Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 4 / 12C (Dynamics)

For constant particle masses define the centre of mass by $M\mathbf{X}={\sum }_i{m}_i{\mathbf{r}}_i$, with $M={\sum }_i{m}_i$. Summing Newton's second law over all particles givesEach pair of internal forces cancels by Newton's third law. HenceOnly the action-reaction condition is needed for this translation law; centrality of the internal forces is needed for the angular momentum result in part (ii).

About a fixed inertial origin, the total angular momentum is $\mathbf{L}={\sum }_i{m}_i{\mathbf{r}}_i\times {\dot{\mathbf{r}}}_i$. Differentiation givesbecause ${\dot{\mathbf{r}}}_i\times {\dot{\mathbf{r}}}_i=0$. Separate the external and internal contributions. The combined torque of an internal pair isThe first equality uses Newton's third law, and the second uses the given central-force condition. All internal torques therefore cancel, leavingAn equal-and-opposite pair which is not central need not have zero combined torque; this explains why both internal-force hypotheses matter.

Write ${\mathbf{r}}_i=\mathbf{X}+{\mathbf{r}}_i^{\prime }$ and ${\mathbf{v}}_i=\mathbf{V}+{\mathbf{v}}_i^{\prime }$, where $\mathbf{V}=\dot{\mathbf{X}}$. The definition of the centre of mass and its derivative giveExpand the total angular momentum:The two middle terms vanish. Thus the angular momentum decomposition about the centre of mass isThe first term is orbital angular momentum of the centre; the second is angular momentum about the centre of mass.

Let $r=|{\mathbf{r}}_1-{\mathbf{r}}_2|$ and $\hat{\mathbf{r}}=({\mathbf{r}}_1-{\mathbf{r}}_2)/r$. A differentiable central potential energy gives the internal forcesThere are no external forces. Differentiating the total kinetic energy and using Newton's second law therefore givesSince $\dot{r}=\hat{\mathbf{r}}\cdot ({\mathbf{v}}_1-{\mathbf{v}}_2)$, the last expression is $-U^{\prime }(r)\dot{r}=-dU/dt$. ThusThis conserves total mechanical energy, including the kinetic energy of centre-of-mass motion, not merely the relative kinetic energy.