\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\Tr}{Tr}
\begin{document}
\begin{eqnarray*}
\mbox{(1a)} \qquad r_{1}
 & = & x \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(1b)} \qquad r_{2}
 & = & y \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(1c)} \qquad r_{3}
 & = & z \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2a)} \qquad \vec v
 & = & \frac{\mbox{d} \vec r}{\mbox{d}t} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2b)} \qquad v_{i}
 & = & \frac{\mbox{d}r_{i}}{\mbox{d}t} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2c)} \qquad v_{r}
 & = & \frac{\mbox{d}r}{\mbox{d}t} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(3)} \qquad \vec F
 & = & m \vec a \\
\vec a
 & = & \frac{\vec F}{m} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4a)} \qquad \vec a
 & = & \frac{\mbox{d} \vec v}{\mbox{d}t} \\
& = & \frac{\mbox{d}^{2} \vec r}{\mbox{d}t^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4b)} \qquad a_{i}
 & = & \frac{\mbox{d}v_{i}}{\mbox{d}t} \\
& = & \frac{\mbox{d}^{2} r_{i}}{\mbox{d}t^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4c)} \qquad a_{r}
 & = & \frac{\mbox{d}v_{r}}{\mbox{d}t} \\
& = & \frac{\mbox{d}^{2} r}{\mbox{d}t^{2}} \\
& = & a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5a)} \qquad \vec j
 & = & \frac{\mbox{d} \vec a}{\mbox{d}t} \\
& = & \frac{\mbox{d}^{2} \vec v}{\mbox{d}t^{2}} \\
& = & \frac{\mbox{d}^{3} \vec r}{\mbox{d}t^{3}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5b)} \qquad j_{i}
 & = & \frac{\mbox{d}a_{i}}{\mbox{d}t} \\
& = & \frac{\mbox{d}^{2} v_{i}}{\mbox{d}t^{2}} \\
& = & \frac{\mbox{d}^{3} r_{i}}{\mbox{d}t^{3}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5c)} \qquad j_{r}
 & = & \frac{\mbox{d}a_{r}}{\mbox{d}t} \\
& = & \frac{\mbox{d}^{2} v_{r}}{\mbox{d}t^{2}} \\
& = & \frac{\mbox{d}^{3} r}{\mbox{d}t^{3}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(6)} \qquad m_{t}
 & = & m_{1} + m_{2} \\
& = & M \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(7)} \qquad r
 & = & \frac{a (1 - e^{2})}{1 + e \cos \theta} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(8)} \qquad v
 & = & \sqrt{G M \left ( \frac{2}{r} - \frac{1}{a} \right )} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9)} \qquad v_{r}
 & = & \sqrt{\frac{G M}{a (1 - e^{2})}} \, e \sin \theta \\
& = & \frac{\mbox{d}r}{\mbox{d}t} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(10)} \qquad \mu
 & = & \frac{m_{1} m_{2}}{m_{1} + m_{2}} \\
& = & \frac{m_{1} m_{2}}{M} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(11)} \qquad F_{g}
 & = & \frac{G m_{1} m_{2}}{r^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(12)} \qquad a
 & = & a_{r} \\
& = & - \frac{G M}{r^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(13)} \qquad \vec a
 & = & - \frac{G M}{r^{2}} \hat r \\
& = & - \frac{G M}{r^{3}} \vec r \\
& = & \frac{a}{r} \vec r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14)} \qquad \vec j
 & = & \frac{\mbox{d} \vec a}{\mbox{d}t} \\
& = & \frac{\mbox{d}}{\mbox{d}t} \left ( \frac{a}{r} \vec r \, \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}t} \left ( - \frac{G M}{r^{3}} \vec r \, \right ) \\
