingly small errors in modern data would have kept that conjunction from happening.
- PaulHoule 17d agoIn the case of a planet going around the sun we know that the planet travels in an ellipse, more generally a conic section including interstellar comets. Orbital periods and everything else can be computed straightforwardly with formulas. In the three body problem you can always do a numerical integration (e.g. simulation) and this is valid for a certain amount of time but will not be valid forever because of: (1) chaotic motion which amplifies small errors exponentially over time and (2) celestial mechanics is symplectic which means it conserves certain geometric properties and most integrators are not symplectic and must have different long term dynamics. There are symplectic integrators but they don’t work as well overall as ordinary integrators. We do not know, for instance, if the solar system is stable. In the short term the planets seem to be basically doing their own thing in their orbits with just minor perturbations. We know the orbits vary a bit over millions of years. We aren’t sure which side of the sun Pluto will be on in 30 million years. It’s very believable that the planets are doing the same thing in 4 billion years but we can’t rule out that the orbits could change in a big way or one could get ejected.
- kurthr 17d agoIt's worth noting that the Sun-Earth-Moon "system" is much more complex than a three-body problem just due to momentum transfer from tides/bending. That's ignoring the effect of all of the other planets and asteroids or solar wind. The 3BP is just the simplest chaotic system showing the limits of simple models and approximations.
- PaulHoule 17d agoWell there are simpler chaotic Hamiltonians like Henon–Heiles. There are a lot of things wrong with how we teach classical mechanics and one of them is that the two index problems that are used in the undergraduate course are the harmonic oscilator and celestial dynamics and these are the worst non-generic problems that there are. The harmonic oscillator is generic in quantum theory and is a good place to start doing perturbation theory from but the fact that the frequency doesn't change with amplitude makes it non-starter in classical perturbation theory. In the 2-body case the periods to: (1) go around the sun, (2) go in and out towards the sun, and (3) go up and down out of the plane are all the same which again breaks perturbation theory... and of course classical perturbation theory is difficult and doesn't work that well even in cases where it does work. Contrast that to quantum mechanics where you can start doing simple calculation with perturbation theory, like to calculate the lifetime of an excited state, right away with paper and pencil. Note that, in general, you can get chaos with as few as two position variables. In the case of the two body problem you have six position variables, but because of conservation of momentum the center of mass doesn't really matter, so you can pick a coordinate frame where the total angular momentum is zero and there are just three variables that matter which is the vector between the two bodies. There are a lot of conserved quantities here, especially angular momentum so the (1) and (2) and (3) motions all do the same thing coincidentally with the same period! One you add a third body you are adding three more variables but not any more conserved quantities so it is a problem with a lot of dimensions. Mapping out the orbits of the three body problem has gone pretty slowly because, compared to simpler Hamiltonians, we have to search for those orbits in a high dimensional space. You can find a periodic orbit numerically if you know where it is, but it takes a systematic approach to find many of them.
- coderenegade 17d agoThere is a general solution to the three body problem due to Sundman, in the form of an infinite series. Unfortunately it converges so slowly that it's useless in practice. Being unable to state the solution of a problem in closed form doesn't prevent solutions from existing.
- RALaBarge 17d agoThis is an amazing looking website, I like it a lot.
- inatreecrown2 17d agoVery cool visuals and site! Could I make a suggestion: You show the masses (1,1,1), but not the starting positions, which alter the course of events too.
- MarkusQ 16d agoAny set of points (w. velocities) along the trajectories could be the "starting positions".
- nautilus12 17d agoIs this assumed to be 2D? I was going to ask if there are any observed examples of 3 body equilibrium observed in nature.
- raverbashing 17d agoYes. Because 3 points are coplanar, so every "3D problem" with 3 objects can be turned into a 2D problem on the correspondent plane (of course in real life your plane would keep changing, and probably some other complicated math I can't think right now)
- raincole 17d agoWhy would the plane keep changing? If there are only these three objects, won't the vectors of their gravitational pull to each other all be on this plane too?
- raverbashing 17d agoNot from an external point of view, as you might have a momentum component perpendicular to that plane (but yes I think you might be right if we're centered on the CG)
- twnettytwo 17d agoOne of the necessary conditions here is that the three objects return to their exact initial position, and so does the centre of mass. Initial conditions with non-zero momentum must trivially be ruled out. But this doesn't stop them from having velocities perpendicular to the initial plane that cancel out perfectly, so this doesn't refute the assertion that the planes keep changing.
- btilly 17d agoSorry, but this is a word salad. Most of the solutions always have non-zero momentum, including in the initial conditions. And https://numericaltank.sjtu.edu.cn/three-body/three-body.htm https://numericaltank.sjtu.edu.cn/three-body/three-body.htm includes periodic solutions that move in all three dimensions.
- MeteorMarc 17d agoI assume some of the solutions are stable against small perturbations, while others are not. That would be interesting to see.
