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W. Herr and E. Forest
coordinates and angles at a given position (x, x , y, y ), e.g. at the end of a beam
line or a small spot one an electron microscope. When “accelerator physicists” talk
about concepts such as tune, resonances, β-functions, equilibrium emittances etc.,
all these are irrelevant for single pass machine. There is no need to study iterating
systems. In these cases the use of the trace space is fully adequate, in fact preferred
because the quantities can be measured. In the end, the mathematical tools are very
different from the ones discussed in this article.
3.2.2 Curved Coordinate System
For a “curved” trajectory, in general not circular, with a local radius of curvature
ρ(s) in the horizontal (X–Z plane), we have to transform to a new coordinate system
(x, y, s) (co-moving frame) with:
X = (x + ρ) cos
s
ρ
− ρ
Y = y
Z = (x + ρ) sin
s
ρ
(3.4)
The new canonical momenta become:
p x = P X cos
s
ρ
+ P Z sin
s
ρ
p y = P Y
p s = P Z
1 +
x
ρ
cos
s
ρ
− P X
1 +
x
ρ
sin
s
ρ
(3.5)
3.3 Sources of Non-linearities
Any object creating non-linear electromagnetic fields on the trajectory of the
beam can strongly influence the beam dynamics. They can be generated by the
environment or by the beam itself.
3.3.1 Non-linear Machine Elements
Non-linear elements can be introduced into the machine on purpose or can be
the result of field imperfections. Both types can have adverse effects on the beam
stability and must be taken into account.