Submitted by FishFollower74 t3_z3qfy2 in askscience
I just watched a YouTube video that explains the three body problem and it states that the problem is unsolvable. But I don’t understand why.
As I understand it we can run computer simulations that can show what happens with 3 bodies rotating around each other. But if we can simulate it why can’t it be solved for with a function?
This is really helpful and informative…thank you!
I think the issue is deeper than just that we want to write the solution in the specific form of using only elementary functions. The real problem is that the solution is chaotic, which is an inherent mathematical fact that has nothing to do with what functions we want to use. The reason why we want to solve with elementary functions in the first place is because they have very predictable behavior; for example, you can make long range predictions without much difficulty without increasingly large errors. We could introduce new functions (and people do, these are often taught under special functions), but other "nice" functions won't solve the problem, and the one that does solve the problem would be chaotic and hard to analyze. Ultimately, the main issue is that the general solutions to the 3-body problem is just too chaotic that it is resistance to analysis.
Thanks for the insights.
Can you explain what makes the problem chaotic in nature? I'm a software developer and I understand that a physics simulation might not be deterministic due to floating point math etc, but I also know that if I take a bit of special care I can program fully deterministic physics simulation in a sense that it will repeatedly give me same results.
So I can't wrap my head around why a 2-body problem is not chaotic and will always run the same thing at, but 3-body is not? What am I missing?
Chaotic is different from stochastic. Stochastic means there are randomness involved in the evolution of the state. 2-body and 3-body problems are deterministic, not stochastic. The state always evolve the same given the exact same initial condition.
But for 3-body problem, it's chaotic. If you don't have exact values for the initial condition, the error became exponentially large as time go by, so after a certain amount of time the state became essentially unpredictable (but there are special exceptions). If you do have exact values for initial condition, then you can make arbitrarily accurate prediction for arbitrary long period of time, but you will need to perform a lot more calculations compare to non-chaotic case to control this exponentially growing error (you always acquire error due to numerical imprecision). Chaotic implies a few properties. One is the butterfly effect, as I mentioned above. Another one is mixing: it's not merely that you can't predict precisely if time is long enough and you don't have the exact initial condition, you can't even make a vague estimate that carry any useful information at all.
Why is 2-body problem not chaotic? Essentially, it has too few variables compared to the amount of symmetry. It's known that if you have at most 2 free variables you can't be chaotic. A 2-body problem has 12 variables (position and velocity for each body), but standard physics 10 symmetries gives you 10 constant of motions (center of mass, linear momentum, angular momentum, energy) so the problem is reduced to 2 dimensions. Actually, there is a special 11th constant of motion specific to this problem: the LRL vector, but only its direction on the plane of motion matter, because the plane of motion and length is determined by angular momentum and energy, so the problem is reduced to just 1 dimension. 1 dimensional system is very easily solvable explicitly.
For 3-body problems, you start with 18 variables, but you only have the usual 10 symmetry. It's proven that there are no other algebraic constant of motions, so at most you can reduce this to an 8 dimensional problem using this technique. Actually showing that this problem is chaotic (and hence you really can't reduce further) is harder.
Thank you! This was a mind-opener. I knew I was missing something and you gave an excellent explanation!
Most interesting problems are unsolvable in the algebraic sense. Some are very hard, and we don't know whether a solution exists (tethered goat), others are proven not to have a formulaic solution.
Wolfram likes to write about that a lot, check him out
This may interest you. Apparently the Tethered Goat problem was solved.
Solved with caveat - new math symbols were introduced, iirc. Could just call the area of the grazing goat "Ґ", much like we do π.
That's the problem with the Poincare conjecture solution as well: a new area of math was developed just for it.
There are probably 10-100 people in the world who can follow Perelman's proof, and fewer who are qualified to find flaws in it.
Offloading interesting problems to AI will make the problem worse, there will be a lot of "trust me, I am a computer* moments with AI
> Solved with caveat - new math symbols were introduced, iirc.
The closed-form solution found by Ingo Ullisch is on the Wikipedia page for The Goat Problem. It's expressed in terms of pi, sin, cos, and contour integrals.
