Add introduction and (a somewhat messy) method part

latex/appendix/appendix.tex

@@ -0,0 +1,44 @@
+\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}

latex/main.tex

@@ -1,5 +1,5 @@
 \documentclass[english,notitlepage,reprint,nofootinbib]{revtex4-1}  % defines the basic parameters of the document
-\usepackage{subfiles}
+% 
 % For preview: skriv i terminal: latexmk -pdf -pvc filnavn
 % If you want a single-column, remove "reprint"
 
@@ -18,7 +18,7 @@
 \usepackage{physics,amssymb}  % mathematical symbols (physics imports amsmath)
 
 \usepackage{graphicx}         % include graphics such as plots
-\def\resources{\graphicspath{{"./images/"}}}
+\graphicspath{{images/}}
 \usepackage{xcolor}           % set colors
 \usepackage{hyperref}         % automagic cross-referencing
 \usepackage{listings}         % display code
@@ -29,7 +29,7 @@
 \usepackage[noend]{algpseudocode}
 \usepackage{subfigure}
 \usepackage{tikz}
-\usetikzlibrary{quantikz}
+% \usetikzlibrary{quantikz2}
 % defines the color of hyperref objects
 % Blending two colors:  blue!80!black  =  80% blue and 20% black
 \hypersetup{ % this is just my personal choice, feel free to change things
@@ -39,8 +39,12 @@
     urlcolor={blue!80!black}}
 
 % Biblio stuff
-\def\biblio{\bibliographystyle{plain}\bibliography{./references}}
+% \def\biblio{\bibliographystyle{plain}\bibliography{../references/references}}
+\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}}
 
+\usepackage{xr}
+\usepackage{subfiles}
+% \externaldocument[M-]{\subfix{main}}
 
 \begin{document}
 
@@ -50,28 +54,31 @@
 \noaffiliation                            % ignore this, but keep it.
 
 % Abstract
-\subfile{./sections/abstract.tex}
+\subfile{sections/abstract}
 
 
 \maketitle
 
 
 % Introduction
-\subfile{./sections/introduction.tex}
+\subfile{sections/introduction}
 
 % Methods
-\subfile{./sections/methods.tex}
+\subfile{sections/methods}
 
 % Results
-\subfile{./sections/results.tex}
+\subfile{sections/results}
 
 % Conclusion
-\subfile{./sections/conclusion.tex}
+\subfile{sections/conclusion}
+
+
+\subfile{appendix/appendix}
 
 \onecolumngrid
 
-\bibliographystyle{plain}
+\bibliographystyle{unsrt}
-\bibliography{.references/references}
+\bibliography{references/references}
 
 
 \end{document}

latex/references/references.bib

@@ -1,9 +1,77 @@
-
+@book{vogel:2018,
-@inbook{midpoint_rule,
+  author    = {Manuel Vogel},
-	author	= "A.C Faul",
+  title     = {Particle Confinement in Penning Traps},
-	title	= "A Concise Introduction to Numerical Analysis",
+  subtitle  = {An Introduction},
-	chapter	= "5",
+  publisher = {Springer Cham},
-	pages	= "131",
+  year      = {2018},
-	publisher= "CRC Press",
+  edition   = {1},
-	year	= "2016",
+  abstract  = {An introduction to the field of Penning traps and related experimental techniques. The book is motivated by the observation that often a vast number of different resources have to be explored to gain a good overview of Penning trap principles. This is especially true for students who experience additional difficulty due to the different styles of presentation and notation.}
+}
+
+@inbook{vogel:2018:ch5,
+  author    = {Manuel Vogel},
+  title     = {Particle Confinement in Penning Traps},
+  subtitle  = {An Introduction},
+  publisher = {Springer Cham},
+  year      = {2018},
+  edition   = {1},
+  pages     = {45--50}
+}
+
+@inbook{vogel:2018:ch1,
+  author    = {Manuel Vogel},
+  title     = {Particle Confinement in Penning Traps},
+  subtitle  = {An Introduction},
+  publisher = {Springer Cham},
+  year      = {2018},
+  edition   = {1--6}
+}
+
+@article{hunter:2007,
+  author    = {Hunter, J. D.},
+  title     = {Matplotlib: A 2D graphics environment},
+  journal   = {Computing in Science \& Engineering},
+  volume    = {9},
+  number    = {3},
+  pages     = {90--95},
+  abstract  = {Matplotlib is a 2D graphics package used for Python for
+               application development, interactive scripting, and publication-quality
+               image generation across user interfaces and operating systems.},
+  publisher = {IEEE COMPUTER SOC},
+  doi       = {10.1109/MCSE.2007.55},
+  year      = 2007
+}
+
+@inbook{lindstrom:2016:ch3,
+  author    = {Tom Lindstrøm},
+  title     = {Kalkulus},
+  publisher = {Universitetsforlaget},
+  year      = {2016},
+  edition   = {4},
+  pages     = {133--137}
+}
+
+@inbook{lindstrom:2016:ch10:5,
+  author    = {Tom Lindstrøm},
+  title     = {Kalkulus},
+  publisher = {Universitetsforlaget},
+  year      = {2016},
+  edition   = {4},
+  pages     = {589--606}
+}
+
+@article{britannica:2023:matter,
+  author  = {The Editors of Encyclopaedia Britannica},
+  title   = {matter},
+  year    = {2023},
+  url     = {https://www.britannica.com/science/matter},
+  urldate = {2023-10-17}
+}
+
+@inbook{gjevik:2014:appendix,
+  author    = {Bjørn Gjevik and Morten Wang Fagerland},
+  title     = {Feltteori og vektoranalyse},
+  publisher = {Farleia Forlag},
+  year      = {2018},
+  pages     = {162--163}
 }

