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