The Linearization of the Equations of Motion
101
to focusing.
We illustrate the resulting motion in Fig. 4.10, which shows the resulting
transversal orbits for six particles on both sides of the origin that were initially
moving parallel to the axis of the solenoid without any transverse motion.
Each of these particles performs a circular motion with a radius that equals
half of its entrance position. In particular this entails that the particles do not
have concentric orbits like in a cyclotron, but rather that after a half period
of their oscillations, all these particles will pass through the exact center of
the device.
This observation now leads to focusing; on their path from their initial
position x i to the position x = 0 after a half period, each particle loses distance
to the axis and is thus focused.
We further observe that all particles rotate with the same angular frequency
given by
ω c =
v t
r
=
Ze
γm
B 0 .
This entails that all initially parallel particles reach the origin at the same
time, and also that the relative loss of distance to the axis is the same. So the
speed with which they move towards the origin is proportional to their initial
position, and it is a hallmark of thin lens focusing.
The picture suggests another interesting feature; initially axis-parallel particles that are on a common line upon entering remain on a common line,
shown dashed in Fig. 4.10, in their further motion through the region of
constant magnetic field. Elementary geometry shows that this is actually the
case: by connecting any particle in Fig. 4.10 (dot) and the corresponding
center of its orbit (cross), we can see immediately that the angle between the
dashed line and the x-axis is always half of that between the radius and the
x-axis. We further observe that when the particles reach the origin and have
performed a half revolution around their orbits, the dashed line will coincide
with the vertical axis, and thus will have performed a quarter revolution; in
fact elementary geometry shows that the dashed line performs a rotation with
one half of the rotational frequency of particles.
4.4.2.3 The Rotating Coordinate System
These observations of the idealized case now motivate the treatment of the
general case, in which fields do not jump abruptly but rather gently depend
on s. In this case the appearing momentary rotation frequencies of both the
particles as well as a possible suitable rotating coordinate system are not
constant but will change with the position s. Thus, we attempt the Ansatz of
introducing new variables
Z that describe the motion in a rotating coordinate
system via
z(s) = ˆ
R(θ(s)) ·
Z(s),
(4.24)
and we will arrive at the expected result at eqs. (4.28). Here, ˆ
R(θ) is a
rotation matrix, and for the further discussion, the following matrix properties
101
to focusing.
We illustrate the resulting motion in Fig. 4.10, which shows the resulting
transversal orbits for six particles on both sides of the origin that were initially
moving parallel to the axis of the solenoid without any transverse motion.
Each of these particles performs a circular motion with a radius that equals
half of its entrance position. In particular this entails that the particles do not
have concentric orbits like in a cyclotron, but rather that after a half period
of their oscillations, all these particles will pass through the exact center of
the device.
This observation now leads to focusing; on their path from their initial
position x i to the position x = 0 after a half period, each particle loses distance
to the axis and is thus focused.
We further observe that all particles rotate with the same angular frequency
given by
ω c =
v t
r
=
Ze
γm
B 0 .
This entails that all initially parallel particles reach the origin at the same
time, and also that the relative loss of distance to the axis is the same. So the
speed with which they move towards the origin is proportional to their initial
position, and it is a hallmark of thin lens focusing.
The picture suggests another interesting feature; initially axis-parallel particles that are on a common line upon entering remain on a common line,
shown dashed in Fig. 4.10, in their further motion through the region of
constant magnetic field. Elementary geometry shows that this is actually the
case: by connecting any particle in Fig. 4.10 (dot) and the corresponding
center of its orbit (cross), we can see immediately that the angle between the
dashed line and the x-axis is always half of that between the radius and the
x-axis. We further observe that when the particles reach the origin and have
performed a half revolution around their orbits, the dashed line will coincide
with the vertical axis, and thus will have performed a quarter revolution; in
fact elementary geometry shows that the dashed line performs a rotation with
one half of the rotational frequency of particles.
4.4.2.3 The Rotating Coordinate System
These observations of the idealized case now motivate the treatment of the
general case, in which fields do not jump abruptly but rather gently depend
on s. In this case the appearing momentary rotation frequencies of both the
particles as well as a possible suitable rotating coordinate system are not
constant but will change with the position s. Thus, we attempt the Ansatz of
introducing new variables
Z that describe the motion in a rotating coordinate
system via
z(s) = ˆ
R(θ(s)) ·
Z(s),
(4.24)
and we will arrive at the expected result at eqs. (4.28). Here, ˆ
R(θ) is a
rotation matrix, and for the further discussion, the following matrix properties