Why is it the 3 body "problem"?

[Cikomyr2]

So i understand (roughly) the n-body problem regarding laws of gravitation and the concept of stable orbit. I also understand that the aforementioned "problem" can refer to the impossibility (unlikeliness?) to properly calculate a stable orbit for system with more than 2 bodies.

So, just want to make sure why its a "problem". Like.. is it that we have seen stable orbits in n-body systems and have been incapable of explaining them? Is it simply about the base impossibility of forecasting the movement of celestial bodies when the system gets too complicated?

I just want to understand if this "problem" is a hurdle to science at the moment that prevent us from properly understanding more things, or its simply a matter of being unable to model stable complex systems.

[NichG]

It's a 'problem' in the sense of solving a mathematical problem.

People had wanted to write down closed-form mathematical solutions for the motions of arbitrary numbers of bodies, but found they could only do it in general for 2, and then furthermore found that a general closed-form solution for 3+ is provably impossible (even if you can do it in special cases).

[Cikomyr2]

It's a 'problem' in the sense of solving a mathematical problem.

People had wanted to write down closed-form mathematical solutions for the motions of arbitrary numbers of bodies, but found they could only do it in general for 2, and then furthermore found that a general closed-form solution for 3+ is provably impossible (even if you can do it in special cases).

It has been ***proven*** to be impossible?!

All right, thanks!!

[NichG]

I should say, you can absolutely approximate the orbits of n bodies under classical gravitation, and its generally pretty easy to do so, and we do it a lot for very large n. And we can write down iterative approximations that provably converge to the true motion (so its not like, yeah this looks legit, but it starts to deviate in the 150th digit in some way that becomes dominant after a billion years - though chaos is an issue, there's stuff like KAM theory that gives a semi-statistical description that works as long as the orbits are stable-ish).

Its just that, we can't write down some finite expression of known transcendental and algebraic functions that's just 'this is the orbits, for all time' for all possible initial values. It's always going to have a limit, or an infinite sum, or an integral.

[kyoryu]

Also the 3bp can be solved for some configurations, but there is not a general solution for it, and I believe provably so.