& = & - G M \left ( - \frac{3}{r^{4}} \, \frac{\mbox{d}r}{\mbox{d}t} \, \vec r + \frac{1}{r^{3}} \, \frac{\mbox{d} \vec r}{\mbox{d}t} \, \right ) \\
& = & - \frac{G M}{r^{3}} \left ( \vec v - 3 v_{r} \frac{\vec r}{r} \, \right ) \\
& = & \frac{a}{r} (\vec v - 3 v_{r} \hat r \, ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15a)} \qquad \vec r \cdot \vec r
 & = & r^{2} (\hat r \cdot \hat r \, ) \\
& = & r^{2} (1) \\
& = & r^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15b)} \qquad \vec r \cdot \vec v
 & = & r (\hat r \cdot \vec v \, ) \\
& = & r v_{r} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15c)} \qquad \vec r \cdot \vec a
 & = & r (\hat r \cdot \vec a \, ) \\
& = & r a_{r} \\
& = & r a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15d)} \qquad \vec r \cdot \vec j
 & = & \vec r \cdot \left ( \frac{a}{r} (\vec v - 3 v_{r} \hat r \, ) \right ) \\
& = & \frac{a}{r} ((\vec r \cdot \vec v \, ) - 3 v_{r} (\vec r \cdot \hat r \, )) \\
& = & a \left ( \frac{\vec r}{r} \cdot \frac{\mbox{d} \vec r}{\mbox{d}t} - 3 v_{r} \frac{\vec r}{r} \cdot \hat r \, \right ) \\
& = & a \left ( \hat r \cdot \frac{\mbox{d} \vec r}{\mbox{d}t} - 3 v_{r} (\hat r \cdot \hat r \, ) \right ) \\
& = & a \left ( \frac{\mbox{d}r}{\mbox{d}t} - 3 v_{r} (1) \right ) \\
& = & a (v_{r} - 3 v_{r}) \\
& = & a (- 2 v_{r}) \\
& = & - 2 v_{r} a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15e)} \qquad \vec v \cdot \vec v
 & = & v^{2} (\hat v \cdot \hat v \, ) \\
& = & v^{2} (1) \\
& = & v^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15f)} \qquad \vec v \cdot \vec a
 & = & \vec v \cdot \left ( \frac{a}{r} \vec r \, \right ) \\
& = & \frac{a}{r} (\vec v \cdot \vec r \, ) \\
& = & \frac{a}{r} (r v_{r}) \\
& = & v_{r} a \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15g)} \qquad \vec v \cdot \vec j
 & = & \vec v \cdot \left ( \frac{a}{r} (\vec v - 3 v_{r} \hat r \, ) \right ) \\
& = & \frac{a}{r} ((\vec v \cdot \vec v \, ) - 3 v_{r} (\vec v \cdot \hat r \, )) \\
& = & \frac{a}{r} (v^{2} - 3 v_{r}^{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15h)} \qquad \vec a \cdot \vec a
 & = & a^{2} (\hat a \cdot \hat a \, ) \\
& = & a^{2} (1) \\
& = & a^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15i)} \qquad \vec a \cdot \vec j
 & = & \vec a \cdot \left ( \frac{a}{r} (\vec v - 3 v_{r} \hat r \, ) \right ) \\
& = & \frac{a}{r} ((\vec a \cdot \vec v \, ) - 3 v_{r} (\vec a \cdot \hat r \, )) \\
& = & \frac{a}{r} (v_{r} a - 3 v_{r} a) \\
& = & - \frac{2 v_{r} a^{2}}{r} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15j)} \qquad \vec j \cdot \vec j
& = & \left ( \frac{a}{r} (\vec v - 3 v_{r} \hat r \, ) \right ) \cdot \left ( \frac{a}{r} (\vec v - 3 v_{r} \hat r \, ) \right ) \\
& = & \frac{a^{2}}{r^{2}} ((\vec v \cdot \vec v \, ) - 3 v_{r} (\vec v \cdot \hat r \, ) - 3 v_{r} (\hat r \cdot \vec v \, )
 + 9 v_{r}^{2} (\hat r \cdot \hat r \, )) \\
& = & \frac{a^{2}}{r^{2}} (v^{2} - 3 v_{r} (v_{r}) - 3 v_{r} (v_{r}) + 9 v_{r}^{2} (1)) \\
& = & \frac{a^{2}}{r^{2}} (v^{2} - 3 v_{r}^{2} - 3 v_{r}^{2} + 9 v_{r}^{2}) \\
& = & \frac{a^{2}}{r^{2}} (v^{2} + 3 v_{r}^{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(16)} \qquad I
 & = & \int \int\limits_{m} \int r_{\perp}^{2} \, \mbox{d}m \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(17)} \qquad I_{ij}
