3 Non-linear Dynamics in Accelerators
57
Each element at position s acts as a source of forces, i.e. we must write for the
forces K → K(s) which is assumed to be a periodic function, i.e. K(s +
C) = K(s) ring
The solution of this Boundary Value Problem must be periodic too!
It is therefore not applicable in the general case (e.g. Linacs, Beamlines, FFAG,
Recirculators, . . . ), much better to treat it as an Initial Value Problem.
In a more useful approach we do not attempt to solve such an overall equation
but rather consider the fundamental objects of an accelerators, i.e. the machine
elements themselves. These elements, e.g. magnets or other beam elements, are the
basic building blocks of the machine. All elements have a well defined action on a
particle which can be described independent of other elements or concepts such as
closed orbit or β-functions. Mathematically, they provide a “map” from one face of
a building block to the other, i.e. a description of how the particles move inside and
between elements. In this context, a map can be anything from linear matrices to
high order integration routines.
A map based technique is also the basis for the treatment of particle dynamics as
an Initial value Problem (IVP).
It follow immediately that for a linear, 1st order equation of the type
dx(s)
ds
= K(s) x(s)
(and initial values at s 0 )
the solution can always be written as:
x(s) = a · x(s 0 ) + b · x (s 0 )
x (s) = c · x(s 0 ) + d · x (s 0 )
x
x
s
=
A
a b
c d
x
x
s 0
where the function K(s) does not have to be periodic. Furthermore, the determinant
of the matrix A is always 1. Therefore it is an advantage to use maps (matrices)
for a linear systems from the start, without trying to solve a differential equation.
The collection of all machine elements make up the ring pr beam line and
it is the combination of the associated maps which is necessary for the description and analysis of the physical phenomena in the accelerator ring or beam
line.
For a circular machine the most interesting map is the one which describes
the motion once around the machine, the so-called One-Turn-Map. It contains all
necessary information on stability, existence of closed orbit, and optical parameters.
The reader is assumed to be familiar with this concept in the case of linear beam
dynamics (Chap. 2) where all maps are matrices and the Courant-Snyder analysis
of the corresponding one-turn-map produces the desired information such as e.g.
closed orbit or Twiss parameters.
57
Each element at position s acts as a source of forces, i.e. we must write for the
forces K → K(s) which is assumed to be a periodic function, i.e. K(s +
C) = K(s) ring
The solution of this Boundary Value Problem must be periodic too!
It is therefore not applicable in the general case (e.g. Linacs, Beamlines, FFAG,
Recirculators, . . . ), much better to treat it as an Initial Value Problem.
In a more useful approach we do not attempt to solve such an overall equation
but rather consider the fundamental objects of an accelerators, i.e. the machine
elements themselves. These elements, e.g. magnets or other beam elements, are the
basic building blocks of the machine. All elements have a well defined action on a
particle which can be described independent of other elements or concepts such as
closed orbit or β-functions. Mathematically, they provide a “map” from one face of
a building block to the other, i.e. a description of how the particles move inside and
between elements. In this context, a map can be anything from linear matrices to
high order integration routines.
A map based technique is also the basis for the treatment of particle dynamics as
an Initial value Problem (IVP).
It follow immediately that for a linear, 1st order equation of the type
dx(s)
ds
= K(s) x(s)
(and initial values at s 0 )
the solution can always be written as:
x(s) = a · x(s 0 ) + b · x (s 0 )
x (s) = c · x(s 0 ) + d · x (s 0 )
x
x
s
=
A
a b
c d
x
x
s 0
where the function K(s) does not have to be periodic. Furthermore, the determinant
of the matrix A is always 1. Therefore it is an advantage to use maps (matrices)
for a linear systems from the start, without trying to solve a differential equation.
The collection of all machine elements make up the ring pr beam line and
it is the combination of the associated maps which is necessary for the description and analysis of the physical phenomena in the accelerator ring or beam
line.
For a circular machine the most interesting map is the one which describes
the motion once around the machine, the so-called One-Turn-Map. It contains all
necessary information on stability, existence of closed orbit, and optical parameters.
The reader is assumed to be familiar with this concept in the case of linear beam
dynamics (Chap. 2) where all maps are matrices and the Courant-Snyder analysis
of the corresponding one-turn-map produces the desired information such as e.g.
closed orbit or Twiss parameters.