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

The particles undergo a motion for which $\mathbf{x}_{i}(t)=a(t) \mathbf{a}_{i}$, where the vectors $\mathbf{a}_{i}$ are independent of time $t$. Show that the equations of motion will be satisfied as long as the function $a(t)$ satisfies

$$\ddot{a}=-\frac{\Lambda}{a^{2}},$$

where $\Lambda$ is a constant and the vectors $\mathbf{a}_{i}$ satisfy

$$\Lambda m_{i} \mathbf{a}_{i}=\mathbf{G}_{i}=\sum_{j \neq i} \frac{G m_{i} m_{j}\left(\mathbf{a}_{i}-\mathbf{a}_{j}\right)}{\left|\mathbf{a}_{i}-\mathbf{a}_{j}\right|^{3}}$$

Show that $(*)$ has as first integral

$$\frac{\dot{a}^{2}}{2}-\frac{\Lambda}{a}=\frac{k}{2}$$

where $k$ is another constant. Show that

$$\mathbf{G}_{i}=\nabla_{i} W$$

where $\boldsymbol{\nabla}_{i}$ is the gradient operator with respect to $\mathbf{a}_{i}$ and

$$W=-\sum_{i} \sum_{j<i} \frac{G m_{i} m_{j}}{\left|\mathbf{a}_{i}-\mathbf{a}_{j}\right|} .$$

Using Euler's theorem for homogeneous functions (see below), or otherwise, deduce that

$$\sum_{i} \mathbf{a}_{i} \cdot \mathbf{G}_{i}=-W .$$

Hence show that all solutions of $(* *)$ satisfy

$$\Lambda I=-W$$

where

$$I=\sum_{i} m_{i} \mathbf{a}_{i}^{2}$$

Deduce that $\Lambda$ must be positive and that the total kinetic energy plus potential energy of the system of particles is equal to $\frac{k}{2} I$.

[Euler's theorem states that if

$$f(\lambda x, \lambda y, \lambda z, \ldots)=\lambda^{p} f(x, y, z, \ldots)$$

then

$$\left.x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}+\ldots=p f .\right]$$