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\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}
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%
\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}