Global Reference System

The global reference orbit of the accelerator is uniquely defined by the sequence of physical elements. The local reference system $(x,y,s)$ may thus be referred to a global Cartesian coordinate system $(X, Y, Z)$. The positions between beam elements are numbered $0, \ldots , i, \ldots n$. The local reference system $(x_i, y_i, s_i)$ at position $i$, i.e. the displacement and direction of the reference orbit with respect to the system $(X, Y, Z)$ are defined by three displacements $(X_i, Y_i, Z_i)$ and three angles $(\Theta_i, \Phi_i, \Psi_i)$.

Figure 1.2: Global Reference System

The above quantities are defined more precisely as follows:

X

Displacement of the local origin in $X$-direction.

Y

Displacement of the local origin in $Y$-direction.

Z

Displacement of the local origin in $Z$-direction.

THETA

Angle of rotation (azimuth) about the global $Y$-axis, between the global $Z$-axis and the projection of the reference orbit onto the $(Z, X)$-plane. A positive angle THETA forms a right-hand screw with the $Y$-axis.

PHI

Pitch angle, i.e. the angle between the reference orbit and its projection onto the $(Z, X)$-plane. A positive angle PHI correspond to $Y$ increasing with $s$. If only horizontal bends are present, the reference orbit remains in the ($Z$, $X$)-plane. In this case PHI is always zero.

PSI

Roll angle about the local $s$-axis, i.e. the angle between the intersection $(x, y)$- and $(Z, X)$-planes and the local $x$-axis. A positive angle PSI forms a right-hand screw with the $s$-axis.

The angles (THETA, PHI, PSI) are not the Euler angles. The reference orbit starts at the origin and points by default in the direction of the positive $Z$-axis. The initial local axes $(x,y,s)$ coincide with the global axes $(X, Y, Z)$ in this order. The six quantities $(X_0, Y_0, Z_0, \Theta_0, \Phi_0, \Psi_0)$ thus all have zero initial values by default. The program user may however specify different initial conditions.

Internally the displacement is described by a vector $V$ and the orientation by a unitary matrix $W$. The column vectors of $W$ are the unit vectors spanning the local coordinate axes in the order $(x,y,s)$. $V$ and $W$ have the values:

$$V=\left(\begin{array}{c} X \\ Y \\ Z \end{array}\right), \qquad W=\Theta\Phi\Psi$$

where

$$\Theta=\left(\begin{array}{ccc} \cos\theta & 0 & \sin\theta... ... \sin\psi & \cos\psi & 0 \\ 0 & 0 & 1 \end{array}\right).$$

The reference orbit should be closed and it should not be twisted. This means that the displacement of the local reference system must be periodic with the revolution frequency of the accelerator, while the position angles must be periodic $\pmod{2\pi}$ with the revolution frequency. If PSI is not periodic $\pmod{2\pi}$, coupling effects are introduced. When advancing through a beam element, MAD computes $V_i$ and $W_i$ by the recurrence relations

$$V_i = W_{i-1}R_i + V_{i-1}, \qquad W_i = w_{i-1}S_i.$$

The vector $R_i$ is the displacement and the matrix $S_i$ is the rotation of the local reference system at the exit of the element $i$ with respect to the entrance of the same element. The values of $R_i$ and $S_i$ are listed below for each physical element type.