190
An Introduction to Beam Physics
x
a
v 1
v 2
FIGURE 8.1: Motion in phase space (left) and in the eigenspace (right)
when | tr M | > 2.
values have to have unit magnitude. Of these, the cases +1 and −1 are
to be excluded too, since even the slightest imperfection in the machine may
otherwise lead to instability. Altogether, in a periodic system, the eigenvalues
must all be complex and of unit magnitude.
It is particularly interesting to study the special case of a matrix with
midplane symmetry. In this case, the x and y motion decouple and can be
described by individual matrices. We obtain for the eigenvalues for the 2 × 2
x sub-matrix, noting that the y sub-matrix is treated similarly:
0 =
ˆ
M − λ ˆ
I
=
(x|x) − λ (x|a)
(a|x) (a|a) − λ
= (x|x) (a|a) − (x|a) (a|x)
1
− λ [(x|x) + (a|a)] + λ
2 ,
and so
λ 1,2 =
[(x|x) + (a|a)] ±
[(x|x) + (a|a)]
2 − 4
2
=
tr ˆ
M
2
±
tr ˆ
M
2
2
− 1.
Hence to have complex eigenvalues requires the very simple condition
−2 < tr ˆ
M < 2.
A quick check of the four cases shows that this excludes the point–to–point
case and the parallel–to–parallel case, as in both of these, the trace just equals
two or exceeds two. The parallel–to–point or point–to–parallel case each have