Write down the equations of motion for a system of $n$ gravitating point particles with masses $m_{i}$ and position vectors $\mathbf{x}_{i}=\mathbf{x}_{i}(t), i=1,2, \ldots, n$.

Assume that $\mathbf{x}_{i}=t^{2 / 3} \mathbf{a}_{i}$, where the vectors $\mathbf{a}_{i}$ are independent of time $t$. Obtain a system of equations for the vectors $\mathbf{a}_{i}$ which does not involve the time variable $t$.

Show that the constant vectors $\mathbf{a}_{i}$ must be located at stationary points of the function

$$\sum_{i} \frac{1}{9} m_{i} \mathbf{a}_{i} \cdot \mathbf{a}_{i}+\frac{1}{2} \sum_{j} \sum_{i \neq j} \frac{G m_{i} m_{j}}{\left|\mathbf{a}_{i}-\mathbf{a}_{j}\right|}$$

Show that for this system, the total angular momentum about the origin and the total momentum both vanish. What is the angular momentum about any other point?