Re: Why is there no general solution to the three-body problem?

In plain language, the answer is that there are not enough conserved
quantities. You can get the 2-body solution by applying the laws of
conservation of energy, momentum, and angular momentum. But to solve the
twelve-dimensional three body problem, you need more conserved quantities
and there just aren't any. 

Generally, people get around this by solving the restricted 3 body problem,
where one of the masses is assumed to be much smaller than the other two.
You can check out:
 http://www.physics.cornell.edu/sethna/teaching/sss/jupiter/Web/Rest3Bdy.htm 

for further details.

If you want to read more about this, there is a whole book on the subject 
by Christian Marchal entitled "The Three-Body Problem", published by
Elsevier science in 1990.