Chapter 6
Linear Phase Space Motion
In this chapter, we want to study the action of transfer matrices on particles
by looking in detail to what happens to entire regions of phase space as
they are transported. This is important because the beam in an accelerator
is just such a region, and of course we want to make sure that at any time,
this region is within the beam pipe.
Let us begin by collecting several observations about two-dimensional transfer maps M.
1. M preserves areas.
2. Different initial points have different final points.
3. Continuous curves stay continuous curves.
4. Closed curves stay closed curves.
5. A point inside a closed curve will stay inside of the closed curve.
Fig. 6.1 illustrates the second item, and Fig. 6.2 illustrates the other items.
The first item led us to giving a name, namely “emittance,” to the preserved
area. The last two items are particularly important, as they tell us that if we
can enclose our beam within any closed boundary curve, then it is sufficient to
study the dynamics of this boundary curve alone. It is interesting to note that
while in the two-dimensional case, closed curves always stay closed curves, it
x 2
a 2
x 1
a 1
M
M
FIGURE 6.1: Mapping of individual points in phase space.
141
DOI:10.1201/b12074-6
Linear Phase Space Motion
In this chapter, we want to study the action of transfer matrices on particles
by looking in detail to what happens to entire regions of phase space as
they are transported. This is important because the beam in an accelerator
is just such a region, and of course we want to make sure that at any time,
this region is within the beam pipe.
Let us begin by collecting several observations about two-dimensional transfer maps M.
1. M preserves areas.
2. Different initial points have different final points.
3. Continuous curves stay continuous curves.
4. Closed curves stay closed curves.
5. A point inside a closed curve will stay inside of the closed curve.
Fig. 6.1 illustrates the second item, and Fig. 6.2 illustrates the other items.
The first item led us to giving a name, namely “emittance,” to the preserved
area. The last two items are particularly important, as they tell us that if we
can enclose our beam within any closed boundary curve, then it is sufficient to
study the dynamics of this boundary curve alone. It is interesting to note that
while in the two-dimensional case, closed curves always stay closed curves, it
x 2
a 2
x 1
a 1
M
M
FIGURE 6.1: Mapping of individual points in phase space.
141
DOI:10.1201/b12074-6