192
An Introduction to Beam Physics
v 1
v 2
complex plane
FIGURE 8.3: Relation between the eigenvalues when | tr M | < 2.
picture in Fig. 8.1). For practical purposes, this case is unstable and hence
useless.
Let us now consider the case | tr ˆ
M | < 2 in more detail. We have the
complex eigenvalues that satisfy
λ 2 = ¯
λ 1 and λ 2 = λ
−1
1 .
So in the complex plane, λ 1 and λ 2 lie on a circle and form conjugate pairs,
as shown in Fig. 8.3.
The eigenvalues can hence be written as
λ 1,2 = e
±iμ ,
where μ is called the tune of the system, which is
μ = arccos
λ 1 + λ 2
2
= arccos
tr ˆ
M
2
.
The eigenvectors v 1,2 belonging to λ 1,2 also form conjugate pairs, since
ˆ
M v 2 = ˆ
MM v 2 = λ 2 v 2 = λ 1 v 2 .
Define now two new basis vectors v + = Re ( v 1 ) , , v − = Im ( v 1 ) as the real
and imaginary parts of the eigenvalues; they define what is called the normal
form basis for stable motion. So we have
v 1 = v + + ii v − , , v 2 = v + − ii v − .
We now observe
ˆ
MM v 1 = λ 1 v 1 = e
iμ ( v + + ii v − )= cos μ· v + −sin μ· v − +i (sin μ · v + + cos μ · v − ) ,