272
An Introduction to Beam Physics
To shorten the equations, let us use again the following symbol
ˆ
R
μ x
μ y
=
⎛
⎜
⎜
⎝
cos μ x sin μ x
0
0
− sin μ x cos μ x
0
0
0
0
cosμ y sin μ y
0
0 − sin μ y cos μ y
⎞
⎟
⎟
⎠ .
After n turns, the coordinates are
⎛
⎜
⎜
⎜
⎝
x n
a n
y n
b n
⎞
⎟
⎟
⎟
⎠
=
⎡
⎢
⎢
⎢
⎣
⎛
⎜
⎜
⎜
⎝
1 0 0 0
0 1 K 0
0 0 1 0
K 0 0 1
⎞
⎟
⎟
⎟
⎠
ˆ
R
μ x
μ y
⎤
⎥
⎥
⎥
⎦
n ⎛
⎜
⎜
⎜
⎝
x 0
a 0
y 0
b 0
⎞
⎟
⎟
⎟
⎠
.
To the first order, the coordinates are
⎛
⎜
⎜
⎜
⎝
x n
a n
y n
b n
⎞
⎟
⎟
⎟
⎠
=
⎡
⎢
⎢
⎢
⎣
ˆ
R
nμ x
nμ y
+
⎛
⎜
⎜
⎜
⎝
0 0 0 0
0 0 K 0
0 0 0 0
K 0 0 0
⎞
⎟
⎟
⎟
⎠
ˆ
R
nμ x
nμ y
+ ˆ
R
μ x
μ y
⎛
⎜
⎜
⎜
⎝
0 0 0 0
0 0 K 0
0 0 0 0
K 0 0 0
⎞
⎟
⎟
⎟
⎠
ˆ
R
(n − 1)μ x
(n − 1)μ y
+ · · ·
+ ˆ
R
(n − 1)μ x
(n − 1)μ y
⎛
⎜
⎜
⎜
⎝
0 0 0 0
0 0 K 0
0 0 0 0
K 0 0 0
⎞
⎟
⎟
⎟
⎠
ˆ
R
μ x
μ y
⎤
⎥
⎥
⎥
⎦
⎛
⎜
⎜
⎜
⎝
x 0
a 0
y 0
b 0
⎞
⎟
⎟
⎟
⎠
=
⎡
⎢
⎢
⎣
ˆ
R
nμ x
nμ y
+ K
⎛
⎜
⎜
⎝
0
0
0
0
0
0
c o s ( nμ y ) sin(nμ y )
0
0
0
0
cos(nμ x ) sin(nμ x )
0
0
⎞
⎟
⎟
⎠
+K
n−1
m=1
⎛
⎜
⎜
⎜
⎜
⎝
0 0
0 0
ˆ
R UR
ˆ
R LL
0 0
0 0
⎞
⎟
⎟
⎟
⎟
⎠
⎤
⎥
⎥
⎥
⎥
⎦
⎛
⎜
⎜
⎜
⎝
x 0
a 0
y 0
b 0
⎞
⎟
⎟
⎟
⎠
,