### 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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