66
W. Herr and E. Forest
If we keep only terms up to first order in L we get:
M s→s+L = L
0
·
1 0
0 1
+ L
1
·
0 1
−K 0
+ O(L
2 )
(3.33)
M s→s+L =
1
L
−K · L 1
+ O(L
2 )
(3.34)
This map is precise to order O(L 1 ), but since we have det M = 1, this truncated
expansion is not symplectic.
3.6.3.3 Symplectic Matrices and Symplectic Integration
However, the map (3.34) can be made symplectic by adding a term −K 2 L 2 . This
term is of order O(L 2 ), i.e. does not deteriorate the approximation because the
inaccuracy is of the same order.
M s→s+L =
1
L
−K · L 1−KL 2
(3.35)
Following the same procedure we can compute a symplectic approximation precise
to order O(L 2 ) from (3.32) using:
M s→s+L =
1 −
1
2 KL 2
L
−K · L 1 −
1
2 KL 2
⇒
1 −
1
2 KL 2 L−
1
4 KL 3
−K · L 1 −
1
2 KL 2
(3.36)
It can be shown that this “symplectification” corresponds to the approximation of a
quadrupole by a single kick in the centre between two drift spaces of length L/2:
1
1
2 L
0 1
1
0
−K · L 1
1
1
2 L
0 1
=
1 −
1
2 KL 2 L −
1
4 KL 3
−K · L 1 −
1
2 KL 2
(3.37)
It may be mentioned that the previous approximation to 1st order corresponds to a
kick at the end of a quadrupole, preceded by a drift space of length L. Both cases
are illustrated in Fig. 3.1.
One can try to further improve the approximation by adding 3 kicks like in
Fig. 3.2 where the distance between kicks and the kick strengths are optimized to
obtain the highest order. The thin lens approximation in Fig. 3.2 with the constants:
a ≈ 0.675602, b ≈ − 0.175602, α ≈ 1.351204, β ≈ − 1.702410
(3.38)
provides an O(L 4 ) integrator [11].
W. Herr and E. Forest
If we keep only terms up to first order in L we get:
M s→s+L = L
0
·
1 0
0 1
+ L
1
·
0 1
−K 0
+ O(L
2 )
(3.33)
M s→s+L =
1
L
−K · L 1
+ O(L
2 )
(3.34)
This map is precise to order O(L 1 ), but since we have det M = 1, this truncated
expansion is not symplectic.
3.6.3.3 Symplectic Matrices and Symplectic Integration
However, the map (3.34) can be made symplectic by adding a term −K 2 L 2 . This
term is of order O(L 2 ), i.e. does not deteriorate the approximation because the
inaccuracy is of the same order.
M s→s+L =
1
L
−K · L 1−KL 2
(3.35)
Following the same procedure we can compute a symplectic approximation precise
to order O(L 2 ) from (3.32) using:
M s→s+L =
1 −
1
2 KL 2
L
−K · L 1 −
1
2 KL 2
⇒
1 −
1
2 KL 2 L−
1
4 KL 3
−K · L 1 −
1
2 KL 2
(3.36)
It can be shown that this “symplectification” corresponds to the approximation of a
quadrupole by a single kick in the centre between two drift spaces of length L/2:
1
1
2 L
0 1
1
0
−K · L 1
1
1
2 L
0 1
=
1 −
1
2 KL 2 L −
1
4 KL 3
−K · L 1 −
1
2 KL 2
(3.37)
It may be mentioned that the previous approximation to 1st order corresponds to a
kick at the end of a quadrupole, preceded by a drift space of length L. Both cases
are illustrated in Fig. 3.1.
One can try to further improve the approximation by adding 3 kicks like in
Fig. 3.2 where the distance between kicks and the kick strengths are optimized to
obtain the highest order. The thin lens approximation in Fig. 3.2 with the constants:
a ≈ 0.675602, b ≈ − 0.175602, α ≈ 1.351204, β ≈ − 1.702410
(3.38)
provides an O(L 4 ) integrator [11].