An atlas of periodic solutions to the three-body problem (threebodyorbits.com)

258 pointsby danielmorozoff2 days ago56 comments

pixelpoet 5 hours 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!

mr_mitm 5 hours ago

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!

ironSkillet 5 hours ago

Something cool and interesting for its own sake. A rare find.

addaon 2 hours ago

Pretty useful if you’re a Puppeteer, though.

Espressosaurus 2 hours ago

Or trying the solve a Three Body Problem.

doormatt 3 minutes ago

Klemperer rosette FTW.

cobbzilla 2 hours ago

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?

chris_st 2 hours ago

A 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 an hour ago

further 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.

larodi 17 minutes ago

So proud to see the prof. who oversaw my masters thesis has their paper (and solutions) featured here. :D

deskamess 7 hours ago

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?

incognito124 7 hours ago

That's exactly the case

isolli 6 hours ago

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).

layer8 5 hours ago

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

Sharlin 5 hours ago

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.

mr_mitm 5 hours ago

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.

dreamcompiler 5 hours ago

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.

IAmBroom 2 hours 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 2 hours ago

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.

PaulHoule an hour ago

In 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 12 minutes ago

It'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.

sajithdilshan 35 minutes ago

Very nicely animated. Also didn’t know there can be so many stable solutions for 3 body problem

inatreecrown2 6 hours ago

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.

MeteorMarc 6 hours ago

I assume some of the solutions are stable against small perturbations, while others are not. That would be interesting to see.

summa_tech 6 hours ago

I think that's what "STABLE ONLY" clickable text filters by.

QuesnayJr 4 hours ago

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.

jrflo 5 hours ago

Quick, someone tell the Trisolarans!

m4rtink 3 hours ago

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. ;-)

hakuseki 3 hours ago

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.

pletnes 3 hours ago

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.

RALaBarge 6 hours ago

This is an amazing looking website, I like it a lot.

dreamcompiler 5 hours ago

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.

moritzwarhier 5 hours ago

Wow. This is really cool. Deterministic chaos is my absolute favorite in all the nerdy things there are to like in the abstract world.

edbaskerville 3 hours ago

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

boringg 4 hours ago

This is pretty cool to be able to see all the varieties.

glitchbot 5 hours ago

SPIROGRAPH, lowtech!

shagie 2 hours ago

Wild Gears for the modern physical version.

There's also Spirograph Simulator (2014) https://news.ycombinator.com/item?id=13256222 from... oh, a decade ago (I feel old). It still works. https://inspiral-web.nathanfriend.com

kmitz 5 hours ago

Fantastic UX, so smooth even on smartphone ! Congrats

khalic 6 hours ago

I'm going to spend so much time on this website, very good work

freakynit 4 hours ago

This is beautiful. And fast!!!

ur-whale 4 hours ago

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?

RugnirViking 7 hours ago

very ai but also pretty cool. wheres the data source? could I find my own periodic solution?

IshKebab 5 hours ago

I assume this is at least partially vibe coded, but this is the first good vibe coded website I've seen. Amazing work.

nautilus12 6 hours ago

Is this assumed to be 2D? I was going to ask if there are any observed examples of 3 body equilibrium observed in nature.

raverbashing 6 hours ago

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)

raincole 6 hours ago

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?

raverbashing 6 hours ago

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)

twnettytwo 5 hours ago

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.

btilly 3 hours ago

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.

hammock 5 hours ago

It’s an arbitrary plane, chosen at each moment just so you can flatten it

mr_mitm 5 hours ago

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.

crisf an hour ago

Each orbit is a spinning top. You pull on a top from the side, it's spin axis precesses.

nautilus12 3 hours ago

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

petsfed 2 hours ago

Put another way, while their positions are one set of 3 points, their momenta are another set of 3 points, and there is no requirement that 6 points will always lay on the same plane.

I wonder what phantom forces would appear when the reference frame changes in some complicated fashion. We get centrifugal "force" when we reconstruct F=dP/dt in a rotating reference frame, what would the 3-body "force" look like?

btilly 3 hours ago

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.

hanw040519 3 hours ago

cool!