An atlas of periodic solutions to the three-body problem
threebodyorbits.com167 points by danielmorozoff 2 days ago
167 points by danielmorozoff 2 days ago
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!
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 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.
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).
Quick, someone tell the Trisolarans!
This is beautiful. And fast!!!
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.
This is an amazing looking website, I like it a lot.
This is pretty cool to be able to see all the varieties.
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.
Fantastic UX, so smooth even on smartphone ! Congrats
I assume some of the solutions are stable against small perturbations, while others are not. That would be interesting to see.
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.
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?
I'm going to spend so much time on this website, very good work
SPIROGRAPH, lowtech!
I assume this is at least partially vibe coded, but this is the first good vibe coded website I've seen. Amazing work.
very ai but also pretty cool. wheres the data source? could I find my own periodic solution?
Is this assumed to be 2D? I was going to ask if there are any observed examples of 3 body equilibrium observed in nature.
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.
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.