32
An Introduction to Beam Physics
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FIGURE 2.1: Reference orbit, arc length s along it and local coordinates.
2.1 Coordinates and Maps
Usually when studying dynamics, the time t plays the role of the independent variable, and we study the motion of positions x and velocities v or momenta
p as coordinates. Using the Lagrange mechanism, it is easy to transfer
to new coordinates, in particular the coordinates that describe the relative
dynamics around the reference orbit. Furthermore, instead of using t, we
usually use the arc length s along the reference orbit as an independent
variable. Fig. 2.1 illustrates the concept.
For the understanding of the motion in relative coordinates, let us assume
we have studied and understood the motion of the reference orbit. In case
there is no field at all, this reference orbit will merely follow a straight line.
Furthermore, there are many devices used in accelerators that have fields, but
along one given straight line, all the fields vanish, and the device is lined up
in such a way that the reference particle follows this line. Another important
device uses magnetic fields, and along the reference orbit one tries to hold
the magnetic fields constant, in which case the reference orbit is circular, at
least within the element. In all other cases, it is usually necessary to describe
the reference orbit by numerically integrating the equations of motion.
We assume the position and momenta of the reference particle r ref (s), ,
p ref (s)
are known. Here the momentum p is the dynamical momentum as in eq.
(1.4). As a technical detail, let us also assume that for all points s, we have
p ref (s) ∦ e zLab , i.e., the motion is never pointing vertically straight (which
for most real accelerators is no limitation whatsoever). Let furthermore r tube
be smaller than the minimum radius of curvature that the reference orbit
experiences in the section of the device that we want to study. We now
consider a “flexible tube” of radius r tube centered around the reference orbit,
and restrict the particles that we want to describe to only those within the
tube, as Fig. 2.2 illustrates the situation. Again, for practical devices this
represents hardly a limitation; for example, in the LHC (see Table 1.1), the
“tube” would be more than 2 km wide, much larger than the region required
An Introduction to Beam Physics
5HIHUHQFH2UELW
V
U UHI V
H
H
H
\
V
[
FIGURE 2.1: Reference orbit, arc length s along it and local coordinates.
2.1 Coordinates and Maps
Usually when studying dynamics, the time t plays the role of the independent variable, and we study the motion of positions x and velocities v or momenta
p as coordinates. Using the Lagrange mechanism, it is easy to transfer
to new coordinates, in particular the coordinates that describe the relative
dynamics around the reference orbit. Furthermore, instead of using t, we
usually use the arc length s along the reference orbit as an independent
variable. Fig. 2.1 illustrates the concept.
For the understanding of the motion in relative coordinates, let us assume
we have studied and understood the motion of the reference orbit. In case
there is no field at all, this reference orbit will merely follow a straight line.
Furthermore, there are many devices used in accelerators that have fields, but
along one given straight line, all the fields vanish, and the device is lined up
in such a way that the reference particle follows this line. Another important
device uses magnetic fields, and along the reference orbit one tries to hold
the magnetic fields constant, in which case the reference orbit is circular, at
least within the element. In all other cases, it is usually necessary to describe
the reference orbit by numerically integrating the equations of motion.
We assume the position and momenta of the reference particle r ref (s), ,
p ref (s)
are known. Here the momentum p is the dynamical momentum as in eq.
(1.4). As a technical detail, let us also assume that for all points s, we have
p ref (s) ∦ e zLab , i.e., the motion is never pointing vertically straight (which
for most real accelerators is no limitation whatsoever). Let furthermore r tube
be smaller than the minimum radius of curvature that the reference orbit
experiences in the section of the device that we want to study. We now
consider a “flexible tube” of radius r tube centered around the reference orbit,
and restrict the particles that we want to describe to only those within the
tube, as Fig. 2.2 illustrates the situation. Again, for practical devices this
represents hardly a limitation; for example, in the LHC (see Table 1.1), the
“tube” would be more than 2 km wide, much larger than the region required