[adonovan]

Is it possible that we might discover some new branch of (say) calculus that produces an exact formula for position as a function of time? (To my amateur eyes it looks like a tricky integral.) If so, this does appear to be merely a weakness of iterative simulation.

Or can we prove that there is no simple exact formula?

[freemint]

This is a good question and the answer depends on what you mean by formula.

One could probably show that there exists no formula using a finite number of +,*,/,^. However one might be able to define functions which together with a finite number of operations from above allow to express solutions. However it is likely that calculating those helper functions even when they are well studied and known is basically solving a slightly different differential equation (or as you said integration problem).