- summa_tech 17d agoI think that's what "STABLE ONLY" clickable text filters by.
- QuesnayJr 17d agoIf you go to the individual solutions, the text description tells you if it's stable. There's also a slider that allows you to perturb the orbit so you can see for yourself when you perturb it.
- IshKebab 17d agoI assume this is at least partially vibe coded, but this is the first good vibe coded website I've seen. Amazing work.
- moritzwarhier 17d agoWow. This is really cool. Deterministic chaos is my absolute favorite in all the nerdy things there are to like in the abstract world.
- edbaskerville 17d agoDeterministic chaos is cool. But this is even more special in a sense: for a problem where random initial conditions are almost always chaotic, this is a catalog of periodic orbits—these are all non-chaotic. It would be cool to pair this with a numerical simulator that shows what happens when you perturb any of them. EDIT: oh, it already does this, thanks other comments
- mr_mitm 17d agoIt's well organized, the animations are smooth, and it looks beautiful... I'm not sure what to do with the information, but it's mesmerizing and fascinating. Great find!
- ironSkillet 17d agoSomething cool and interesting for its own sake. A rare find.
- addaon 17d agoPretty useful if you’re a Puppeteer, though.
- doormatt 17d agoKlemperer rosette FTW.
- Espressosaurus 17d agoOr trying the solve a Three Body Problem.
- ivanjermakov 17d ago> I'm not sure what to do with the information Get inspired and spend a rest of your life looking for new 3bp solutions.
- dice 17d agoHandy if you get really good at it because "the rest of your life" will be indicated by the unavoidable countdown timer that shows up in your vision!
- dreamcompiler 17d agoI took graduate orbital mechanics from Roger Broucke. He was one of my best professors. Not only did I learn from him what orbital elements were, but he also taught me the Runge-Kutta numerical integration method. I didn't learn until years later that he had discovered several of the periodic solutions to the three-body problems. You'll see his name on this page.
- jrflo 17d agoQuick, someone tell the Trisolarans!
- m4rtink 17d agoGiven all the illogical insanity they are doing (like, trying to fight an interstellar war instead of just the simple stuff like moving to habitats, improving their bodies or even so,me stellar lifting) I don't think the will listen. ;-)
- criemen 17d agoWhile I love the books as much as everyone else, the pedant in me has to point out that the Trisolareans have a four body problem at hand (3 suns and their planet), not a three body problem.
- hypersoar 17d agoThe Trisolarians are supposedly at Alpha Centauri, a trinary star that isn't actually that chaotic. It has two sun-like stars orbiting each other quite close, and then a red dwarf, Proxima Centauri, orbiting that pair really really far away.
- glitchbot 17d agoSPIROGRAPH, lowtech!
- shagie 17d agoWild Gears for the modern physical version. There's also Spirograph Simulator (2014) https://news.ycombinator.com/item?id=13256222 https://news.ycombinator.com/item?id=13256222 from... oh, a decade ago (I feel old). It still works. https://inspiral-web.nathanfriend.com https://inspiral-web.nathanfriend.com
- kmitz 17d agoFantastic UX, so smooth even on smartphone ! Congrats
- pixelpoet 17d agoI have something similar at https://gravitoy.xyz https://gravitoy.xyz With it I have discovered up to 11-dimensional choreographies, see https://lycium.github.io/hyperchoreography/ https://lycium.github.io/hyperchoreography/ and code at https://github.com/lycium/hyperchoreography/ https://github.com/lycium/hyperchoreography/ Exposition video: https://youtube.com/watch?v=sIfff10hYZA https://youtube.com/watch?v=sIfff10hYZA Example rendered output from Gravitoy (not of a choreography though): https://www.youtube.com/watch?v=N3BwCoiwsGk https://www.youtube.com/watch?v=N3BwCoiwsGk Looks like I need to update my catalogue to take into account the many different 2D choreographies from the references on this site!
- emmelaich 17d agoThat Pythagorean one with the ejection is cool. I wonder whether you could use it to accelerate a spacecraft.
- SiempreViernes 16d agoCool, are any of these particularly stable? Like, is there any choreography that could plausibly survive in practice?
- pixelpoet 16d agoMost of them are extremely unstable, which is what the Morse number encodes - the number of directions you can roll down doing action minimisation. I don't think any N-body choreographies can exist in nature, because it requires point masses, exactly equal masses, Newtonian gravity etc. BTW, I've wondered if it's possible to have N-body choreographies in general relativity; maybe some brave person can explore that :)
- ur-whale 17d agoOh, this is so nice, but man is it hugely frustrating not to be able to rotate the thing in 3D. Or did I not find the controls?
- boringg 17d agoThis is pretty cool to be able to see all the varieties.
- freakynit 17d agoThis is beautiful. And fast!!!
- hanw040519 17d agocool!
- hakuseki 17d agoI was surprised that I couldn't find any simple-looking solutions in this atlas. At first I was looking for Lagrange orbits, but maybe it makes sense to exclude them if zero-mass bodies aren't allowed. I think the equilateral triangle ought to be included though.
