So for the proof of Kepler's first law they make a simplification of a differential equation, but I don't get how they come up with that simplification. It goes like this
$$2\frac{dr}{dt}\frac{d\theta}{dt}+r\frac{d^2\theta}{dt^2}=0$$ once simplified (which I don't understand) they came up with this answer $$\frac{1}{2r}\frac{d}{dt}\left(r\left(\frac{d\theta}{dt}\right)^2\right)=0$$ could someone please explain to me what they did?
thanks in advance.
$$\frac{dr}{dt}\frac{d\theta}{dt}+\color {red} {2r}\frac{d^2\theta}{dt^2}=0$$ More simply: $$r'\theta '+2r\theta ''=0$$ Multiply by $\theta ':$ $$r'(\theta ')^2+2 r \theta '\theta ''=0$$ Then you have: $$r' (\theta ')^2+r ((\theta ')^2)'$$ Finally: $$(r((\theta ')^2))'=0$$
But this $$2\frac{dr}{dt}\frac{d\theta}{dt}+r\frac{d^2\theta}{dt^2}=0$$ Is : $$2r'\theta ' +r \theta ''=0$$ $$2r'r\theta ' +r^2 \theta ''=0$$ $$(r^2 \theta ')'=0$$ The two equations you wrote are not giving the same result.