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Great, thanks!
One outside perspective I've heard describing a potential solution was actually in relation to the Fermi Paradox.
IIRC, in short, it basically states that humanity would be better off separating the small and the big because our ability to accept different forms of intelligent life is so vast.
For example: if intelligent life was capable of living with the type of stars that were present earlier in the universe, their skin color would become saturated in the light from stars of that kind. So, like the idea of people being Blue makes sense if we can take a step back and try to articulate what to generally look for.
I personally feel that once humanity can take this sort of leap, we might be able to work our way back to an understanding that's more in line with why things work the way they do for us here, and them there.
A catalyst is basically missing.
Some think we can reach some solutions to the 3 body problems if we apply a "free fall" catalyst. Others, if we apply the ticking time bomb from those 2 black holes merging 9bya, as if they already merged and it just has yet to hit the rest of the Universe (why everything appears to be accelerating away from each other). Whiplash catalyst at some points. Etc.
How time/space knots up, maybe where the solution of chaos can be broken down into various solutions that are not necessarily repetitive in nature. Meaning the hidden plane of our Solar System, the Milky Way, etc. We don't have all the various planes that impact our physics inserted into the solution yet.
Outside of the solution hiding inside of a black hole where intelligence can only explain it once on the other side, we should figure it out eventually.
L-Points definitely feels like the right idea though. Maybe on a bigger scale though if we want to be able to explain the cosmos. We obviously are sitting on one very special "L-point" of the Cosmos, and we'll figure it out eventually.
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There are so many unsolvable differential equation problems, and the three-body problem is still "abstract", at least the planet can be regarded as a mass point. But some problems cannot be transformed into particles, such as real-time collision detection of two-dimensional space line segments, even if it is only two finite-length line segments with independent angular acceleration functions and linear acceleration functions, and these acceleration functions are simple real coefficients Polynomials, the collision detection equations established with this product are still very complicated.As long as it is an "ordinary" physical problem that is close to reality, especially when it involves collisions, human beings are really helpless. For example, there is a theory of collisions and entropy changes in chemistry, and there has not been much progress so far.
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>As I understand it we can run computer simulations that can show what happens with 3 bodies rotating around each other.
I watched a show where they used the same simulation program to simulate the 3 body problem on three different computers. One was 16 bit, one was 32 bit and one was 64 bit. And the program ran differently on each. Even though it was the same exact program. So it all depends on what computer you run the simulation on. But there is not a single answer, which is why it's unsolvable.
functor7 t1_ixoa1xu wrote
What is meant when we say that the 3-body problem is "unsolvable" really just means that there is no general solution in terms of finite combinations of standard functions, like polynomials, exponents, trig functions, etc. This just means that there isn't a relatively simple expression where you could plug in any initial configuration of 3-bodies and get their trajectories.
What this does NOT means is that:
Solutions don't exist. There is a solution for every configuration, we just can't write them using our favorite functions. This is why we can model the 3-body problem with computers, which approximate these solutions
That all 3-body problems are unsolvable. There are some configurations where we can write the solution using our favorite functions. See here for a list.
That we can't solve them with more complicated functions. For most situations, the 3-body problem can be solved using infinite power series. This vastly expands the number of functions we have access to and so it shouldn't be surprising that we can solve it there.
This notion of "unsolvability" is really down to our preference for what a "solution" looks like. Back in the olden-days, we could only compute some functions really well and so we favored those functions which came to be known as Elementary Functions. But this is a very small sample of what functions can actually be and they are designed around our preferences, and so it makes sense that math/physics won't conform to such tight restrictions. There isn't really anything special about the 3-body problem, it just doesn't care about these restrictions. In fact, we should see the 2-body problem as having something special about it which allows us to write solutions within these restrictions. And that special property is, likely, that 2-body motion takes place in a fixed plane which reduces the complexity of the problem to something elementary.
So, in the end, the 2-body problem is "special" and "mysterious" because we can write it's solutions down using our favorite functions. The 3-body problem is typical in that there's nothing special about it that reduces its solutions to our preferential functions.