latex/sections/methods.tex

@@ -6,7 +6,154 @@
 
 \section{Methods}
 
+Newton's second law of motion is defined as
+\begin{equation}
+    \frac{d^2\vb{r}}{dt^2} = \frac{\vb{F}\left(t, \frac{d\vb{r}}{dt}, \vb{r}\right)}{m}.
+    \label{eq:newtonlaw2}
+\end{equation}
+We can then rewrite the second order ODE from equation~\ref{eq:newtonlaw2} 
+into a set of coupled first order ODEs. 
+We now redefine Newton's second law of motion as
+\begin{equation}
+    \begin{split}
+        \frac{d\vb{r}}{dt} &= \vb{v} \\
+        \frac{d\vb{v}}{dt} &= \frac{\vb{F}(t, \vb{v}, \vb{r})}{m}.
+    \end{split}
+    \label{eq:coupled}
+\end{equation}
 
+\subsection{Forward Euler}
+
+For a single particle, the forward Euler method for a coupled system is 
+expressed as
+
+\begin{equation}
+    \begin{split}
+        \vb{r}_{i+1} &= \vb{r}_i + h \cdot \frac{d\vb{r}_i}{dt} = \vb{r}_i + h \cdot \vb{v}_i \\
+        \vb{v}_{i+1} &= \vb{v}_i + h \cdot \frac{\vb{v}_i}{dt} = \vb{v}_i + h \cdot \frac{\vb{F}\left( t_i, \vb{v}_i, \vb{r}_i \right)}{m},
+    \end{split}
+\end{equation}
+where $\vb{v}_i$ is the current velocity of the particle, $\vb{r}_i$ is the
+current position of the particle, $m$ is the mass of the particle, and $h$
+is a predetermined timestep.
+
+When dealing with a multi-particle system, we need to ensure that we do not
+update the position of any particles until every particle has calculated their
+next step. An easy way of doing this is to create a copy of all the particles,
+then update the copy, and when all the particles have calculated their next 
+step, simply replace the particles with the copies. 
+Algorithm~\ref{algo:forwardeuler} provides an overview on how that can be achieved. % Make this better
+
+\begin{figure}[H]
+    \begin{algorithm}[H]
+    \caption{Forward Euler method}
+    \label{algo:forwardeuler}
+        \begin{algorithmic}
+            \Procedure{Evolve forward Euler}{$particles, dt$}
+            \State $new\_state \leftarrow particles$ \Comment{Create a copy of the particles}
+            \For{ $i = 1, 2, \ldots , \left| particles \right|$ }
+            \State $new\_state_i.\vb{v} \leftarrow particles_i.\vb{v} + dt \cdot \frac{\vb{F}}{m}$
+            \State $new\_state_i.\vb{r} \leftarrow particles_i.\vb{r} + dt \cdot particles_i.\vb{v}$
+            \EndFor
+            \State $particles \leftarrow new\_state$
+            \EndProcedure
+        \end{algorithmic}
+    \end{algorithm}
+\end{figure}
+
+\subsection{4th order Runge-Kutta}
+
+For a single particle, we can express the 4th order Runge-Kutta (RK4) method as
+\begin{equation}
+    \begin{split}
+        \vb{v}_{i+1} &= \vb{v}_i + \frac{h}{6} \left( \vb{k}_{\vb{v},1} 
+            + 2\vb{k}_{\vb{v},2} + 2\vb{k}_{\vb{v},3} + \vb{k}_{\vb{v},4} 
+            \right) \\
+        \vb{r}_{i+1} &= \vb{r}_i + \frac{h}{6} \left( \vb{k}_{\vb{r},1} 
+            + 2\vb{k}_{\vb{r},2} + 2\vb{k}_{\vb{r},3} + \vb{k}_{\vb{r},4} 
+            \right),
+    \end{split}
+\end{equation}
+where
+\begin{equation}
+    \begin{split}
+    \vb{k}_{\vb{v},1} &= \frac{\vb{F}(t_i, \vb{v}_i, \vb{r}_i)}{m} \\
+    \vb{k}_{\vb{r},1} &= \vb{v}_i \\
+    \vb{k}_{\vb{v},2} &= \frac{\vb{F}(t_i+\frac{h}{2}, \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},1}}{2}, \vb{r}_i + h \cdot \frac{\vb{k}_{\vb{r},1}}{2})}{m} \\
+    \vb{k}_{\vb{r},2} &= \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},1}}{2} \\
