Rearrange $\dot{\theta} = \frac{h}{r^2}$ to $\ddot{r} = -\frac{h^2}{r^2}\frac{\mathrm{d}^2}{\mathrm{d} \theta^2}\left( \frac{1}{r} \right)$

Question

I have been trying to rearrange the first equation to the second equation for hours as of now, and I have got absolutely nowhere with it. Any help would be greatly appreciated!

Answer 1

To avoid the more cryptical applications of the chain rule, make the implied composition explicit: Set $r(t)=\frac1{u(\theta(t))}$. Then $$ \dot r(t)=-\frac{u'(θ(t))}{u(θ(t))^2}\dot θ(t)=-hu'(θ(t)) $$ The claim now follows from repeating this process of time differentiation and replacing of $\dot θ=hu^2$.