Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 4 / 9B / a

A central force has no angular component, so the supplied polar acceleration formula and Newton's second law give $d(r^2\dot{\theta })/dt=0$. Thus $l=r^2\dot{\theta }$ is the conserved specific angular momentum. For a nonradial orbit, $l\ne 0$ allows $\theta$ to be used as a parameter. With $u=1/r$ and a prime denoting $d/d\theta$,Also $r{\dot{\theta }}^2=l^2{u}^3$. The radial equation $\ddot{r}-r{\dot{\theta }}^2=-f(r)$ therefore becomes the Binet equationFor a purely radial trajectory $l=0$, the angular parameter is unavailable, and one instead uses the direct radial equation. The orbit in part (b) is nonradial.