# 4.II.10E

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

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

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