Past exam of the options I course of the University of Oxford / 2009 / H / H1 / i / Solution

For the first particle, the derivative of its angular momentum about the origin is its torque:This need not vanish. For example take $f(r)=r^2/2$ and instantaneous positions ${\mathbf{q}}_1=(1,0,0)$, ${\mathbf{q}}_2=(0,1,0)$; the torque is $(0,0,1)$. Therefore it is not conserved in general, although it is in special configurations or for a constant potential.