 & = & \mu r_{i} r_{j} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(18)} \qquad \frac{\mbox{d} I_{ij}}{\mbox{d}t}
 & = & \frac{\mbox{d}}{\mbox{d}t} (\mu r_{i} r_{j}) \\
& = & \mu \, \frac{\mbox{d}}{\mbox{d}t} (r_{i} r_{j}) \\
& = & \mu \left ( r_{j} \, \frac{\mbox{d}r_{i}}{\mbox{d}t} + r_{i} \, \frac{\mbox{d}r_{j}}{\mbox{d}t} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(19)} \qquad \frac{\mbox{d}^{2} I_{ij}}{\mbox{d}t^{2}}
 & = & \frac{\mbox{d}}{\mbox{d}t} \left ( \frac{\mbox{d} I_{ij}}{\mbox{d}t} \right ) \\
& = & \mu \, \frac{\mbox{d}}{\mbox{d}t} \left ( r_{j} \, \frac{\mbox{d}r_{i}}{\mbox{d}t} + r_{i} \, \frac{\mbox{d}r_{j}}{\mbox{d}t} \right ) \\
& = & \mu \left ( \frac{\mbox{d}r_{j}}{\mbox{d}t} \, \frac{\mbox{d}r_{i}}{\mbox{d}t}
 + r_{j} \, \frac{\mbox{d}^{2} r_{i}}{\mbox{d}t^{2}}
 + \frac{\mbox{d}r_{i}}{\mbox{d}t} \, \frac{\mbox{d}r_{j}}{\mbox{d}t}
 + r_{i} \, \frac{\mbox{d}^{2} r_{j}}{\mbox{d}t^{2}} \right ) \\
& = & \mu \left ( r_{j} \, \frac{\mbox{d}^{2} r_{i}}{\mbox{d}t^{2}}
 + 2 \, \frac{\mbox{d}r_{i}}{\mbox{d}t} \, \frac{\mbox{d}r_{j}}{\mbox{d}t}
 + r_{i} \, \frac{\mbox{d}^{2} r_{j}}{\mbox{d}t^{2}} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(20)} \qquad \frac{\mbox{d}^{3} I_{ij}}{\mbox{d}t^{3}}
 & = & \frac{\mbox{d}}{\mbox{d}t} \left ( \frac{\mbox{d}^{2} I_{ij}}{\mbox{d}t^{2}} \right ) \\
& = & \mu \, \frac{\mbox{d}}{\mbox{d}t} \left ( r_{j} \, \frac{\mbox{d}^{2} r_{i}}{\mbox{d}t^{2}}
 + 2 \, \frac{\mbox{d}r_{i}}{\mbox{d}t} \, \frac{\mbox{d}r_{j}}{\mbox{d}t}
 + r_{i} \, \frac{\mbox{d}^{2} r_{j}}{\mbox{d}t^{2}} \right ) \\
& = & \mu \left ( \frac{\mbox{d}r_{j}}{\mbox{d}t} \frac{\mbox{d}^{2} r_{i}}{\mbox{d}t^{2}}
 + r_{j} \, \frac{\mbox{d}^{3} r_{i}}{\mbox{d}t^{3}}
 + 2 \, \frac{\mbox{d}^{2} r_{i}}{\mbox{d}t^{2}} \, \frac{\mbox{d}r_{j}}{\mbox{d}t}
 + 2 \, \frac{\mbox{d}r_{i}}{\mbox{d}t} \, \frac{\mbox{d}^{2} r_{j}}{\mbox{d}t^{2}}
 + \frac{\mbox{d}r_{i}}{\mbox{d}t} \, \frac{\mbox{d}^{2} r_{j}}{\mbox{d}t^{2}}
 + r_{i} \, \frac{\mbox{d}^{3} r_{j}}{\mbox{d}t^{3}} \right ) \\
& = & \mu \left ( r_{j} \, \frac{\mbox{d}^{3} r_{i}}{\mbox{d}t^{3}}
 + 3 \, \frac{\mbox{d}^{2} r_{i}}{\mbox{d}t^{2}} \, \frac{\mbox{d}r_{j}}{\mbox{d}t}
 + 3 \, \frac{\mbox{d}r_{i}}{\mbox{d}t} \, \frac{\mbox{d}^{2} r_{j}}{\mbox{d}t^{2}}
 + r_{i} \, \frac{\mbox{d}^{3} r_{j}}{\mbox{d}t^{3}} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21)} \qquad \frac{\mbox{d}^{3} I_{ij}}{\mbox{d}t^{3}}
 & = & \mu \left ( r_{j} \, \frac{\mbox{d}^{3} r_{i}}{\mbox{d}t^{3}}
 + 3 \, \frac{\mbox{d}^{2} r_{i}}{\mbox{d}t^{2}} \, \frac{\mbox{d}r_{j}}{\mbox{d}t}
 + 3 \, \frac{\mbox{d}r_{i}}{\mbox{d}t} \, \frac{\mbox{d}^{2} r_{j}}{\mbox{d}t^{2}}
 + r_{i} \, \frac{\mbox{d}^{3} r_{j}}{\mbox{d}t^{3}} \right ) \\
& = & \mu (r_{j} j_{i} + 3 a_{i} v_{j} + 3 v_{i} a_{j} + r_{i} j_{j}) \\
& = & \mu ((r_{j} j_{i} + r_{i} j_{j}) + 3 (a_{i} v_{j} + v_{i} a_{j})) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22)} \qquad \left ( \frac{\mbox{d}^{3} I_{ij}}{\mbox{d}t^{3}} \right )^{2}
 & = & (\mu ((r_{j} j_{i} + r_{i} j_{j}) + 3 (a_{i} v_{j} + v_{i} a_{j})))^{2} \\
