Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics IV / 5 / a / Solution

Write $\mu =GM$. In polar coordinates, Newton's second law per unit mass gives the radial and transverse equations of motionThe transverse equation impliesThis is conservation of angular momentum per unit mass. For the nonradial trajectory in the next part, choose orientation so $h>0$. With $u=1/r$ and primes denoting derivatives with respect to $\theta$, the chain rule givesSubstituting into the radial equation yields $-h^2{u}^2(u^{\prime \prime }+u)=-\mu u^2$, henceThe change of independent variable requires $h\ne 0$; a purely radial trajectory instead has constant $\theta$.