First a bit of background on my question. According to Wikipedia In 1887, mathematicians Heinrich Bruns[4] and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases. https://en.wikipedia.org/wiki/Three-body_problem
I disagree the fact that the three-body problem is none repeating does not mean that there is no solution to the problem became there are or at least appear to be none repeating functions.
For example f(t)=Sin(t*2)+sin(t√2)+sin(t)) Because √2 is an irrational number there is only one value you can multiply 2, √2,1 that make them all equal to zero, or a multiple of π. That number being 0. This means that the only place where f(t)=0 f(t)’=2+√2+1 and
|f(t)(n)|= n/2*((-1)n(3+√2)+ 3+√2)) the n repersents nth derivitve
All of these things make 0 a completely unique point in the function. In fact, every point in the function is unique. (Or at least I’m pretty sure they are, I don’t know how to prove it).
If your skeptical or if it is unclear what I’m talking about try graphing Sin(t*2)+sin(t√2)+sin(t)) next to Sin(t*3)+sin(t2)+sin(t)) you will imminently see what I’m talking about. Even looking at the function Sin(t*2)+sin(t√2)+sin(t)) will make is clear that is none repeating.
Here is a good sight to graph at https://www.desmos.com/calculator (note it dose not have to be a sin or cos wave it can be any periodic funtion there also need to three of them and one has to be irrational)
Because of this the differential equation. dx/dt=2cos(t(2)+√2cos(t*2)+cos(t)) Solves for a non-repeating function.
So the question is, is it posable that there is a function that gives the positions of the objects in the three-body problem are made up by three or more periodic function’s similar so the solution in the two body problem?
found something new
according to scholarpedia under three body problem
Two cases of integrability the secular system of the planetary (or lunar) planar three-body problem is four dimensional, hence completely integrable because the angular momentum is the first integral; if one replaces the Newtonian potential, inversely proportional to the distance by the Jacobi potential inversely proportional to the square of the distance, a new first integral 2IH−J2 of the N-body problem exists which was discovered by Jacobi. This implies the complete integrability of the three-body problem on the line with such a potential.
No idea if this useful to anyone but there it is anyway.
You're correct that an aperiodic function can have a closed form in terms of periodic functions, but as you can see on the wikipedia page, there is a proof that no general analytical solution exists for the three-body problem. Perhaps the wording there is misleading, in that the aperiodicity isn't an explanation of why no closed form exists, but that we know the motion is generally non repeating, and also there is no general solution.