The $i$ th particle of a system of $N$ particles has mass $m_{i}$ and, at time $t$, position vector $\mathbf{r}_{i}$ with respect to an origin $O$. It experiences an external force $\mathbf{F}_{i}^{e}$, and also an internal force $\mathbf{F}_{i j}$ due to the $j$ th particle (for each $j=1, \ldots, N, j \neq i$ ), where $\mathbf{F}_{i j}$ is parallel to $\mathbf{r}_{i}-\mathbf{r}_{j}$ and Newton's third law applies.

(i) Show that the position of the centre of mass, $\mathbf{X}$, satisfies

$$M \frac{d^{2} \mathbf{X}}{d t^{2}}=\mathbf{F}^{e}$$

where $M$ is the total mass of the system and $\mathbf{F}^{e}$ is the sum of the external forces.

(ii) Show that the total angular momentum of the system about the origin, $\mathbf{L}$, satisfies

$$\frac{d \mathbf{L}}{d t}=\mathbf{N}$$

where $\mathbf{N}$ is the total moment about the origin of the external forces.

(iii) Show that $\mathbf{L}$ can be expressed in the form

$$\mathbf{L}=M \mathbf{X} \times \mathbf{V}+\sum_{i} m_{i} \mathbf{r}_{i}^{\prime} \times \mathbf{v}_{i}^{\prime}$$

where $\mathbf{V}$ is the velocity of the centre of mass, $\mathbf{r}_{i}^{\prime}$ is the position vector of the $i$ th particle relative to the centre of mass, and $\mathbf{v}_{i}^{\prime}$ is the velocity of the $i$ th particle relative to the centre of mass.

(iv) In the case $N=2$ when the internal forces are derived from a potential $U(|\mathbf{r}|)$, where $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$, and there are no external forces, show that

$$\frac{d T}{d t}+\frac{d U}{d t}=0$$

where $T$ is the total kinetic energy of the system.