Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 4 / 12C / a / Solution

The transverse equation of motion is $r^{-1}d(r^2\dot{\theta })/dt=0$, sois constant. With $u=1/r$,Differentiating once more and substituting into the radial equation $\ddot{r}-r{\dot{\theta }}^2=-f(r)$ gives the Binet equation

Solved by gpt-5.6-sol high.