3 Non-linear Dynamics in Accelerators
73
After square root expansion and sorting the A s contributions:
H =
kinematic
p 2
x + p 2
y
2(1 + δ)
−
dipole
xδ
ρ
bending
+
x 2
2ρ 2
f ocusing
+
quadrupole
k 1
2
(x
2
− y
2 ) +
sextupole
k 2
6
(x
3
− 3xy
2 ) + . . .
(3.61)
using :
k n = k
(n)
n =
1
Bρ
∂ n B y
∂x n
k
(s)
n =
1
Bρ
∂ n B x
∂x n
(3.62)
– The Hamiltonian describes the motion of a particle through an element
– Each element has a component in the Hamiltonian
– Basis to extend the linear to a nonlinear formalism
A short list of Hamiltonians of some machine elements (3D)
In general for multipoles of order n:
H n =
1
1 + n
Re
(k n + ik
(s)
n )(x + iy)
n+1
+
p 2
x + p 2
y
2(1 + δ)
(3.63)
We get for some important types (normal components k n only):
drift space : H = −
(1 + δ) 2 − p 2
x − p 2
y ≈
p 2
x + p 2
y
2(1 + δ)
(3.64)
dipole : H = −
−xδ
ρ
+
x 2
2ρ 2 +
p 2
x + p 2
y
2(1 + δ)
(3.65)
quadrupole : H =
1
2
k 1 (x
2
− y
2 ) +
p 2
x + p 2
y
2(1 + δ)
(3.66)
sextupole : H =
1
3
k 2 (x
3
− 3xy
2 ) +
p 2
x + p 2
y
2(1 + δ)
(3.67)
octupole : H =
1
4
k 3 (x
4
− 6x
2 y
2
+ y
4 ) +
p 2
x + p 2
y
2(1 + δ)
(3.68)