Solving Kepler's second law

Question

Kepler's second law, about equal areas in equal times, is a differential equation: it gives velocity as a function of location.

Where are the best expository accounts of the process of solving this equation, giving position as a function of time?

Answer 1

It's a consequence of the rotation invariance of the system. So, for any given $\hat{z}$ axis, perpendicular to the orbit plane, the $z$ angular momentum component is a constant of motion:

$$ M_z=ymv_x - xmv_y =2m\left(\,{1 \over 2}\,r^2 \dot{\theta}\,\right) =2m\,{{\rm d}\text{Area} \over {\rm d}t} $$