transverse dynamics 29
and correspondingly for the reverse formula
S(s)
φ(s) = arctg β 0 S(s) − α 0 C(s)
1
2
S 2 (s) + [ β
( ) =
0 S(s) − α 0 C(s)]
β s
β 0
β 0 S(s) − α 0 C(s)
or, in a simpler shape
2
1
r
S(s)
β(s) = β 0 sin φ(s)
1
−S ' (s)
)
β(s) + cos φ(s)
β
α(s) =
0
sin φ(s)
Having derived the above equations, we can now see that
we can describe the evolution of the particle trajectories in
a transfer line or in a circular accelerator by means of matrix
formalism (see Fig. 2.10). In other words, linear transformations that are enabled by the linearity of the Hill’s equations
express as
(
y(s)
) (
C(s) S(s)
y(s 0 )
=
'
(2.26)
y ' (s)
C (s) S ' (s)
)(
y ' (s 0 )
)
The matrix elements C(s) and S(s) depend only on the magnetic lattice and not on the initial conditions of the particle.
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FIGURE 2.10
Linear matrix approach for evaluation of the evolution of the
particle coordinates in a transfer line.
The transfer matrix is therefore given by
(
C(s) S(s)
M 1→2 = C ' (s) S ' (s)
)
(2.27)
The described approach allows for the possibility of using the
matrix formalism to describe the evolution of the coordinates
of a charged particle in a magnetic lattice.