Write down an expression for the total momentum $\mathbf{P}$ and angular momentum $\mathbf{L}$ with respect to an origin $O$ of a system of $n$ point particles of masses $m_{i}$, position vectors (with respect to $O) \mathbf{x}_{i}$, and velocities $\mathbf{v}_{i}, i=1, \ldots, n$.

Show that with respect to a new origin $O^{\prime}$ the total momentum $\mathbf{P}^{\prime}$ and total angular momentum $\mathbf{L}^{\prime}$ are given by

$$\mathbf{P}^{\prime}=\mathbf{P}, \quad \mathbf{L}^{\prime}=\mathbf{L}-\mathbf{b} \times \mathbf{P}$$

and hence

$$\mathbf{L}^{\prime} \cdot \mathbf{P}^{\prime}=\mathbf{L} \cdot \mathbf{P},$$

where $\mathbf{b}$ is the constant vector displacement of $O^{\prime}$ with respect to $O$. How does $\mathbf{L} \times \mathbf{P}$ change under change of origin?

Hence show that either

(1) the total momentum vanishes and the total angular momentum is independent of origin, or

(2) by choosing $\mathbf{b}$ in a way that should be specified, the total angular momentum with respect to $O^{\prime}$ can be made parallel to the total momentum.