56 Beam-based Correction and Optimization for Accelerators
In a Hamiltonian system, the time derivative of a function of the phase
space coordinates, f (X), is given by
f
=
∂f
∂s
+
i
[
∂f
∂x i
x
i +
∂f
∂p i
p
i ] =
∂f
∂s
+ [f, H],
(2.99)
where
represents derivative with respect to the free variable, s, summation
is over pairs of conjugate coordinates, H(X; s) is the Hamiltonian, and the
Poisson bracket for functions f and g is defined as
[f, g] =
i
∂f
∂x i
∂g
∂p i
−
∂f
∂p i
∂g
∂x i
≡: f : g,
(2.100)
where : f : g is simply another notation for the Poisson bracket. Assuming no
explicit time dependence in both f and H, the higher order time derivatives
of f can be readily obtained
f
= : −H :f, f
= : (−H)
2 :f, · · · , f
(n) = : (−H)
n :f,
(2.101)
where −H is used because [f, H] = [−H, f ]. For an accelerator element with
constant magnetic field profile over length L, the function f at the exit face
can be expressed in a Taylor series of coordinates at the entrance face
f (X 2 ) = f 1 + f
1 L +
1
2
f
1 L
2 + · · · = e
:−HL: f (X)| X=X1 ,
(2.102)
where subscript 1 indicates values at the entrance face and e
:g: is defined as a
Lie map with the generating function g,
e
:g: ≡ 1+ : g : +
1
2
: g :
2 + · · ·
1
n!
: g :
n + · · · .
(2.103)
When Eq. (2.102) is applied to the phase space coordinates, X, it gives the
transfer map. Hence, for a typical accelerator element the transfer map is a Lie
map with generating function g = −H(X)L. For linear optics elements, the
generating functions are quadratic functions of the phase space coordinates.
For example, the Lie map for a drift space is exp(: −
1
2 (p
2
x + p
2
y )L :). The map
for the linear elements can be expressed in closed forms. However, the maps
for nonlinear elements, such as a sextupole,
f sext = −
p
2
x + p
2
y
2
+
K 2
6
(x
3 − 3xy
2 )
L,
(2.104)
do not have closed forms.
When two elements are joined together, the Lie map for the section consisting the two elements can be obtained by concatenating the two individual
Lie maps using the Baker-Campbell-Hausdorff (BCH) formula,
e
:f : e
:g: = e
:h: ,
(2.105)
In a Hamiltonian system, the time derivative of a function of the phase
space coordinates, f (X), is given by
f
=
∂f
∂s
+
i
[
∂f
∂x i
x
i +
∂f
∂p i
p
i ] =
∂f
∂s
+ [f, H],
(2.99)
where
represents derivative with respect to the free variable, s, summation
is over pairs of conjugate coordinates, H(X; s) is the Hamiltonian, and the
Poisson bracket for functions f and g is defined as
[f, g] =
i
∂f
∂x i
∂g
∂p i
−
∂f
∂p i
∂g
∂x i
≡: f : g,
(2.100)
where : f : g is simply another notation for the Poisson bracket. Assuming no
explicit time dependence in both f and H, the higher order time derivatives
of f can be readily obtained
f
= : −H :f, f
= : (−H)
2 :f, · · · , f
(n) = : (−H)
n :f,
(2.101)
where −H is used because [f, H] = [−H, f ]. For an accelerator element with
constant magnetic field profile over length L, the function f at the exit face
can be expressed in a Taylor series of coordinates at the entrance face
f (X 2 ) = f 1 + f
1 L +
1
2
f
1 L
2 + · · · = e
:−HL: f (X)| X=X1 ,
(2.102)
where subscript 1 indicates values at the entrance face and e
:g: is defined as a
Lie map with the generating function g,
e
:g: ≡ 1+ : g : +
1
2
: g :
2 + · · ·
1
n!
: g :
n + · · · .
(2.103)
When Eq. (2.102) is applied to the phase space coordinates, X, it gives the
transfer map. Hence, for a typical accelerator element the transfer map is a Lie
map with generating function g = −H(X)L. For linear optics elements, the
generating functions are quadratic functions of the phase space coordinates.
For example, the Lie map for a drift space is exp(: −
1
2 (p
2
x + p
2
y )L :). The map
for the linear elements can be expressed in closed forms. However, the maps
for nonlinear elements, such as a sextupole,
f sext = −
p
2
x + p
2
y
2
+
K 2
6
(x
3 − 3xy
2 )
L,
(2.104)
do not have closed forms.
When two elements are joined together, the Lie map for the section consisting the two elements can be obtained by concatenating the two individual
Lie maps using the Baker-Campbell-Hausdorff (BCH) formula,
e
:f : e
:g: = e
:h: ,
(2.105)