288
An Introduction to Beam Physics
non-symplectic version of A
−1 .
After a straightforward yet rather lengthy derivation, the final result is
⎛
⎝
x 1
a 1
⎞
⎠ =
⎛
⎝
x 0 cos μ + a 0 sin μ
−
x 0 sin μ + a 0 cos μ
⎞
⎠ +
⎛
⎝
A 111
x
3
0 + A 112
x
2
0
a 0 + A 122
x 0 a
2
0 + A 222 a
3
0
B 111
x
3
0 + B 112
x
2
0
a 0 + B 122
x 0 a
2
0 + B 222 a
3
0
⎞
⎠ ,
where
A 111 = −k
2
s β
3 cos
3 (μ/2) sin μ cos
2 μ
sin (3μ/2)
,
A 112 = k
2
s β
3 cos
3 (μ/2) cos μ
2 sin (3μ/2)
[3 cos (2μ) − 1] ,
A 122 = k
2
s β
3 cos
4 (μ/2) sin (μ/2)
sin (3μ/2)
[1 + 3 cos (2μ)] ,
A 222 = k
2
s β
3 4 cos
5 (μ/2) sin
2 (μ/2) cos μ
sin (3μ/2)
,
(11.9)
and
B 111 = k
2
s β
3
4 cos
5 (μ/2) sin
2 (μ/2) cos μ
sin (3μ/2)
+
cos (μ/2) cos
4 μ
sin (3μ/2)
,
B 112 = k
2
s β
3
−
cos
4 (μ/2) sin (μ/2)
sin (3μ/2)
[1 + 3 cos (2μ)]
+6
sin (μ/2) cos
2 (μ/2) cos
3 μ
sin (3μ/2)
,
B 122 = k
2
s β
3
cos
3 (μ/2) cos μ
2 sin (3μ/2)
[3 cos (2μ) − 1]
+12
sin
2 (μ/2) cos
3 (μ/2) cos
2 μ
sin (3μ/2)
,
B 222 = k
2
s β
3
cos
3 (μ/2) sin μ cos
2 μ
sin (3μ/2)
+
cos (μ/2) cos μ sin
3 μ
sin (3μ/2)
.
(11.10)
The next step is to find a second order transformation (in terms of k s ) that
the third order map in the newest coordinates is a rotation. Unlike the first
order transformation, not all nonlinear terms in the map can be removed even
if μ does not satisfy any resonance condition (i.e., ν is irrational.). It is much
easier to illustrate this in the eigenspace of the linear map, which is complex.
Let us denote (s
+ , s
− ) as the complex coordinates which are related to the
real coordinates by the relations
s
−
s
+
=
1
√
2
1 i
1 −i
x
a
.
(11.11)