& = & \mu^{2} ((r_{j} j_{i} + r_{i} j_{j})^{2} + 6 (r_{j} j_{i} + r_{i} j_{j}) (a_{i} v_{j} + v_{i} a_{j}) + 9 (a_{i} v_{j} + v_{i} a_{j})^{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23)} \qquad \left ( \frac{\mbox{d}^{3} I}{\mbox{d}t^{3}} \right )^{2}
 & = & \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} \left ( \frac{\mbox{d}^{3} I_{ij}}{\mbox{d}t^{3}} \right )^{2} \\
& = & \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3}
 \mu^{2} ((r_{j} j_{i} + r_{i} j_{j})^{2} + 6 (r_{j} j_{i} + r_{i} j_{j}) (a_{i} v_{j} + v_{i} a_{j}) + 9 (a_{i} v_{j} + v_{i} a_{j})^{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24a)} \qquad \mbox{Eerste term}
 & = & \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} (r_{j} j_{i} + r_{i} j_{j})^{2} \\
& = & \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} r_{j} j_{i} r_{j} j_{i} + 2 r_{j} j_{i} r_{i} j_{j} + r_{i} j_{j} r_{i} j_{j} \\
& = & (\vec r \cdot \vec r \, ) (\vec j \cdot \vec j \, ) + 2 (\vec r \cdot \vec j \, ) (\vec r \cdot \vec j \, )
 + (\vec r \cdot \vec r \, ) (\vec j \cdot \vec j \, ) \\
& = & 2 (\vec r \cdot \vec r \, ) (\vec j \cdot \vec j \, ) + 2 (\vec r \cdot \vec j \, )^{2} \\
& = & 2 (r^{2}) (j^{2}) + 2 (\vec r \cdot \vec j \, )^{2} \\
& = & 2 r^{2} j^{2} + 2 (\vec r \cdot \vec j \, )^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24b)} \qquad \mbox{Tweede term}
 & = & \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} 6 (r_{j} j_{i} + r_{i} j_{j}) (a_{i} v_{j} + v_{i} a_{j}) \\
& = & \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} 6 (r_{j} j_{i} a_{i} v_{j} + r_{j} j_{i} v_{i} a_{j} + r_{i} j_{j} a_{i} v_{j} + r_{i} j_{j} v_{i} a_{j}) \\
& = & 6 ((\vec r \cdot \vec v \, ) (\vec a \cdot \vec j \, ) + (\vec r \cdot \vec a \, ) (\vec v \cdot \vec j \, )
 + (\vec r \cdot \vec a \, ) (\vec v \cdot \vec j \, ) + (\vec r \cdot \vec v \, ) (\vec a \cdot \vec j \, )) \\
& = & 6 (2 (\vec r \cdot \vec v \, ) (\vec a \cdot \vec j \, ) + 2 (\vec r \cdot \vec a \, ) (\vec v \cdot \vec j \, )) \\
& = & 12 (\vec r \cdot \vec v \, ) (\vec a \cdot \vec j \, ) + 12 (\vec r \cdot \vec a \, ) (\vec v \cdot \vec j \, ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24c)} \qquad \mbox{Derde term}
 & = & \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} 9 (a_{i} v_{j} + v_{i} a_{j})^{2} \\
& = & \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} 9 (a_{i} v_{j} a_{i} v_{j} + 2 a_{i} v_{j} v_{i} a_{j} + v_{i} a_{j} v_{i} a_{j}) \\
& = & 9 ((\vec v \cdot \vec v \, ) (\vec a \cdot \vec a \, ) + 2 (\vec v \cdot \vec a \, ) (\vec v \cdot \vec a \, )
 + (\vec v \cdot \vec v \, ) (\vec a \cdot \vec a \, )) \\
& = & 9 (2 (\vec v \cdot \vec v \, ) (\vec a \cdot \vec a \, ) + 2 (\vec v \cdot \vec a \, )^{2}) \\
& = & 9 (2 (v^{2}) (a^{2}) + 2 (\vec v \cdot \vec a \, )^{2}) \\
& = & 18 v^{2} a^{2} + 18 (\vec v \cdot \vec a \, )^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25)} \qquad \left ( \frac{\mbox{d}^{3} I}{\mbox{d}t^{3}} \right )^{2}
 & = & \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3}
 \mu^{2} ((r_{j} j_{i} + r_{i} j_{j})^{2} + 6 (r_{j} j_{i} + r_{i} j_{j}) (a_{i} v_{j} + v_{i} a_{j}) + 9 (a_{i} v_{j} + v_{i} a_{j})^{2}) \\
& = & \mu^{2} (2 r^{2} j^{2} + 2 (\vec r \cdot \vec j \, )^{2}
 + 12 (\vec r \cdot \vec v \, ) (\vec a \cdot \vec j \, ) + 12 (\vec r \cdot \vec a \, ) (\vec v \cdot \vec j \, )
