It'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!
I 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?
The 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?#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.
The 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.
There 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.
I'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.
Yes. 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.
It'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).
I 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.
They’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.
Very 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.
Given 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. ;-)
If 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.
I 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.
Deterministic 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
We have no observed examples in nature of three body equilibrium. But then again, all places we have looked are either influenced by the chaotic orbits around them of the Solar System, our surrounding galaxy, or nearby galaxies in a cluster.
There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.
Why 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?
One 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.
Oh makes perfect sense, do you have thoughts about real life examples? I did some research using AI and it said there were examples of restricted 3 body problems like the trojan asteroids, but no examples in real life similar to what is in this web app
It’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?
I have something similar at https://gravitoy.xyz
With it I have discovered up to 11-dimensional choreographies, see https://lycium.github.io/hyperchoreography/ and code at https://github.com/lycium/hyperchoreography/
Exposition video: https://youtube.com/watch?v=sIfff10hYZA
Example rendered output from Gravitoy (not of a choreography though): 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!
It'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!
Something cool and interesting for its own sake. A rare find.
Pretty useful if you’re a Puppeteer, though.
Or trying the solve a Three Body Problem.
I 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?
The 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?#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.
The 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
There 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.
I'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.
Yes. 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.
> 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.
That's exactly the case
It'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).
I 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.
They’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.
Very 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.
Quick, someone tell the Trisolarans!
Given 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. ;-)
This is an amazing looking website, I like it a lot.
I assume some of the solutions are stable against small perturbations, while others are not. That would be interesting to see.
I think that's what "STABLE ONLY" clickable text filters by.
If 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.
I 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.
Wow. This is really cool. Deterministic chaos is my absolute favorite in all the nerdy things there are to like in the abstract world.
Deterministic 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
This is pretty cool to be able to see all the varieties.
SPIROGRAPH, lowtech!
Fantastic UX, so smooth even on smartphone ! Congrats
This is beautiful. And fast!!!
I'm going to spend so much time on this website, very good work
Oh, 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?
very ai but also pretty cool. wheres the data source? could I find my own periodic solution?
cool!
I assume this is at least partially vibe coded, but this is the first good vibe coded website I've seen. Amazing work.
Is this assumed to be 2D? I was going to ask if there are any observed examples of 3 body equilibrium observed in nature.
https://numericaltank.sjtu.edu.cn/three-body/three-body.htm shows that there are three dimensional solutions.
We have no observed examples in nature of three body equilibrium. But then again, all places we have looked are either influenced by the chaotic orbits around them of the Solar System, our surrounding galaxy, or nearby galaxies in a cluster.
There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.
Yes. 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)
Why 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?
If you define the initial conditions such that their relative velocity is zero or parallel to the plane, yes. But that's not the case in general.
It’s an arbitrary plane, chosen at each moment just so you can flatten it
Not 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)
One 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.
Sorry, 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 includes periodic solutions that move in all three dimensions.
Oh makes perfect sense, do you have thoughts about real life examples? I did some research using AI and it said there were examples of restricted 3 body problems like the trojan asteroids, but no examples in real life similar to what is in this web app