# Why isn't the 3 body problem solvable?

I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of freedom, then the system is solvable in terms of elementary functions. Is this correct? I get that for each linearly independent constant of motion you can reduce the degree of freedom by one, but I don't understand why the theorem

Theorem (First integrals of the n-body problem) The only linearly independent integrals of the $n$-body problem, which are algebraic with respect to $q$, $p$ and $t$ are the $10$ described above. (http://en.wikipedia.org/wiki/N-body_problem#Three-body_problem)

implies that there is no analytic solution (I think this is synonymous with closed-form solution, and solution in terms of elementary functions). I've been trying to think about it, but I can't reason it, and apparently integrability implies no chaos, which I can't see either.

• Have you read en.wikipedia.org/wiki/Integrable_system ?
– lhf
Commented May 17, 2012 at 20:21
• I had understood that a perfectly elastic simultaneous collision between three bodies could not be solved - momentum and energy are conserved, but this does not determine the subsequent motion. Commented May 17, 2012 at 20:47
• Fun fact: the quantum mechanics 3-body problem is solvable! Commented May 17, 2012 at 21:53
• "Integrable by quadrature", which is the classical (Liouville) notion of integrability, does not mean "integrable in terms of elementary functions". It means that you can in principle write down the solution, provided that you are able to compute all antiderivatives and inverses of functions that you happen to come across along the way. But unfortunately not all antiderivatives and inverses of elementary functions are elementary, as you probably know... Commented May 28, 2012 at 7:08
• All of which leads to the curious, and unanswered question, is the solar system stable? Commented Jul 16, 2012 at 19:04