An atlas of periodic solutions to the three-body problem(threebodyorbits.com) |
An atlas of periodic solutions to the three-body problem(threebodyorbits.com) |
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!
BTW, I've wondered if it's possible to have N-body choreographies in general relativity; maybe some brave person can explore that :)
Get inspired and spend a rest of your life looking for new 3bp solutions.
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.
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?
- 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.
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
Or did I not find the controls?
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
(of course in real life your plane would keep changing, and probably some other complicated math I can't think right now)
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?
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.
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.
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
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.
https://www.threebodyorbits.com/orbit/freefall_part2_f2310_1... (change to a Short trail without a full trace)
Further reading: https://en.wikipedia.org/wiki/Three-body_problem#Solutions
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.
That's a Newtonian approximation. Under General Relativity it is not true - the possible solutions are time varying, and I'm not even sure there are periodic orbits, nor even stable orbits. All N-body patterns radiate gravitational energy till the system collapses. I'm pretty sure there are no general closed form solution, and I think it's true there is not even a single case where the path is a closed form solution.
The 3BP is just the simplest chaotic system showing the limits of simple models and approximations.
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.
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.
Proving that statement is true might be harder than either of the other two issues.
The "about" info states that all of these are computed numerically.
Being unable to state the solution of a problem in closed form doesn't prevent solutions from existing.
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.
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?
Classical motion under an 1/r^2 field is really strange as a dynamical system because the periods of rotation, in-out and up-down motion are all the same which is why the motion closes as an ellipse. It is beautiful in a lot of ways but a godawful mess from the viewpoint of perturbation theory because these are always in resonance as opposed to only in resonance occasionally.
Ignoring gravitational radiation, in GR (or with secular perturbations from other planets) those periods no longer match up so the orbit goes in-and-out in not quite the same time it goes around and then you get a precession so it is still basically an ellipse but the angle of the ellipse changes and you'd see something like a spirograph if you draw it. It's a quasi-periodic orbit.
You can write down closed forms (often with special-function integrals) for things like 1/r^3 and it is the same story, the variables separate nicely because angular momentum is conserved.
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.
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.
For anyone else interested in the math behind the shape sphere coordinates and topological free group classification, I put together an agent skill with code examples here: https://github.com/suvakov/vibes/blob/main/skills/shape_sphe...
(but yes I think you might be right if we're centered on the CG)
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.
[1] ignoring the parameters of that ellipse changing slightly and slowly thanks to the other planets
All of the things in the solution can constantly have momentum.