Resonances in Repetitive Systems
291
Defining A 23 = 3 A 3 ◦ A 2 , we have
N 3 = 3 A 23 ◦ M 2 ◦ A
−1
23 .
When carrying out the transformations, we usually make sure that A 23 is
symplectic up to the third order. Therefore, we have
ˆ
N 3 ˆ
J ˆ
N
T
3 = 2 ˆ
A 23 ˆ
M 2 ˆ
A
−1
23
ˆ
J
ˆ
A 23 ˆ
M 2 ˆ
A
−1
23
T
= 2 ˆ
A 23 ˆ
M 2 ˆ
A
−1
23
ˆ
J
ˆ
A
−1
23
T ˆ
M
T
2
ˆ
A
T
23
= 2 ˆ
A 23 ˆ
M 2 ˆ
J ˆ
M
T
2
ˆ
A
T
23 = 2 ˆ
A 23 ˆ
J ˆ
A
T
23 = 2 ˆ
J.
Now that we have shown that N 3 is symplectic up to the third order, we
can find out the relations between the terms. The Jacobian of N 3 can be
written as
ˆ
N 3 = ˆ
R + ˆ
S 3 ,
where
ˆ
R =
e
−iμ 0
0 e
iμ
and ˆ
S 3 =
S
−
32 s
−
0 s
+
0
S
−
32
s
−
0
2
S
+
31
s
+
0
2 S
+
31 s
−
0 s
+
0
.
The symplectic condition becomes
ˆ
R + ˆ
S 3
ˆ
J
ˆ
R
T + ˆ
S
T
3
= 2 ˆ
J,
which leads to the relation
ˆ
S 3 ˆ
J ˆ
R
T + ˆ
R ˆ
J ˆ
S
T
3 = 0.
Plugging in the matrices ˆ
R and ˆ
S 3 , we have
S
−
32 s
−
0 s
+
0
S
−
32
s
−
0
2
S
+
31
s
+
0
2 S
+
31 s
−
0 s
+
0
0 1
−1 0
e
−iμ 0
0 e
iμ
+
e
−iμ 0
0 e
iμ
0 1
−1 0
S
−
32 s
−
0 s
+
0
S
+
31
s
+
0
2
S
−
32
s
−
0
2 S
+
31 s
−
0 s
+
0
= 0,
which can be simplified to
S
−
32 e
iμ + S
+
31 e
−iμ
s
−
0 s
+
0
0 1
−1 0
= 0.
Defining
T S = iS
−
32 e
iμ ,
we obtain
S
−
32 = −iT S e
−iμ , S
+
31 = iT S e
iμ .