at.physics.amat

A-matrix construction

Functions

a_matrix(M)	Find the \(\mathbf{A}\) matrix from one turn map \(\mathbf{M}\)
amat(M)	Find the A matrix from one turn map matrix T
jmat(ind)	antisymetric block diagonal matrix [[0, 1][-1, 0]]
jmatswap(ind)	Modified antisymetric block diagonal matrix to deal with the swap of the longitudinal coordinates
get_tunes_damp(M[, R])	Computes the mode emittances, tunes and damping times
get_mode_matrices(A)	Derives the R-matrices from the A-matrix
symplectify(M)	Makes A matrix more symplectic

a_matrix(M)

Find the \(\mathbf{A}\) matrix from one turn map \(\mathbf{M}\)

\(\mathbf{A}\) represents a change of referential which converts the one-turn transfer matrix \(\mathbf{M}\) into a set of rotations:

\[\mathbf{M} = \mathbf{A} \cdot \mathbf{R} \cdot \mathbf{A}^{-1}\]

with \(\mathbf{R}\) A block-diagonal matrix:

\[\begin{split}\mathbf{R}=\begin{pmatrix}rot_1 & 0 & 0 \\ 0 & rot_2 & 0 \\ 0 & 0 & rot_3\end{pmatrix} \text{, and } rot_i = \begin{pmatrix}\cos{\mu_i} & \sin{\mu_i} \\ -\sin{\mu_i} & cos{\mu_i}\end{pmatrix}\end{split}\]

With radiation damping, the diagonal blocks are instead damped rotations:

\[\begin{split}rot_i = \begin{pmatrix}\exp{(-\alpha_i}) & 0 \\ 0 & \exp{(-\alpha_i)} \end{pmatrix} \cdot \begin{pmatrix}\cos{\mu_i} & \sin{\mu_i} \\ -\sin{\mu_i} & cos{\mu_i}\end{pmatrix}\end{split}\]

The order of diagonal blocks it set so that it is close to the order of x,y,z.

Parameters:

M – (m, m) transfer matrix for 1 turn

m, the dimension of \(\mathbf{M}\), may be 2 (single plane), 4 (betatron motion) or 6 (full motion)

Returns:

• A – (m, m) A-matrix

• eigval – (m/2,) Vector of Eigen values of T

References

[1] Etienne Forest, Phys. Rev. E 58, 2481 – Published 1 August 1998

amat(M)

Find the A matrix from one turn map matrix T

Provided for backward compatibility, see a_matrix()

Parameters:

M – (m, m) transfer matrix for 1 turn

Returns:

A – (m, m) A-matrix

get_mode_matrices(A)

Derives the R-matrices from the A-matrix

Parameters:

A – A-matrix

Returns:

R

References

[1] Andrzej Wolski, Phys. Rev. ST Accel. Beams 9, 024001 – Published 3 February 2006

get_tunes_damp(M, R=None)

Computes the mode emittances, tunes and damping times

Parameters:

• M – (m, m) transfer matrix for 1 turn

• R – (m, m) beam matrix (optional), allows computing the mode emittances

m, the dimension of \(\mathbf{M}\), may be 2 (single plane), 4 (betatron motion) or 6 (full motion)

Returns:

V – record array with the following fields:

tunes (m/2,) tunes of the m/2 normal modes

damping_rates (m/2,) damping rates of the m/2 normal modes

mode_matrices (m/2, m, m) the R-matrices of the m/2 normal modes

mode_emittances Only if R is specified: (m/2,) emittance of each of the m/2 normal modes

jmat(ind)

antisymetric block diagonal matrix [[0, 1][-1, 0]]

Parameters:

ind (int) – Matrix dimension: 1, 2 or 3

Returns:

S – Block diagonal matrix, (2, 2) or (4, 4) or (6, 6)

jmatswap(ind)

Modified antisymetric block diagonal matrix to deal with the swap of the longitudinal coordinates

symplectify(M)

Makes A matrix more symplectic

following the Healy algorithm described by MacKay

Parameters:

M – Almost symplectic matrix

Returns:

MS – Symplectic matrix

References

[1] W.W.MacKay, Comment on Healy’s symplectification algorithm, Proceedings of EPAC 2006