From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
Date: Mon, 2 Sep 2019 13:46:59 +0900
Subject: Initial commit

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 macros/latex/contrib/exam-n/sample/dynamical3.tex | 59 +++++++++++++++++++++++
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+\documentclass[compose]{exam-n}
+\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}
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