 + 18 v^{2} a^{2} + 18 (\vec v \cdot \vec a \, )^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26)} \qquad \Tr I
 & = & \sum_{k = 1}^{k = 3} I_{kk} \\
& = & \sum_{k = 1}^{k = 3} \mu r_{k} r_{k} \\
& = & \mu \sum_{k = 1}^{k = 3} r_{k}^{2} \\
& = & \mu r^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27)} \qquad \frac{\mbox{d} \Tr I}{\mbox{d}t}
 & = & \frac{\mbox{d}}{\mbox{d}t} (\mu r^{2}) \\
& = & 2 \mu r \, \frac{\mbox{d}r}{\mbox{d}t} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(28)} \qquad \frac{\mbox{d}^{2} \Tr I}{\mbox{d}t^{2}}
 & = & \frac{\mbox{d}}{\mbox{d}t} \left ( \frac{\mbox{d} \Tr I}{\mbox{d}t} \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}t} \left ( 2 \mu r \, \frac{\mbox{d}r}{\mbox{d}t} \right ) \\
& = & 2 \mu \, \frac{\mbox{d}r}{\mbox{d}t} \, \frac{\mbox{d}r}{\mbox{d}t} + 2 \mu r \, \frac{\mbox{d}^{2} r}{\mbox{d}t^{2}} \\
& = & 2 \mu \left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2} + 2 \mu r \, \frac{\mbox{d}^{2} r}{\mbox{d}t^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(29)} \qquad \frac{\mbox{d}^{3} \Tr I}{\mbox{d}t^{3}}
 & = & \frac{\mbox{d}}{\mbox{d}t} \left ( \frac{\mbox{d}^{2} \Tr I}{\mbox{d}t^{2}} \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}t} \left ( 2 \mu \left ( \frac{\mbox{d}r}{\mbox{d}t} \right )^{2} + 2 \mu r \, \frac{\mbox{d}^{2} r}{\mbox{d}t^{2}} \right ) \\
& = & 4 \mu \, \frac{\mbox{d}r}{\mbox{d}t} \, \frac{\mbox{d}^{2} r}{\mbox{d}t^{2}}
 + 2 \mu \, \frac{\mbox{d}r}{\mbox{d}t} \, \frac{\mbox{d}^{2} r}{\mbox{d}t^{2}}
 + 2 \mu r \, \frac{\mbox{d}^{3} r}{\mbox{d}t^{3}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(30)} \qquad \frac{\mbox{d}^{3} \Tr I}{\mbox{d}t^{3}}
 & = & 4 \mu \, \frac{\mbox{d}r}{\mbox{d}t} \, \frac{\mbox{d}^{2} r}{\mbox{d}t^{2}}
 + 2 \mu \, \frac{\mbox{d}r}{\mbox{d}t} \, \frac{\mbox{d}^{2} r}{\mbox{d}t^{2}}
 + 2 \mu r \, \frac{\mbox{d}^{3} r}{\mbox{d}t^{3}} \\
& = & 4 \mu v_{r} a_{r} + 2 \mu v_{r} a_{r} + 2 \mu r j_{r} \\
& = & 6 \mu v_{r} a_{r} + 2 \mu r j_{r} \\
& = & \mu (2 r j_{r} + 6 v_{r} a_{r}) \\
& = & \mu (2 (\vec r \cdot \vec j \, ) + 6 (\vec v \cdot \vec a \, )) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(31)} \qquad \left ( \frac{\mbox{d}^{3} \Tr I}{\mbox{d}t^{3}} \right )^{2}
 & = & (\mu (2 (\vec r \cdot \vec j \, ) + 6 (\vec v \cdot \vec a \, )))^{2} \\
& = & \mu^{2} (2 (\vec r \cdot \vec j \, ) + 6 (\vec v \cdot \vec a \, ))^{2} \\
& = & \mu^{2} (4 (\vec r \cdot \vec j \, )^{2} + 24 (\vec r \cdot \vec j \, ) (\vec v \cdot \vec a \, ) + 36 (\vec v \cdot \vec a \, )^{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(32)} \qquad P
 & = & \frac{G}{5 c^{5}} \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} \left ( \dddot Q_{ij}^{T} \right )^{2} \\
& = & \frac{G}{5 c^{5}} \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} \left ( \frac{\mbox{d}^{3} Q_{ij}^{T}}{\mbox{d}t^{3}} \right )^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(33)} \qquad P
 & = & \frac{G}{5 c^{5}} \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} \left ( \frac{\mbox{d}^{3} Q_{ij}^{T}}{\mbox{d}t^{3}} \right )^{2} \\
& = & \frac{G}{5 c^{5}} \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} \left (
