Commit 76e88ed3 authored by snuverink_j's avatar snuverink_j
Browse files

replace matr with mathcal

parent 75242662
......@@ -95,10 +95,10 @@ The local coordinates $(x_i, y_i, z_i)$ at element $e_i$ with respect to the glo
$\Psi$ is the roll angle about the global $Z$-axis. $\Phi$ is the pitch angle about the global $Y$-axis. Lastly, $\Theta$ is the yaw angle about the global $X$-axis. All three angles form right-handed screws with their corresponding axes. The angles ($\Theta,\Phi,\Psi$) are the Tait-Bryan angles \ref{bib:tait-bryan}.
The displacement is described by a vector $\mathbf{v}$
and the orientation by a unitary matrix $\matr{W}$.
The column vectors of $\matr{W}$ are unit vectors spanning
and the orientation by a unitary matrix $\mathcal{W}$.
The column vectors of $\mathcal{W}$ are unit vectors spanning
the local coordinate axes in the order $(x, y, z)$.
$\mathbf{v}$ and $\matr{W}$ have the values:
$\mathbf{v}$ and $\mathcal{W}$ have the values:
\begin{equation}
\mathbf{v} =\left(\begin{array}{c}
X \\
......@@ -106,24 +106,24 @@ $\mathbf{v}$ and $\matr{W}$ have the values:
Z
\end{array}\right),
\qquad
\matr{W}=\matr{S}\matr{T}\matr{U}
\mathcal{W}=\mathcal{S}\mathcal{T}\mathcal{U}
\end{equation}
where
\begin{equation}
\matr{S}=\left(\begin{array}{ccc}
\mathcal{S}=\left(\begin{array}{ccc}
\cos\Theta & 0 & n\Theta \\
0 & 1 & 0 \\
-n\Theta & 0 & \cos\Theta
\end{array}\right),
\quad
\matr{T}=\left(\begin{array}{ccc}
\mathcal{T}=\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos\Phi & n\Phi \\
0 & -n\Phi & \cos\Phi
\end{array}\right),
\end{equation}
\begin{equation}
\matr{U}=\left(\begin{array}{ccc}
\mathcal{U}=\left(\begin{array}{ccc}
\cos\Psi & -n\Psi & 0 \\
n\Psi & \cos\Psi & 0 \\
0 & 0 & 1
......@@ -132,18 +132,18 @@ where
We take the vector $\mathbf{r}_i$ to be the displacement and the matrix
$\matr{S}_i$ to be the rotation of the local reference system
$\mathcal{S}_i$ to be the rotation of the local reference system
at the exit of the element $i$ with respect to the entrance
of that element.
Denoting with $i$ a beam line element,
one can compute $\mathbf{v}_i$ and $\matr{W}_i$
one can compute $\mathbf{v}_i$ and $\mathcal{W}_i$
by the recurrence relations
\begin{equation} \label{eq:surv}
\mathbf{v}_i = \matr{W}_{i-1}\mathbf{r}_i + \mathbf{v}_{i-1}, \qquad
\matr{W}_i = \matr{W}_{i-1}\matr{S}_i,
\mathbf{v}_i = \mathcal{W}_{i-1}\mathbf{r}_i + \mathbf{v}_{i-1}, \qquad
\mathcal{W}_i = \mathcal{W}_{i-1}\mathcal{S}_i,
\end{equation}
where $\mathbf{v}_0$ corresponds to the origin of the \texttt{LINE} and $\matr{W}_0$ to its orientation. In \textit{OPAL-t} they can be defined using either \texttt{X}, \texttt{Y}, \texttt{Z}, \texttt{THETA}, \texttt{PHI} and \texttt{PSI} or \texttt{ORIGIN} and \texttt{ORIENTATION}, see Section~\ref{line:simple}.
where $\mathbf{v}_0$ corresponds to the origin of the \texttt{LINE} and $\mathcal{W}_0$ to its orientation. In \textit{OPAL-t} they can be defined using either \texttt{X}, \texttt{Y}, \texttt{Z}, \texttt{THETA}, \texttt{PHI} and \texttt{PSI} or \texttt{ORIGIN} and \texttt{ORIENTATION}, see Section~\ref{line:simple}.
\subsubsection{Space Charge Coordinate System}
In order to calculate space charge in the electrostatic approximation, we introduce a co-moving coordinate system $K_{\text{sc}}$, in which the origin coincides with the mean position of the particles and the mean momentum is parallel to the z-axis.
......
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