76
W. Herr and E. Forest
Interlude 4
The exact Hamiltonian in two transverse dimensions and with a relative
momentum deviation δ is (full Hamiltonian with
A( x, t) = 0):
H = −
(1 + δ) 2 − p 2
x − p 2
y
−→ f drif t = L · H
The exact map for a drift space is now:
x
new
= x + L ·
p x
(1 + δ) 2 − p 2
x − p 2
y
p
new
x
= p x
y
new
= y + L ·
p y
(1 + δ) 2 − p 2
x − p 2
y
p
new
y
= p y
In 2D and with δ = 0 it is more complicated than Eq. (3.78). In practice the
map can (often) be simplified to the well known form.
More general, acting on the phase space coordinates:
e
:f : (x, p) 1 = (x, p) 2
(3.79)
is the Lie transformation which describes how to go from one point to another.
While a Lie operator propagates variables over an infinitesimal distance, the Lie
transformation propagates over a finite distance.
To illustrate this technique with some simple examples, it can be shown easily,
using the formulae above, that the transformation:
e
: −
1
2f x 2 :
(3.80)
corresponds to the map of a thin quadrupole with focusing length f , i.e.
x 2 = x 1
p 2 = p 1 −
1
f
x 1