 \left ( \frac{\mbox{d}^{3} I_{ij}}{\mbox{d}t^{3}} \right )^{2} - \frac{1}{3} \left ( \frac{\mbox{d}^{3} \Tr I}{\mbox{d}t^{3}} \right )^{2} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(34)} \qquad P
 & = & \frac{G}{5 c^{5}} \sum_{i = 1}^{i = 3} \sum_{j = 1}^{j = 3} \left (
 \left ( \frac{\mbox{d}^{3} I_{ij}}{\mbox{d}t^{3}} \right )^{2} - \frac{1}{3} \left ( \frac{\mbox{d}^{3} \Tr I}{\mbox{d}t^{3}} \right )^{2} \right ) \\
& = & \frac{G}{5 c^{5}}
 \Biggl ( \mu^{2} (2 r^{2} j^{2} + 2 (\vec r \cdot \vec j \, )^{2}
 + 12 (\vec r \cdot \vec v \, ) (\vec a \cdot \vec j \, ) + 12 (\vec r \cdot \vec a \, ) (\vec v \cdot \vec j \, )
 + 18 v^{2} a^{2} + 18 (\vec v \cdot \vec a \, )^{2} \\
 & & \qquad \qquad \qquad - \, \frac{1}{3} \mu^{2}
 (4 (\vec r \cdot \vec j \, )^{2} + 24 (\vec r \cdot \vec j \, ) (\vec v \cdot \vec a \, ) + 36 (\vec v \cdot \vec a \, )^{2}) \Biggr ) \\
& = & \frac{\mu^{2} G}{5 c^{5}}
 \Biggl ( 2 r^{2} j^{2} + 2 (\vec r \cdot \vec j \, )^{2}
 + 12 (\vec r \cdot \vec v \, ) (\vec a \cdot \vec j \, ) + 12 (\vec r \cdot \vec a \, ) (\vec v \cdot \vec j \, )
 + 18 v^{2} a^{2} + 18 (\vec v \cdot \vec a \, )^{2} \\
 & & \qquad \qquad \qquad - \,
 \frac{4}{3} (\vec r \cdot \vec j \, )^{2} - 8 (\vec r \cdot \vec j \, ) (\vec v \cdot \vec a \, ) - 12 (\vec v \cdot \vec a \, )^{2} \Biggr ) \\
& = & \frac{\mu^{2} G}{5 c^{5}}
 \Biggl ( 2 r^{2} j^{2} + \frac{2}{3} (\vec r \cdot \vec j \, )^{2}
 + 12 (\vec r \cdot \vec v \, ) (\vec a \cdot \vec j \, ) + 12 (\vec r \cdot \vec a \, ) (\vec v \cdot \vec j \, )
 + 18 v^{2} a^{2} + 6 (\vec v \cdot \vec a \, )^{2} \\
 & & \qquad \qquad \qquad - \, 8 (\vec r \cdot \vec j \, ) (\vec v \cdot \vec a \, ) \Biggr ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(35)} \qquad P
 & = & \frac{\mu^{2} G}{5 c^{5}}
 \Biggl ( 2 r^{2} j^{2} + \frac{2}{3} (\vec r \cdot \vec j \, )^{2}
 + 12 (\vec r \cdot \vec v \, ) (\vec a \cdot \vec j \, ) + 12 (\vec r \cdot \vec a \, ) (\vec v \cdot \vec j \, )
 + 18 v^{2} a^{2} + 6 (\vec v \cdot \vec a \, )^{2} \\
 & & \qquad \qquad \qquad - \, 8 (\vec r \cdot \vec j \, ) (\vec v \cdot \vec a \, ) \Biggr ) \\
& = & \frac{\mu^{2} G}{5 c^{5}}
 \Biggl ( 2 r^{2} (\vec j \cdot \vec j \, ) + \frac{2}{3} (- 2 v_{r} a)^{2}
 + 12 r v_{r} \left ( - \frac{2 v_{r} a^{2}}{r} \right ) + 12 r a \left ( \frac{a}{r} (v^{2} - 3 v_{r}^{2}) \right ) + 18 v^{2} a^{2} \\
 & & \qquad \qquad \qquad + \, 6 (\vec v \cdot \vec a \, )^{2} - 8 (- 2 v_{r} a) (\vec v \cdot \vec a \, ) \Biggr ) \\
& = & \frac{\mu^{2} G}{5 c^{5}}
 \Biggl ( 2 r^{2} \left ( \frac{a}{r} (\vec v - 3 v_{r} \hat r \, ) \right ) \cdot \left ( \frac{a}{r} (\vec v - 3 v_{r} \hat r \, ) \right )
 + \frac{8}{3} (\vec v \cdot \vec a \, )^{2} - 24 v_{r}^{2} a^{2} + 12 a^{2} (v^{2} - 3 v_{r}^{2}) + 18 v^{2} a^{2} \\
 & & \qquad \qquad \qquad + \, 6 (\vec v \cdot \vec a \, )^{2} + 16 v_{r} a (\vec v \cdot \vec a \, ) \Biggr ) \\
& = & \frac{\mu^{2} G}{5 c^{5}}
 \Biggl ( 2 r^{2} \frac{a^{2}}{r^{2}} (\vec v \cdot \vec v - 3 v_{r} (\vec v \cdot \hat r \, )
 - 3 v_{r} (\hat r \cdot \vec v \, ) + 9 v_{r}^{2} (\hat r \cdot \hat r \, ))
 + \frac{26}{3} (\vec v \cdot \vec a \, )^{2} - 24 (\vec v \cdot \vec a \, )^{2} + 12 v^{2} a^{2} \\
