\documentclass[../main.tex]{subfiles}
\graphicspath{{\subfix{../images/}}}

\begin{document}
\section{Conclusion}
We studied the movement of particles confined by an ideal Penning trap, where we used iterative methods to simulate the particle behavior. We included the magnetic and electric field of the Penning trap, in addition to simulating the particles behavior when interaction with other each other. When we introduced the interaction, the movement in both radial direction and z-direction changed. From a circular path, to a more elliptical path, where the particles initial condition determine how it is affecting other particles path. 

We also compared iterative methods, with the analytical solution, and found that the forward Euler \(FE\) method result in an approximation with a large relative error compared to the relative error of the 4th order Runge-kutta \(RK4\) method. In addition, we also found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. Which suggest RK4 reach the solution faster than what FE, however, when we increase the number of time steps both methods result in similar relative error. When the number of calculations increase, and the number of time steps is sufficient, FE can be the better choise to conserve computational resources.

When we explored the particles behavior at angular frequencies $\omega_{V} \in (0.2, 2.5)$ MHz, we found that particles are pushed out of the Penning trap when the amplitude of the applied time-dependent potential increase. The amplitude $f = 0.7$ result in particles being pushed out at most of the range of angular frequencies, whereas an amplitude $f = 0.1$ result in particles being pushed in a more narrow range. Since particles are being pushed out when the amplitude is low, there is likely a resonance frequency at around $1.4$ MHz.
\end{document}