+    \vb{k}_{\vb{v},3} &= \frac{\vb{F}(t_i+\frac{h}{2}, \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},2}}{2}, \vb{r}_i + h \frac{\cdot \vb{k}_{\vb{r},2}}{2})}{m} \\
+    \vb{k}_{\vb{r},3} &= \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},2}}{2} \\
+    \vb{k}_{\vb{v},4} &= \frac{\vb{F}(t_i+h, \vb{v}_i + h \cdot \vb{k}_{\vb{v},3}, \vb{r}_i + h \cdot \vb{k}_{\vb{r},3})}{m} \\
+    \vb{k}_{\vb{r},4} &= \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},1}}{2}.
+    \end{split}
+\end{equation}
+
+\newpage
+
+\begin{figure}
+    \begin{algorithm}[H]
+    \caption{RK4 method}
+    \label{algo:rk4}
+        \begin{algorithmic}
+            \Procedure{Evolve RK4}{$dt$}
+            \State $N \leftarrow \text{Number of particles inside the Penning trap}$
+            \State $orig\_p \leftarrow \text{Copy of particles}$ \Comment{Keep the original particles}
+            \State $tmp\_p \leftarrow \text{Array of particles of size }N$
+            \State $\vb{k}_{\vb{r}} \leftarrow \text{2D array of vectors of size } 4 \cross N$
+            \State $\vb{k}_{\vb{v}} \leftarrow \text{2D array of vectors of size } 4 \cross N$
+
+            \For{ $i = 1, 2, \ldots, N$ }
+            \State $\vb{k}_{\vb{r},1,i} \leftarrow \vb{v}$
+            \State $\vb{k}_{\vb{v},1,i} \leftarrow \frac{\vb{F}}{m}$
+
+            \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{2} \cdot \vb{k}_{\vb{r},1,i}$
+            \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{2} \cdot \vb{k}_{\vb{v},1,i}$
+            \EndFor
+
+            \State $particles \leftarrow tmp\_p$ \Comment{Update particles}
+
+            \For{ $i = 1, 2, \ldots, N$ }
+            \State $\vb{k}_{\vb{r},2,i} \leftarrow \vb{v}$
+            \State $\vb{k}_{\vb{v},2,i} \leftarrow \frac{\vb{F}}{m}$
+
+            \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{2} \cdot \vb{k}_{\vb{r},2,i}$
+            \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{2} \cdot \vb{k}_{\vb{v},2,i}$
+            \EndFor
+
+            \State $particles \leftarrow tmp\_p$ \Comment{Update particles}
+
+            \For{ $i = 1, 2, \ldots, N$ }
+            \State $\vb{k}_{\vb{r},3,i} \leftarrow \vb{v}$
+            \State $\vb{k}_{\vb{v},3,i} \leftarrow \frac{\vb{F}}{m}$
+
+            \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + dt \cdot \vb{k}_{\vb{r},3,i}$
+            \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + dt \cdot \vb{k}_{\vb{v},3,i}$
+            \EndFor
+
+            \State $particles \leftarrow tmp\_p$ \Comment{Update particles}
+
+            \For{ $i = 1, 2, \ldots, N$ }
+            \State $\vb{k}_{\vb{r},4,i} \leftarrow \vb{v}$
+            \State $\vb{k}_{\vb{v},4,i} \leftarrow \frac{\vb{F}}{m}$
+
+            \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{6} 
+            \cdot \left( \vb{k}_{\vb{r},1,i} + \vb{k}_{\vb{r},2,i} 
+            + \vb{k}_{\vb{r},3,i} + \vb{k}_{\vb{r},4,i} \right)$
+
+            \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{6} 
+            \cdot \left( \vb{k}_{\vb{v},1,i} + \vb{k}_{\vb{v},2,i} 
+            + \vb{k}_{\vb{v},3,i} + \vb{k}_{\vb{v},4,i} \right)$
+            
+            \EndFor
+
+            \State $particles \leftarrow tmp\_p$ \Comment{Final update}
+            \EndProcedure
+        \end{algorithmic}
+    \end{algorithm}
+    $\vb{F}$ in the algorithm does not take any arguments as it uses the 
+    velocities and positions of the particles inside the array $particles$ to
+    calculate the total force acting on particle $i$.
+\end{figure}
 
 
 \biblio