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!
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...
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.
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.
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.
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.
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.
Well 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.
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).
While 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.
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. ;-)
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.
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 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.
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.
> There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.
This is of course a relative statement. Every object affects every other object, subject to the limitations of lightspeed propagation of gravity waves through expanding space.
But as you point out, we still haven't noticed any examples that are stable short-term.
That depends on what you mean by stable short-term.
The Sun, Jupiter, and any small asteroid at L4 of L5 is, by itself, linearly stable. Meaning that any small perturbation will only grow linearly.
But, of course, this system interacts with Saturn. The arrangement of those three objects is still remarkably stable. But the interaction with Saturn makes for a chaotic system again.
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?
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.
They presumably meant non-zero total momentum. If the total momentum were non-zero, then the center of mass will be moving in a straight line, and will not return to where it began, and therefore the orbit would not be periodic.
Big picture the center of mass momentum is concerned and it does not matter if the system as a whole is moving up or down or to the right or the left. Like the Earth is basically orbiting the sun in an ellipse [1] so far as the sun is concerned and from the viewpoint of the solar system not care so much that it is moving around the galaxy unless we are interested that orbit being perturbed by other stars that we pass near over millions and milions of years.
[1] ignoring the parameters of that ellipse changing slightly and slowly thanks to the other planets
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
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.
Klemperer rosette FTW.
Or trying the solve a Three Body Problem.
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?
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...
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.
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.
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.
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.
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.
Well 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.
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).
Quick, someone tell the Trisolarans!
While 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.
I think we can consider the planet(s) negligible from a mass perspective.
What's the chance that either one of the four bodies crash into another?
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. ;-)
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.
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.
This is an amazing looking website, I like it a lot.
So proud to see the prof. who oversaw my masters thesis has their paper (and solutions) featured here. :D
Very nicely animated. Also didn’t know there can be so many stable solutions for 3 body problem
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.
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.
I'm going to spend so much time on this website, very good work
SPIROGRAPH, lowtech!
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
Fantastic UX, so smooth even on smartphone ! Congrats
This is beautiful. And fast!!!
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?
It has a list of data sources here: 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.
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.
> There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.
This is of course a relative statement. Every object affects every other object, subject to the limitations of lightspeed propagation of gravity waves through expanding space.
But as you point out, we still haven't noticed any examples that are stable short-term.
That depends on what you mean by stable short-term.
The Sun, Jupiter, and any small asteroid at L4 of L5 is, by itself, linearly stable. Meaning that any small perturbation will only grow linearly.
But, of course, this system interacts with Saturn. The arrangement of those three objects is still remarkably stable. But the interaction with Saturn makes for a chaotic system again.
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)
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?
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.
Each orbit is a spinning top. You pull on a top from the side, it's spin axis precesses.
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.
They presumably meant non-zero total momentum. If the total momentum were non-zero, then the center of mass will be moving in a straight line, and will not return to where it began, and therefore the orbit would not be periodic.
Big picture the center of mass momentum is concerned and it does not matter if the system as a whole is moving up or down or to the right or the left. Like the Earth is basically orbiting the sun in an ellipse [1] so far as the sun is concerned and from the viewpoint of the solar system not care so much that it is moving around the galaxy unless we are interested that orbit being perturbed by other stars that we pass near over millions and milions of years.
[1] ignoring the parameters of that ellipse changing slightly and slowly thanks to the other planets
Oh. That makes sense. Yes. The total momentum has to be zero.
All of the things in the solution can constantly have momentum.
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
cool!