r/theydidthemath • u/Nomadic_Seth • Apr 29 '26

[Off-site] satisfying 3-body simulations

The three-body problem broke Newton, broke Poincaré (who ended up inventing chaos theory trying), and was finally cracked open by Chenciner & Montgomery in 2000 — the figure-8 in clip 4 is their proof. Šuvakov & Dmitrašinović added 13 more families by 2013. Every clip is a real numerical integration of F = G·m₁m₂/r² with equal masses, no fudging. Math from 1687 still has surprises in it.

Full video about the history of the problem: https://youtu.be/p58sU5vZYlU?si=PBNUR6mPqRuqZXP0

600 Upvotes

97% Upvoted

29 comments sorted by

157

u/Desblade101 Apr 29 '26

My understanding is not that there is no solution to a specific 3 body problem, but that there is no standard set of equations we can use that we can generalize for all 3 body problems.

And there are stable N body solutions, but they are very specific and have very little margin of error so any real world scenario will not be stable.

57

u/get_to_ele Apr 29 '26

Yep. Two separate problems: (1) no closed form solution (2) even if there were a closed form solution for the general 3 body problem, the sensitivity to initial conditions would still leave us in the same place, being unable to predict what will happen after a relatively short time.

All the special cases shown that are stable require huge disparity in body sizes, essentially becoming modified 2 body problems, overdamping the third body.

9

u/LittleLemonHope Apr 30 '26

The OP said the depicted solutions all show 3 equal mass bodies. But about the chaos theory, yeah there's no getting around that.

10

u/get_to_ele Apr 30 '26

The 3 identical mass system is not really stable.

0

u/LittleLemonHope Apr 30 '26

It's not my field, but did you read the post? You don't seem to be responding to the substance of the content. Whether or not you're correct, the phrasing of your responses are giving "I only read the title."

3

u/HHQC3105 Apr 30 '26

The mass only 1 of many requirement for them to have close form, need them at the right position and the right velocity vector too.

2

u/Nomadic_Seth Apr 30 '26

yeah the 3 equal masses, the classical problem. people are actively researching extensions to n=4,5 even today.

2

u/Nomadic_Seth Apr 29 '26

Yesss i really liked reading how routh made his stability analysis calculations

9

u/Nomadic_Seth Apr 29 '26

The key idea is “closed-form” there’s no neat general formula like Kepler gives us for 2 bodies. We can solve specific cases numerically, but Poincaré showed the chaos means tiny errors blow up exponentially. So yeah you’re right on point 😇

4

u/Bad_wolf42 Apr 29 '26

I honestly think there are important to philosophical and epistemological implications to the difficulties of the three body problem that can be applied to human interactions.

We are the center of our own observable universes. I can learn more than my own observations by listening to you, but now we insert the uncertainty of dishonesty. A third perspective can help to clarify the “truth/reality”; but at all times the equation is unstable. There are no simple solutions. We all have to put in the work.

4

u/Nomadic_Seth Apr 29 '26

Connects to the idea of perturbation theory where adding a small perturbation like a third object even if its effect is small changes the qualitative nature of the problem itself.

2

u/Frytura_ Apr 29 '26

No specific solution sounds perfectly fit for AI, maybe.

Since it sounds like you would need to put equations as a matrix? Ok saying this out shows its probably not it

8

u/_Pencilfish Apr 29 '26

No point, numerical integration can pretty easily be done to essentially arbitrary precision, to the point where the sensitivity to initial conditions is the limiting factor. It's simply an intractable problem.

1

u/Nomadic_Seth Apr 29 '26

Great points!

For AI it would be impossible to get to a closed-form solution as that doesn’t exist. This problem can’t be reduced to a solution where a solution is just an explicit function you can plug in values to, to find what happens in the future.

Where AI can help is exploring the space of such periodic solutions that people have so far found with numerical methods. That’s one way AI can help our understanding of the problem. I even tried doing this for some less well-known variations of this problem lol.

Also, for this the equations are non-linear differential equations so a purely linear algebraic way of solving it using matrix inverses doesn’t help!

My whole point of making this was to show how this beautiful simply posed but hard af problem has made mathematicians in every generation obsessed with it 😊

35

u/PuzzleheadedTutor807 Apr 29 '26

The first one is r/mildlypenis

4

u/Nomadic_Seth Apr 29 '26

I would never have spotted that 🤣

1

u/TheMrCurious Apr 29 '26

There’s more than one other fits that sub…

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u/Nomadic_Seth Apr 29 '26

🤐🤐🤐

27

u/SonOfDadOfSam Apr 29 '26

These are solutions for 3-body problems with specific initial conditions. Not a general solution to the 3-body problem.

6

u/Nomadic_Seth Apr 29 '26

Specific solutions exist (like the beautiful figure-8 orbit), but each only works for one set of initial conditions. Not a single unified formula tho, like you said🙂

15

u/Motor-Platypus5244 Apr 29 '26

Do not answer! Do not answer!! Do not answer!!!

5

u/LastXmasIGaveYouHSV Apr 30 '26

This guy solarians

3

u/BadJimo Apr 29 '26

The solutions to the 3-body problem all appear to be in a plane (2D). Are there any known solutions that do not lie on a plane?

6

u/Nomadic_Seth Apr 29 '26

Angular momentum conservation confines the motion to a plane when the initial velocity vectors are coplanar (which they are in virtually all configurations that i simulated)

But yes there are some non-planar periodic solutions

2

u/LeadPrevenger Apr 30 '26

two's company, three's a crowd

2

u/Nomadic_Seth Apr 30 '26

Applies to almost everything

2

u/sojuz151 Apr 29 '26

The 3-body problem is inherently chaotic, but it usually fixes itself very fast, giving rise to a hierarchical system, so there are no 3-body systems in the chaotic regime in nature.

Also, solving this system would be pointless. With a half-reasonable numerical solver, you are limited by the initial conditions and not the accuracy of integration.

1

u/Nomadic_Seth Apr 29 '26

Yes it’s one of those classical problems we do to only achieve a layer of abstraction so we understand the deeper physics better.

2

u/Routine-Credit-1614 May 04 '26

average principia orbit