64
W. Herr and E. Forest
3.6.3.1 Taylor Maps
A Taylor map can be written using higher order matrices and in the case of two
dimensions we have:
z j (s 2 ) =
4
k=1
R jk z k (s 1 ) +
4
k=1
4
l=1
T jkl z k (s 1 )z l (s 1 )
(3.26)
(where z j , j = 1, . . . , 4, stand for x, x , y, y ). Let us call the collection: A 2 = (R, T ) the second order map A 2 . Higher orders can be defined
as needed, e.g. for the 3rd order map A 3 = (R, T , U) we add a third order matrix:
+
4
k=1
4
l=1
4
m=1
U jklm z k (s 1 )z l (s 1 )z m (s 1 )
(3.27)
Since Taylor expansions are not matrices, to provide a symplectic map, it is the
associated Jacobian matrix J which must fulfill the symplecticity condition:
J ik =
∂z i (s 2 )
∂z k (s 1 )
and J must fulfill : J
t
· S · J = S
(3.28)
However, in general J ik = const and for a truncated Taylor map it can
be difficult to fulfill this condition for all z. As a consequence, the number of
independent coefficients in the Taylor expansion is reduced and the complete,
symplectic Taylor map requires more coefficients than necessary [7].
The explicit maps for a sextupole is:
x 2 = x 1 + Lx
1 − k 2
L 2
4 (x 2
1 − y 2
1 ) +
L 3
12 (x 1 x
1 − y 1 y
1 ) +
L 4
24 (x
1 − y
1 )
x
2 = x
1
− k 2
L
2 (x 2
1 − y 2
1 ) +
L 2
4 (x 1 x
1 − y 1 y
1 ) +
L 3
6 (x
1 − y
1 )
y 2 = y 1 + Ly
1 + k 2
L 2
4 x 1 y 1 +
L 3
12 (x 1 y
1 + y 1 x
1 ) +
L 4
24 (x
1 y
1 )
y
2 = y
1
+ k 2
L
2 x 1 y 1 +
L 2
4 (x 1 y
1 + y 1 x
1 ) +
L 3
6 (x
1 y
1 )
(3.29)
Writing the explicit form of the Jacobian matrix:
J ik =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
∂x 2
∂x 1
∂x 2
∂x
1
∂x 2
∂y 1
∂x 2
∂y
1
∂x
2
∂x 1
∂x
2
∂x
1
∂x
2
∂y 1
∂x
2
∂y
1
∂y 2
∂x 1
∂y 2
∂x
1
∂y 2
∂y 1
∂y 2
∂y
1
∂y
2
∂x 1
∂y
2
∂x
1
∂y
2
∂y 1
∂y
2
∂y
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
→ k 2 = 0
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 L 0 0
0 1 0 0
0 0 1 L
0 0 0 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3.30)