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An Introduction to Beam Physics
x
a
FIGURE 6.4: Action of a thin lens in phase space.
x
a
FIGURE 6.5: Action of a quadrupole or dipole in phase space.
6.2 Polygon–like Phase Space
In order to study the motion of ensembles of particles under linear transformations, it is useful to characterize them by certain simple geometric forms
in which the particles are contained and requiring only few parameters. The
two most useful such forms are the polygon and the ellipse.
A polygon in phase space is uniquely defined by its corner points; and since
straight lines stay straight lines, it is sufficient to study just the motion of
the corner points. Fig. 6.6 shows the motion of a polygon under a linear
transformation.
Frequently a polygon with just four points is chosen; if its lines are initially
symmetrically arranged around the origin, they will stay symmetrically ar-
An Introduction to Beam Physics
x
a
FIGURE 6.4: Action of a thin lens in phase space.
x
a
FIGURE 6.5: Action of a quadrupole or dipole in phase space.
6.2 Polygon–like Phase Space
In order to study the motion of ensembles of particles under linear transformations, it is useful to characterize them by certain simple geometric forms
in which the particles are contained and requiring only few parameters. The
two most useful such forms are the polygon and the ellipse.
A polygon in phase space is uniquely defined by its corner points; and since
straight lines stay straight lines, it is sufficient to study just the motion of
the corner points. Fig. 6.6 shows the motion of a polygon under a linear
transformation.
Frequently a polygon with just four points is chosen; if its lines are initially
symmetrically arranged around the origin, they will stay symmetrically ar-