Resonances in Repetitive Systems
287
-0.4
-0.2
0
0.2
0.4
0.6
-0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
˜ a
˜
x
[
a
FIGURE 11.3: Invariant obtained through first order perturbation theory
(left) and tracking (right), with k s β
3
2 = 1 and ν = 130/360.
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
˜ a
˜
x
[
a
FIGURE 11.4: Invariant obtained through first order perturbation theory
(left) and tracking (right), with k s β
3
2 = 1 and ν = 110/360.
The last line is the function A
−1 to the third order. It is straightforward to
verify that it is true using the relation B 12 = −2A 11 . Yet A and A
−1 are not
symplectic to the third order, which is easily verified using eq. (5.6). The
details of obtaining the symplectic version of A and A
−1 are beyond the scope
of this book. The result is actually very simple, which is
A =
⎛
⎝
x + A 11 x
2 + A 22 a
2 + A
2
11
x
3
0 + A 22 B 11
x
2
0
a 0 − A 11 A 22
x 0 a
2
0
a + B 11 x
2
− 2A 11 xa + A
2
11
x
2
0
a 0 + A 22 B 11
x 0 a
2
0 − A 11 A 22 a
3
0
⎞
⎠ ,
A
−1 =
⎛
⎝
x − A 11 x
2
− A 22 a
2 + A
2
11
x
3
0 + A 22 B 11
x
2
0
a 0 − A 11 A 22
x 0 a
2
0
a − B 11 x
2 + 2A 11 xa + A
2
11
x
2
0
a 0 + A 22 B 11
x 0 a
2
0 − A 11 A 22 a
3
0
⎞
⎠ .
Again, it is straightforward to verify the symplecticity of the above map using
eq. (5.6). Note that the third order part in A and A
−1 is half of that in the