Linear Normal Form Around Ring

Consider positions $s_1, s_2, \dots, s_n$ around the ring. We have a phase space at
each of these positions which we notate by $\vec Z_j$. We write $M_j$ for the one turn map at $s_j$ which gives

(1)
\begin{align} \vec Z_j^{k+1}=M_j \vec Z_j^{k} \end{align}

Now consider the transfer matrices $T_{j\rightarrow k}$ that bring $\vec Z_j$ to $\vec Z_k$:

(2)
\begin{align} T_{j\rightarrow k} \vec Z_j = \vec Z_k \end{align}

The transfer matrix for a section of ring is then noted

(3)
\begin{align} \bar T_j = T_{j+1} T_{j}^{-1} \end{align}

For normal form from A, see coupling-from-one-turn-map-matrix.

In the normalized coordinates, we have for each section of ring

(4)
\begin{align} \bar R_j = A_{j+1}^{-1}\bar T_j A_j \end{align}