74
W. Herr and E. Forest
Interlude 3
A few remarks are required after this list of Hamiltonian for particular
elements.
– Unlike said in many introductory textbooks and lectures, a multipole of
order n is not required to drive a nth order resonance—nothing could be
more wrong!!
– In leading order perturbation theory, only elements with an even order
(and larger than 2) in the Hamiltonian can produce an amplitude dependent tune shift and tune spread.
3.7.3.1 Lie Maps and Transformations
In this chapter we would like to introduce Lie algebraic tools and Lie transformations [15–17]. We use the symbol z i = (x i , p i ) where x and p stand for canonically
conjugate position and momentum. We let f (z) and g(z) be any function of x, p and
can define the Poisson bracket for a differential operator [18]:
[f, g] =
n
i=1
∂f
∂x i
∂g
∂p i
−
∂f
∂p i
∂g
∂x i
(3.69)
Assuming that the motion of a dynamic system is defined by a Hamiltonian H , we
can now write for the equations of motion [18]:
[x i , H ] =
∂H
∂p i
=
dx i
dt
(3.70)
[p i , H ] = −
∂H
∂x i
=
dp i
dt
(3.71)
If H does not explicitly depend on time then:
[f, H ] = 0
(3.72)
implies that f is an invariant of the motion. To proceed, we can define a Lie operator
: f : via the notation:
: f : g = [f, g]
(3.73)
where : f : is an operator acting on the function g.
W. Herr and E. Forest
Interlude 3
A few remarks are required after this list of Hamiltonian for particular
elements.
– Unlike said in many introductory textbooks and lectures, a multipole of
order n is not required to drive a nth order resonance—nothing could be
more wrong!!
– In leading order perturbation theory, only elements with an even order
(and larger than 2) in the Hamiltonian can produce an amplitude dependent tune shift and tune spread.
3.7.3.1 Lie Maps and Transformations
In this chapter we would like to introduce Lie algebraic tools and Lie transformations [15–17]. We use the symbol z i = (x i , p i ) where x and p stand for canonically
conjugate position and momentum. We let f (z) and g(z) be any function of x, p and
can define the Poisson bracket for a differential operator [18]:
[f, g] =
n
i=1
∂f
∂x i
∂g
∂p i
−
∂f
∂p i
∂g
∂x i
(3.69)
Assuming that the motion of a dynamic system is defined by a Hamiltonian H , we
can now write for the equations of motion [18]:
[x i , H ] =
∂H
∂p i
=
dx i
dt
(3.70)
[p i , H ] = −
∂H
∂x i
=
dp i
dt
(3.71)
If H does not explicitly depend on time then:
[f, H ] = 0
(3.72)
implies that f is an invariant of the motion. To proceed, we can define a Lie operator
: f : via the notation:
: f : g = [f, g]
(3.73)
where : f : is an operator acting on the function g.