Add introduction and (a somewhat messy) method part

latex/appendix/appendix.tex

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+\documentclass[../main.tex]{subfiles}
+\graphicspath{{\subfix{../images/}}}
+
+\begin{document}
+\appendix
+\section*{Appendix A}\label{sec:appendix_a}
+Equations given 
+\begin{equation}\label{eq:newton_second}
+    m \ddot{\mathbf{r}} = \sum_{i} \mathbf{F}_{i}
+\end{equation}
+%
+
+%
+\begin{equation}\label{eq:e_field_point}
+    \mathbf{E} = k_{e} \sum_{j=1}^{n} q_j \frac{\mathbf{r} - \mathbf{r}_{j}}{|\mathbf{r} - \mathbf{r}_{j}|^{3}}
+\end{equation}
+%
+\begin{equation}\label{eq:e_field_potential}
+    \mathbf{E} = - \nabla V
+\end{equation}
+
+\section*{Appendix B}\label{sec:appendix_b}
+Sum of all forces
+\begin{align*}
+    sum
+\end{align*}
+
+We find the physical coordinates from $x(t) = \text{Re} f(t)$ and $y(t) = \text{Im} f(t)$, where $f(t)$ is given by equation \eqref{eq:general_solution}. 
+
+We can rewrite $f(t)$ using the definition as
+\begin{align*}
+    % f(t) =& A_{+}e^{-i(\omega_{+} t + \phi_{+})} + A_{-}e^{-i(\omega_{-} t + \phi_{-})} \\
+    f(t) =& A_{+}(\cos{\omega_{+} t + \phi_{+}} - i \sin{\omega_{+} t + \phi_{+}}) \\
+    \numberthis \label{eq:general_solution_trig}
+    &+ A_{-}(\cos{\omega_{-} t + \phi_{-}} - i \sin{\omega_{-} t + \phi_{-}})
+\end{align*}
+%
+If we rearrange the right side of equation \eqref{eq:general_solution_trig}, we find the physical coordinates
+\begin{align}\label{eq:physical_coord}
+    x(t) &= A_{+}(\cos{\omega_{+} t + \phi_{+}}) + A_{-}(\cos{\omega_{-} t + \phi_{-}}) \\
+    y(t) &= - A_{+}(i \sin{\omega_{+} t + \phi_{+}}) - A_{-}(i \sin{\omega_{-} t + \phi_{-}})
+\end{align}
+
+\end{document}