latex/sections/abstract.tex

@@ -1,6 +1,5 @@
 \documentclass[../main.tex]{subfiles}
-\resources % search for graphics in ../res/imgs/
+\graphicspath{{\subfix{../images/}}}
-
 
 \begin{document}
 
@@ -8,5 +7,4 @@
     Add an abstract for the project?  
 \end{abstract}
 
-\biblio
 \end{document}

latex/sections/conclusion.tex

@@ -1,6 +1,5 @@
 \documentclass[../main.tex]{subfiles}
-\resources % search for graphics in ../res/imgs/
+\graphicspath{{\subfix{../images/}}}
-
 
 \begin{document}
 
@@ -9,5 +8,5 @@
 
 
 
-\biblio
+
 \end{document}

latex/sections/introduction.tex

@@ -1,13 +1,13 @@
 \documentclass[../main.tex]{subfiles}
-\resources % search for graphics in ../res/imgs/
+\graphicspath{{\subfix{../images/}}}
-
 
 \begin{document}
 
 \section{Introduction}
+We are surrounded by matter, which are made up of elementary particles. In the field of physics we want to understand the properties of these particles, measure their physical quantities, not to mention explain the origin of mass \cite{britannica:2023:matter}. 
 
+However, to study a particle, it is necessary isolate and contain it over time. The Penning trap is a device, able to confine charged particles for a period of time. This concept was evolved from F. M. Penning's implementation of magnetic fiels to a vaccum gauge, and J. R. Pierce's work with electron beams, and put into practice by Hans Dehmelt. In 1973 Dehmelt and his group of researchers were able to contain a particle and store it over several months \cite{vogel:2018:ch1}. 
 
+We want to study the effects of the Penning trap, when it is used to confine charged particles. Our focus will be on an ideal Penning trap, where an electrostatic field confines the particle in z-direction, and a magnetic field confines it in the radial direction. Using a numerical approach we will model a single particle, to study the particle motion in radial direction. In addition, we will model a system of particles, to study their motion both with and without particle interaction.
 
-
-\biblio
 \end{document}

latex/sections/methods.tex

@@ -1,13 +1,69 @@
 \documentclass[../main.tex]{subfiles}
-\resources % search for graphics in ../res/imgs/
+\graphicspath{{\subfix{../images/}}}
-
 
