Betatron Oscillations

We can evaluate A and B at s 1 , let ψ=0, y=y 1 , and y ' =y 1‘ . Then A = y 1
,
and B = w 1
y ′ 1
− w ′ 1
y 1
. Introduce the values of A and B into 4.26 and
collect coefficients of y 1 and y 1
'
to find
4.27
M( s 2
s 1 )=
⎡
⎢
⎢ ⎛
cosψ⎜
⎣
⎢ ⎝
w ′ 2
−
w 1
cosψ w 2
− w 2
w 1
′ sinψ
w 1
w 1
′
w 2
⎞
⎟ − sinψ
⎠
w 1
⎤
w 1
w 2
sinψ
⎥
⎛
⎜
1 ⎞
+ w 1
′ w ′ 2
⎟ cosψ w 1
⎥
+ sinψ w 1
′ w
2′
⎝ w 1
w 2
⎠ w 2
⎥
⎦
Now we evaluate M in the case s 2 = s 1 +L, and require w 1 = w 2 = w
4.28
M( s 2
)=
⎡ cosψ − w w ′ sinψ w 2 sinψ ⎤
⎢
− ⎛ 1 ⎝ w + ( w ′ ⎞
2 )2
⎠
sinψ cosψ+w w ′
⎣
sinψ ⎥
⎦
19 Jun 2007 Accelerators: Theory and
Applications
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