- pletnes 17d agoThey’re not stable except at L4 and L5, and they all assume oke body to be massless. Arguably they are 2-body orbits for that reason. Not sure but suspect that this atlas contains non-massless bodies.
- cobbzilla 17d agoIt’s my understanding that most 3-body orbits are unstable; minor random perturbations will set them adrift. Are there any 3-body orbits with a natural resonance that maintains the shape of the orbits? If so, how large of a disturbance can the most-stable 3-body orbit withstand?
- chris_st 17d agoA few that I looked at are annotated, "It is linearly stable (largest Floquet multiplier 1.000), so a small nudge only makes it wobble." Doesn't define "small nudge", alas...
- cobbzilla 17d agofurther research indicates some do exist: - equilateral triangle is the simple case - some figure eight configurations - some configurations that act like two nested binary systems there might be others. fascinating stuff.
- slwvx 17d agoOne interesting scenario is where the Sun, Jupiter, and another object are the three bodies in question. I think we could put an object at the L4 or L5 point of Jupiter's orbit (see the Trojans) and have it be stable. How large could this object be? Would an Earth-sized object be stable there? If so, how close would a passing star have to be to disturb that stability?
- sajithdilshan 17d agoVery nicely animated. Also didn’t know there can be so many stable solutions for 3 body problem
- larodi 17d agoSo proud to see the prof. who oversaw my masters thesis has their paper (and solutions) featured here. :D
- amenghra 17d agoImagine the tides if our system were https://www.threebodyorbits.com/orbit/simo2001_sc002_1_1_1 https://www.threebodyorbits.com/orbit/simo2001_sc002_1_1_1
- shagie 17d agoI find the freefall ones to be... more wild. https://www.threebodyorbits.com/orbit/freefall_part2_f2310_10_01 https://www.threebodyorbits.com/orbit/freefall_part2_f2310_1... (change to a Short trail without a full trace)
- threebodyorbits 17d agoHi all, I just added a new feature to the site. You can now contribute compute via your browser to help find novel orbits. If you find a linearly stable orbit, you get to name it! https://www.threebodyorbits.com/hunt https://www.threebodyorbits.com/hunt Also thank you so much for all the positive feedback, I truly appreciate it. If you have any additional suggestions please let me know
- altairprime 17d agoHow did you end up choosing the color scheme for this site?
- threebodyorbits 16d agoThis whole project was inspired by the Cixin Liu Three Body Problem books. I tried to capture the essence and feelings that I had when reading them in the color scheme and design of the website. I wanted a black background, representing the darkness, vastness, and mystery of space. For the orbits and other elements, I wanted the dominant color to represent danger and death, but also feel like a celebration of life and beauty. I thought blood-related colors best fit that purpose
- msuvakov 17d agoReally cool project! On a personal note, thanks for the reference! I'm one of the authors of the 2013 paper you cited, and it's awesome seeing how many new solutions have been found since then. I have a quick question: how are you currently checking if a new candidate is genuinely novel rather than a rotated, time-shifted, or rescaled duplicate of an existing solution? If you are not already computing it, topological classification (mapping orbits to free group words) would be a great way to quickly prune duplicates before heavy numerical checks, and it's also really useful metadata to display in the atlas itself. Also, showing shape sphere projections of the orbits would be a nice visual addition.
- threebodyorbits 16d agoThank you very much, I'm so excited that an original author of one of the three body problem papers that this atlas is based on found this. After you sent your question I noticed an issue with the "hunt" settings. Orbit novelty was not properly assessed, and it just kept re-discovering already existing orbit families. I updated the search based on your suggestions and just submitted an update that now checks candidates like this: - Does the motion really repeat? An independent more high precision periodicity check is performed to confirm that bodies truly return to their starting positions and velocities - Is it a different form of an existing orbit? Compares against reference orbits from the atlas, allowing for rotation, different starting points in time, rescaling, reflection, and relabeling of bodies -Is it just the same orbit traced several times? Checks for shorter repeated periods -Now topological information is taken into account as well. In the current version, I use reduced syzygy sequences that record which body is in the middle whenever the three bodies are lined up. This was easier to implement than the familiar a, b, A, B notation. Also, new candidates here are not automatically declared as new discoveries. They will undergo additional validation before they are added to the atlas. The shape-sphere suggestion sounds great - would you suggest adding this to the "hunt" page, or to every orbit in the atlas?
- sscaryterry 17d agoHoly fuck this is one of the coolest things I've seen.
- r3trohack3r 17d agoI would totally run hunt as an electric sheep[1] style screensaver [1] https://en.wikipedia.org/wiki/Electric_Sheep https://en.wikipedia.org/wiki/Electric_Sheep
- threebodyorbits 16d agoGreat suggestion, I added an option to have it in full screen mode that prevents sleep - click "Open Observatory" to enter full screen mode