Wikipedia says:
In 1887, mathematicians Heinrich Bruns 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.
I have two questions:
What does an "analytical solution" mean here -- analytic in the sense of an analytic function? Or in the sense of being built out of elementary functions?
At a high level, how does the proof go? How does one rule out the existence of a solution like this?
[As a sidebar: is there a mathematical connection between Wikipedia's second sentence ("generally non-repeating") and its first? I can certainly imagine non-repeating solutions which are analytical. Are repeating solutions necessarily analytical?]