 & & \qquad \qquad \qquad - \, 36 v_{r}^{2} a^{2} + 18 v^{2} a^{2} + 16 (\vec v \cdot \vec a \, ) (\vec v \cdot \vec a \, ) \Biggr ) \\
& = & \frac{\mu^{2} G}{5 c^{5}} \left ( 2 a^{2} (v^{2} - 6 v_{r}^{2} + 9 v_{r}^{2})
 + \frac{2}{3} (\vec v \cdot \vec a \, )^{2} + 30 v^{2} a^{2} - 36 (\vec v \cdot \vec a \, )^{2} \right ) \\
& = & \frac{\mu^{2} G}{5 c^{5}} \left ( 2 v^{2} a^{2} + 6 v_{r}^{2} a^{2} - \frac{106}{3} (\vec v \cdot \vec a \, )^{2} + 30 v^{2} a^{2} \right ) \\
& = & \frac{\mu^{2} G}{5 c^{5}} \left ( 32 v^{2} a^{2} + 6 (\vec v \cdot \vec a \, )^{2} - \frac{106}{3} (\vec v \cdot \vec a \, )^{2} \right ) \\
& = & \frac{\mu^{2} G}{5 c^{5}} \left ( 32 v^{2} a^{2} - \frac{88}{3} (\vec v \cdot \vec a \, )^{2} \right ) \\
& = & \frac{8 \mu^{2} G}{15 c^{5}} (12 v^{2} a^{2} - 11 (\vec v \cdot \vec a \, )^{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(36)} \qquad P
 & = & \frac{8 \mu^{2} G}{15 c^{5}} (12 v^{2} a^{2} - 11 (\vec v \cdot \vec a \, )^{2}) \\
& = & \frac{8 \mu^{2} G}{15 c^{5}} (12 v^{2} a^{2} - 11 v_{r}^{2} a^{2}) \\
& = & \frac{8 \mu^{2} G}{15 c^{5}} a^{2} (12 v^{2} - 11 v_{r}^{2}) \\
& \Rightarrow & \mbox{Let op: hiervoor is $a$ de versnelling, maar hierna is $a$ de halve lange baanas.} \, \, \, \Leftarrow \\
& = & \frac{8 \mu^{2} G}{15 c^{5}} \left ( - \frac{G M}{r^{2}} \right )^{2}
 \left ( 12 \left ( \sqrt{G M \left ( \frac{2}{r} - \frac{1}{a} \right )} \, \right )^{2}
 - 11 \left ( \sqrt{\frac{G M}{a (1 - e^{2})}} \, e \sin \theta \right )^{2} \right ) \\
& = & \frac{8 \mu^{2} G}{15 c^{5}} \frac{G^{2} M^{2}}{r^{4}}
 \left ( 12 G M \left ( \frac{2}{r} - \frac{1}{a} \right ) - \frac{11 G M e^{2} \sin^{2} \theta}{a (1 - e^{2})} \right ) \\
& = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5}} \frac{1}{r^{4}}
 \left ( \frac{24}{r} - \frac{12}{a} - \frac{11 e^{2} (1 - \cos^{2} \theta)}{a (1 - e^{2})} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(37)} \qquad P
 & = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5}} \frac{1}{r^{4}}
 \left ( \frac{24}{r} - \frac{12}{a} - \frac{11 e^{2} (1 - \cos^{2} \theta)}{a (1 - e^{2})} \right ) \\
& = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5}} \left ( \frac{a (1 - e^{2})}{1 + e \cos \theta} \right )^{-4}
 \left ( 24 \left ( \frac{a (1 - e^{2})}{1 + e \cos \theta} \right )^{-1} - \frac{12}{a} - \frac{11 e^{2} (1 - \cos^{2} \theta)}{a (1 - e^{2})} \right ) \\
& = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{4}} \frac{(1 + e \cos \theta)^{4}}{(1 - e^{2})^{4}}
 \left ( \frac{24 (1 + e \cos \theta)}{a (1 - e^{2})} - \frac{12}{a}
 - \frac{11 e^{2}}{a (1 - e^{2})} + \frac{11 e^{2} \cos^{2} \theta}{a (1 - e^{2})} \right ) \\
& = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e \cos \theta)^{4}}{(1 - e^{2})^{5}}
 (24 (1 + e \cos \theta) - 12 (1 - e^{2}) - 11 e^{2} + 11 e^{2} \cos^{2} \theta) \\
& = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e \cos \theta)^{4}}{(1 - e^{2})^{5}}
 (24 + 24 e \cos \theta - 12 + 12 e^{2} - 11 e^{2} + 11 e^{2} \cos^{2} \theta) \\
& = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e \cos \theta)^{4}}{(1 - e^{2})^{5}}
 (12 + e^{2} + 24 e \cos \theta + 11 e^{2} \cos^{2} \theta) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(38)} \qquad f (e, \theta)
