What is a Three Body Problem?
In: Physics
In (simplified) gravity things pull each other towards each other.
If you have two objects each will attract the other. Which will make them accelerate, which will change where they are, which may change the direction they pull the other one in. This makes the maths of working out where each is going to be for all time a big complicated, but it is solvable (particularly if you are careful in which reference frame you choose).
With *three* objects, each will attract the other two towards itself, and each will be attracted towards the other two. So rather than just having to worry about two interactions (A on B and B on A) we have 6 (A on B, A on C, B on A, B on C, C on A, C on B). The maths becomes a lot more complicated, and we get what is called a *chaotic* system, where small changes in the initial conditions (where the objects start and how fast they are going) leads to very different outcomes. You can get some [really weird solutions](https://en.wikipedia.org/wiki/File:Three-body_Problem_Animation_with_COM.gif).
Unlike the two-body problem the three-body problem has no “general closed-form solution” – “general” meaning it applies in all cases (you just have to put in your initial conditions and will get an answer), and “closed-form” meaning it is made up of a finite number of expressions.
There is an infinite-series solution to the three-body problem, and it turns out that it converges in all non-trivial cases. There are also a bunch of special case solutions (including [some stable ones](https://en.wikipedia.org/wiki/File:Three_body_problem_figure-8_orbit_animation.gif)).
The three-body problem is a special case of the more general n-body problem, where you have n objects interacting.
The n-body problem is important in astrophysics because most gravitational systems have more than 2 things in them (the classic 3-body one being the Sun, Moon and Earth). Often they are arranged so you can ignore one of them at a time and reduce it to a series of 2-body problems, but sometimes you can’t – particularly not if you want good answers.
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