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?

functor7t1_ixoa1xu wroteWhat is meant when we say that the 3-body problem is "unsolvable" really just means that there is no

generalsolution 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

all3-body problems are unsolvable. There are some configurations where wecanwrite 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

somefunctions 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 aboutitwhich 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.