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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.

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MagicSquare8-9 t1_ixqabux wrote

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.

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Garo5 t1_iy4ydz4 wrote

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?

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MagicSquare8-9 t1_iy6w2gt wrote

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.

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Garo5 t1_iy7vdx8 wrote

Thank you! This was a mind-opener. I knew I was missing something and you gave an excellent explanation!

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