 \begin{document}
-
 \section{Methods}
+% problem 1
+When we study the Penning traps effect on a particle with a charge $q$, we need to consider the forces acting on the particle. The sum of all forces acting on the particle, is given by the Lorentz force \eqref{eq:lorentz_force}.
+\begin{equation}\label{eq:lorentz_force}
+    \mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B},
+\end{equation}
+We can use Newton's second law \eqref{eq:newton_second} to determine this sum by 
+\begin{align*}
+    \ddot{\mathbf{r}} &= \frac{1}{m} \sum_{i} \mathbf{F}_{i} \\
+    &= \frac{1}{m} (q \mathbf{E} + q \mathbf{v} \times \mathbf{B}) \\
+    &= \frac{q}{m} \big(\frac{V_{0}}{d^{2}} (x, y, -2z) + (B_{0}\dot{y}, -B_{0}\dot{x}, 0) \big),
+\end{align*}
+where the complete derivation can be found in \ref{sec:appendix_b}. We can now write the particle's position as
+\begin{align}
+    \label{eq:motion_x}
+    \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x &= 0, \\
+    \label{eq:motion_y}
+    \ddot{y} + \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y &= 0, \\
+    \label{eq:motion_z}
+    \ddot{z} + \omega_{z}^{2} z &= 0,
+\end{align}
+% define w_0 og w_z i appendix?
+In addition, we can find the general solution for eq. \eqref{eq:motion_z}, when we consider the characteristic equation of a second order differential equation \cite{lindstrom:2016:ch10:5}. 
+\begin{align*}
+    r^{2} + \omega_{z}^{2} &= 0 \\
+    r &= \pm \sqrt{- \omega_{z}^{2}} = \mp i \omega_{z},
+\end{align*}
+two complex roots which gives us solutions in the form of
+\begin{equation}\label{eq:all_diff_sol}
+    z = c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t},
+\end{equation}
+For a complex number $z = a + ib$, we can define $e^{z} \equiv e^{a} (\cos{b} + i \sin{b})$ \cite{lindstrom:2016:ch3}. We can rewrite \eqref{eq:all_diff_sol} as
+\begin{align*}
+    c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t} &= c_{1} (\cos{\omega_{z} t} + i \sin{\omega_{z} t}) + c_{2} (\cos{\omega_{z} t} - i \sin{\omega_{z} t} \\
+    &= E \cos{\omega_{z} t} + i F \sin{\omega_{z} t}
+\end{align*}
+%
+Since \eqref{eq:motion_x} and \eqref{eq:motion_y} are coupled, we want to rewrite it as a single differential equation. We can obtain this by introducing $f(t) = x(t) + iy(t)$,
+\begin{align*}
+    (\ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x) + i (\ddot{y} + \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y) &= 0 \\
+    \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x + i\ddot{y} + i\omega_{0} \dot{x} - i \frac{1}{2} \omega_{z}^{2} y &= 0 \\
+    \ddot{x} + i\ddot{y} + i\omega_{0} \dot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x - i \frac{1}{2} \omega_{z}^{2} y &= 0 \\ 
+    \ddot{x} + i\ddot{y} + i\omega_{0} (\dot{x} + i \dot{y}) - \frac{1}{2} \omega_{z}^{2} x - i \frac{1}{2} \omega_{z}^{2} y &= 0 \text{where $i \omega_{0} \dot{x} + (-1) \omega_{0} \dot{y} = i \omega_{0} \dot{x} + i^{2} \omega_{0} \dot{y}$} \\
+    \ddot{f} + i \omega_{0} \dot{f} - \frac{1}{2} \omega_{z}^{2} f &= 0
+\end{align*}
 
  
 
 
-\biblio
+
+
+
+Physical properties given by newtons second law \eqref{eq:newton_second}
+\begin{equation}\label{eq:general_solution}
+    f(t) = A_{+}e^{-i(\omega_{+} t + \phi_{+})} + A_{-}e^{-i(\omega_{-} t + \phi_{-})}
+\end{equation}
+The particle moves and its position can be determined using newton. where the electric field
+
+\subsection*{Algorithm}
+
+
+\subsection*{Tools}
+We used matplotlib
+
 \end{document}

latex/sections/results.tex

@@ -1,13 +1,10 @@
 \documentclass[../main.tex]{subfiles}
-\resources % search for graphics in ../res/imgs/
+\graphicspath{{\subfix{../images/}}}
-
 
 \begin{document}
 
-\section{Results}
+\section{Results and Discussion}
+From equation \eqref{eq:general_solution} we can find
 
-
+Problem 9: we have to deal with $|\mathbf{r}| = d$
-
-
-\biblio
 \end{document}