62
W. Herr and E. Forest
Other definitions based on handwaving arguments or those approximately valid
only in special cases, should be discarded, in particular those relying on presumed
distributions, e.g. Gaussian.
3.6 Techniques and Tools to Evaluate and Correct
Non-linear Effects
The key to a more modern approach shown in this section is to avoid the prejudices
about the stability and other properties of the ring. Instead, we must describe the
machine in terms of the objects it consists of with all their properties, including
the non-linear elements. The analysis will reveal the properties of the particles
such as e.g. stability. In the simplest case, the ring is made of individual machine
elements such as magnets which have an existence on their own, i.e. the interaction
of a particle with a given element is independent of the motion in the rest of
the machine. Also for the study of non-linear effects, the description of elements
should be independent of concepts such as tune, chromaticity and closed orbit. To
successfully study single particle dynamics, one must be able to describe the action
of the machine element on the particle as well as the machine element.
3.6.1 Particle Tracking
The ring being a collection of maps, a particle tracking code, i.e. an integrator of
the equation of motion, is the most reliable map for the analysis of the machine. Of
course, this requires an appropriate description of the non-linear maps in the code.
It is not the purpose of this article to describe the details of tracking codes and the
underlying philosophy, such details can be found in the literature (see e.g. [6]). Here
we review and demonstrate the basic principles and analysis techniques.
3.6.1.1 Symplecticity
If we define a map through
z 2 = M 12 (
z 1 ) as a propagator from a location “1” to a
location “2” in the ring, we have to consider that not all possible maps are allowed.
The required property of the map is called “symplecticity” and in the simplest case
where M 12 is a matrix, the symplecticity condition can be written as:
M ⇒ M
T
· S · M = S where S =
⎛
⎜
⎜
⎝
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
⎞
⎟
⎟
⎠
(3.25)
W. Herr and E. Forest
Other definitions based on handwaving arguments or those approximately valid
only in special cases, should be discarded, in particular those relying on presumed
distributions, e.g. Gaussian.
3.6 Techniques and Tools to Evaluate and Correct
Non-linear Effects
The key to a more modern approach shown in this section is to avoid the prejudices
about the stability and other properties of the ring. Instead, we must describe the
machine in terms of the objects it consists of with all their properties, including
the non-linear elements. The analysis will reveal the properties of the particles
such as e.g. stability. In the simplest case, the ring is made of individual machine
elements such as magnets which have an existence on their own, i.e. the interaction
of a particle with a given element is independent of the motion in the rest of
the machine. Also for the study of non-linear effects, the description of elements
should be independent of concepts such as tune, chromaticity and closed orbit. To
successfully study single particle dynamics, one must be able to describe the action
of the machine element on the particle as well as the machine element.
3.6.1 Particle Tracking
The ring being a collection of maps, a particle tracking code, i.e. an integrator of
the equation of motion, is the most reliable map for the analysis of the machine. Of
course, this requires an appropriate description of the non-linear maps in the code.
It is not the purpose of this article to describe the details of tracking codes and the
underlying philosophy, such details can be found in the literature (see e.g. [6]). Here
we review and demonstrate the basic principles and analysis techniques.
3.6.1.1 Symplecticity
If we define a map through
z 2 = M 12 (
z 1 ) as a propagator from a location “1” to a
location “2” in the ring, we have to consider that not all possible maps are allowed.
The required property of the map is called “symplecticity” and in the simplest case
where M 12 is a matrix, the symplecticity condition can be written as:
M ⇒ M
T
· S · M = S where S =
⎛
⎜
⎜
⎝
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
⎞
⎟
⎟
⎠
(3.25)