 & = & \frac{(1 + e \cos \theta)^{4}}{(1 - e^{2})^{5}} (12 + e^{2} + 24 e \cos \theta + 11 e^{2} \cos^{2} \theta) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(39)} \qquad P_{\theta = 0}
 & = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e \cos 0)^{4}}{(1 - e^{2})^{5}}
 (12 + e^{2} + 24 e \cos 0 + 11 e^{2} \cos^{2} 0) \\
& = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e)^{4}}{(1 - e^{2})^{5}} (12 + e^{2} + 24 e + 11 e^{2}) \\
& = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e)^{4}}{(1 - e^{2})^{5}} (12 + 24 e + 12 e^{2}) \\
& = & \frac{96 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e)^{4}}{(1 - e^{2})^{5}} (1 + 2 e + e^{2}) \\
& = & \frac{96 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e)^{4}}{(1 - e^{2})^{5}} (1 + e)^{2} \\
& = & \frac{96 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e)^{6}}{(1 - e^{2})^{5}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(40)} \qquad \Delta E
 & = & P \, \Delta t \\
& = & \frac{96 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e)^{6}}{(1 - e^{2})^{5}} \, \Delta t \\
& = & \frac{96 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e)^{6}}{(1 - e^{2})^{5}} \cdot 1 \\
& = & \frac{96 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e)^{6}}{(1 - e^{2})^{5}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(41)} \qquad \Delta E
 & = & \Delta m \, c^{2} \\
\Delta m
 & = & \frac{\Delta E}{c^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(42)} \qquad \Delta m
 & = & \frac{\Delta E}{c^{2}} \\
& = & \frac{1}{c^{2}} \frac{96 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e)^{6}}{(1 - e^{2})^{5}} \\
& = & \frac{96 \mu^{2} G^{4} M^{3}}{15 c^{7} a^{5}} \frac{(1 + e)^{6}}{(1 - e^{2})^{5}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(43)} \qquad P_{e = 0}
 & = & \left ( \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5} a^{5}} \frac{(1 + e \cos \theta)^{4}}{(1 - e^{2})^{5}}
 (12 + e^{2} + 24 e \cos \theta + 11 e^{2} \cos^{2} \theta) \right )_{e = 0} \\
& = & \frac{8 \mu^{2} G^{4} M^{3}}{15 c^{5} r^{5}} \frac{(1 + 0)^{4}}{(1 - 0)^{5}} (12 + 0 + 0 + 0) \\
& = & \frac{32 \mu^{2} G^{4} M^{3}}{5 c^{5} r^{5}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(44)} \qquad v
 & = & \sqrt{G M \left ( \frac{2}{r} - \frac{1}{a} \right )} \\
& = & \sqrt{G M \left ( \frac{2}{r} - \frac{1}{r} \right )} \\
& = & \sqrt{\frac{G M}{r}} \\
v^{2}
 & = & \frac{G M}{r} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(45)} \qquad P_{e = 0}
 & = & \frac{32 \mu^{2} G^{4} M^{3}}{5 c^{5} r^{5}} \\
& = & \frac{32 \mu^{2} G}{5 c^{5} r^{2}} \frac{G^{3} M^{3}}{r^{3}} \\
& = & \frac{32 \mu^{2} G v^{6}}{5 c^{5} r^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(46)} \qquad \omega
 & = & \frac{v}{r} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(47)} \qquad P_{e = 0}
 & = & \frac{32 \mu^{2} G v^{6}}{5 c^{5} r^{2}} \\
& = & \frac{32 \mu^{2} G r^{4}}{5 c^{5}} \frac{v^{6}}{r^{6}} \\
& = & \frac{32 \mu^{2} G r^{4} \omega^{6}}{5 c^{5}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(48)} \qquad I
 & = & \mu r^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(49)} \qquad P_{e = 0}
 & = & \frac{32 \mu^{2} G r^{4} \omega^{6}}{5 c^{5}} \\
& = & \frac{32 G I^{2} \omega^{6}}{5 c^{5}} \\
\end{eqnarray*}
\end{document}
