\documentclass[compose]{guexam}
\begin{document}

\begin{question}{30} \comment{by Declan Diver}
For a system of  $N$ objects, each having mass  $m_i$ and position
vector $\mathbf{R}_i$ with respect to a fixed co-ordinate system,
use the moment of inertia
\[
I=\sum_{i=1}^N m_i R_i^2
\]
to deduce the virial theorem in the forms
\[
\ddot{I}=4E_k+2E_G=2E_k+2E
\]
where $E_k$  and $E_G$  are respectively the total kinetic and
gravitational potential energy, and $E$  is the total energy of
the system.
\partmarks{8}

Given the inequality
\ifbigfont
  \begin{multline*}
  \left(\sum_{i=1}^N
  a_i^2\right) \left(\sum_{i=1}^N b_i^2\right) \\
\ge \left(\sum_{i=1}^N \mathbf{a}_i\cdot\mathbf{b}_i\right)^2 \\
+ \left(\sum_{i=1}^N \mathbf{a}_i\times\mathbf{b}_i\right)^2
  \end{multline*}
\else
  \begin{equation*}
  \left(\sum_{i=1}^N
  a_i^2\right) \left(\sum_{i=1}^N b_i^2\right) \ge \left(\sum_{i=1}^N
  \mathbf{a}_i\cdot\mathbf{b}_i\right)^2 + \left(\sum_{i=1}^N
  \mathbf{a}_i\times\mathbf{b}_i\right)^2
  \end{equation*}
\fi
for arbitrary vectors $\mathbf{a}_i$, $\mathbf{b}_i$,
$i=1,\ldots,N$, deduce the following relationship for the $N$-body
system
\begin{equation*}
\frac{1}{4}\dot{I}^2+J^2\le 2IE_k,
\end{equation*}
where $\mathbf{J}$ is the total angular momentum of the system.
\partmarks{8}

Assuming the system is isolated, use the virial theorem to deduce
further the generalised Sundman inequality
\begin{equation*}
\frac{\dot{\sigma}}{\dot{\rho}}\ge 0,
\end{equation*}
in which $\rho^2=I$  and
$\displaystyle\sigma=\rho\dot{\rho}^2+\frac{J^2}{\rho}-2\rho E $.
\partmarks{8}

Why  does  this  inequality  preclude  the  possibility  of an
$N$-fold collision for a system with finite angular momentum?
\partmarks{6}

\end{question}
\end{document}
