Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / H / 1 (H1)

The Lagrangian is $L={\sum }_i{m}_i|{\dot{q}}_i{|}^2/2-f(r)$. Away from a collision, its equations of motion areSumming forces proves conservation of total linear momentum. The total mechanical energy $E={\sum }_i{m}_i|{\dot{q}}_i{|}^2/2+f(r)$ is also conserved: its derivative is $F\cdot ({\dot{q}}_1-{\dot{q}}_2)+f^{\prime }(r)\dot{r}=0$, because $\dot{r}=(q_1-q_2)\cdot ({\dot{q}}_1-{\dot{q}}_2)/r$.

For the first proposed angular momentum,which need not vanish. Thus particle one's angular momentum about the fixed origin is not generally conserved. Its force is central relative to particle two, not necessarily relative to that origin.

The third particle is free, soThus particle three's angular momentum is conserved.

The derivative of the pair's total angular momentum is $q_1\times F-q_2\times F=(q_1-q_2)\times F=0$, since $F$ is parallel to the separation vector. Therefore the combined angular momentum of particles one and two is conserved, despite the possible exchange between them individually.