Past exam of the options I course of the University of Oxford / 2013 / H / 1 / d / Solution

Away from the coordinate poles, use $\dot{\varphi }=h/{\sin }^2\theta$ in the conserved energy:Rearranging gives the first-order equationWhere the right side is positive, either sign of its square root is possible, according to the direction of motion. Its zeros are turning points. For $h\ne 0$, the centrifugal term excludes the poles; if $h=0$, the polar-coordinate formula is interpreted through the original rod equations.