Recently I was surprised to discover that no exact solution for the position of a planet as a function of time exists. I am referring to the two-body problem in a gravitational field where Newtons law of gravity holds.
Well known are proofs that the planet will move in an ellipse, Keplers laws can be derived fairly easy etc., but for the exact position of the planet on the ellipse as a function of time, no formula exists, only numerical approximations.
Is this correct?
Can anybody elaborate on the deeper reason(s) that this relatively simple case cannot be solved?