3 Non-linear Dynamics in Accelerators
53
new concepts are a natural extension of the Courant-Snyder formalism to non-linear
dynamics. An extensive treatment of these tools and many examples can be found
in [7]. In the last part we summarize the most important physical phenomena caused
by the non-linearities in an accelerator.
3.2 Variables
For what follows one should always use canonical variables!
In Cartesian coordinates:
R = (X, P X , Y, P Y , Z, P Z , t)
(3.1)
If the energy is constant (i.e. P Z = const.), we use:
(X, P X , Y, P Y , Z, t)
(3.2)
This system is rather inconvenient, what we want is the description of the particle
in the neighbourhood of the reference orbit/trajectory:
R d = (X, P X , Y, P Y , Z, t)
(3.3)
which are considered now the deviations from the reference and which are zero for
a particle on the reference trajectory
It is very important that it is the reference not the design trajectory!
(so far it is a straight line along the Z-direction)
3.2.1 Trace Space and Phase Space
A confusion often arises about the terms Phase Space (x, p x , . . .) or Trace Space
(x, x , . . .)
It is not laziness nor stupidity to use one or the other:
– Beam dynamics is strictly correct only with (x, p x , . . .), (see later chapter) but in
general quantities cannot be measured easily
– Beam dynamics with (x, x , . . .) needs special precaution, but quantities based
on these coordinates are much easier to measure
– Some quantities are different (e.g. emittance)
It comes back to a remark made at the beginning, i.e. that we shall use rings
for our arguments. In single pass machine, e.g. linac, beam lines, spectrometers,
the beam is not circulating over many turns and several hours, therefore there is no
interest in stability issues. Instead for most of these applications what counts is the
53
new concepts are a natural extension of the Courant-Snyder formalism to non-linear
dynamics. An extensive treatment of these tools and many examples can be found
in [7]. In the last part we summarize the most important physical phenomena caused
by the non-linearities in an accelerator.
3.2 Variables
For what follows one should always use canonical variables!
In Cartesian coordinates:
R = (X, P X , Y, P Y , Z, P Z , t)
(3.1)
If the energy is constant (i.e. P Z = const.), we use:
(X, P X , Y, P Y , Z, t)
(3.2)
This system is rather inconvenient, what we want is the description of the particle
in the neighbourhood of the reference orbit/trajectory:
R d = (X, P X , Y, P Y , Z, t)
(3.3)
which are considered now the deviations from the reference and which are zero for
a particle on the reference trajectory
It is very important that it is the reference not the design trajectory!
(so far it is a straight line along the Z-direction)
3.2.1 Trace Space and Phase Space
A confusion often arises about the terms Phase Space (x, p x , . . .) or Trace Space
(x, x , . . .)
It is not laziness nor stupidity to use one or the other:
– Beam dynamics is strictly correct only with (x, p x , . . .), (see later chapter) but in
general quantities cannot be measured easily
– Beam dynamics with (x, x , . . .) needs special precaution, but quantities based
on these coordinates are much easier to measure
– Some quantities are different (e.g. emittance)
It comes back to a remark made at the beginning, i.e. that we shall use rings
for our arguments. In single pass machine, e.g. linac, beam lines, spectrometers,
the beam is not circulating over many turns and several hours, therefore there is no
interest in stability issues. Instead for most of these applications what counts is the