3 Non-linear Dynamics in Accelerators
99
and the CBH-formula for the concatenation of the maps (3.92):
e : f 2 : e : F : = e : h : = exp
: f 2 +
: f 2 :
1 − e − : f 2 :
F + O(F
2 ) :
(3.174)
which gives immediately for h:
h = −μJ +
n
c n (J )
inμ
1 − e −inμ
e inn = −μJ +
n
c n (J )
nμ
2 sin(
nμ
2 )
e (inn + i
nμ
2 )
(3.175)
Equation (3.175) is the beam-beam perturbed invariant to first order in the perturbation using (3.92).
From (3.175) we observe that for ν =
μ
2π =
p
n resonances appear for all integers
p and n when c n (J ) = 0.
Away from resonances a normal form transformation gives:
h = − μJ + c 0 (J ) = const.
(3.176)
and the oscillating term disappears. The first term is the linear rotation and the
second term gives the amplitude dependent tune shift (see (3.114)):
μ(J ) = −
1
2π
dc 0 (J )
dJ
(3.177)
The computation of this tuneshift from the equation above can be found in the
literature [7, 23].
3.8.1.3 Phase Space Structure
To demonstrate how this technique can be used to reconstruct the phase space
structure in the presence of non-linearities, we continue with the very non-linear
problem of the beam-beam interaction treated above. To test our result, we compare
the invariant h to the results of a particle tracking program.
The model we use in the program is rather simple:
• linear transfer between interactions
• beam-beam kick for round beams
• compute action J =
β ∗
2σ 2 (
x 2
β ∗ + p 2
x β ∗ )
• compute phase = arctan(
p x
x )
• compare J with h as a function of the phase