transverse dynamics 27
2.4 Matrix formalism
2.4.1 Pseudo-harmonic oscillations
Let’s look for the solution of the Hill’s Eq. 2.18 in following
form
)
[
]
y(s) = ε y β y (s) cos φ y (s) − φ
(2.22)
where
s
ds '
φ y (s) =
(2.23)
β y (s ' )
s 0
which describes pseudo-harmonic oscillations. Here the beta
functions β (in x and z) are proportional to the square of the
envelope of the oscillations. The functions φ(s) (also in x and Beta functions β are proporz) describe the phase of the oscillations.
tional to the square of the enWe can find the differential equation for the beta func- velope of the oscillations
tions, by substituting the form Eq. 2.22 into the Hill’s equation. We use the following equation first
β ' (s)
.

y
' (s) =
ε cos(φ(s) − φ) − φ
' (s) εβ(s) sin(φ(s) − φ)

2
β(s)
and the second derivatives
⎡
⎤
⎢ β '' (s)
β ' 2 (s)
.
⎥ √
⎢
⎥
y
'' (s) = ⎢ ⎢ .
−
− β(s)φ
' 2 (s) ⎥ ⎥ ε cos(φ(s) − φ)−
⎣
⎦
2 β(s) 4β 3/2 (s)
⎡
⎤
⎢
.
β '
.
(s)φ ' (s) ⎥ √
⎢ φ
'' (s)
⎥
⎢
⎥
− ⎢
β(s) +
⎥ ε sin(φ(s) − φ)
⎣
⎦
β(s)
which we substitute to Hill’s equation and proceed by equating the coefficients to zero in front of sin and cos parts. We Alpha function is defined as
therefore obtain
α = −β ' /2
1 ββ
''
−
1 β
'2 + k(s)β
2 = 1
(2.24)
2
4
and
φ
' (s) =
1
(2.25)
y
β y (s)
which represents the differential equations for beta function
and the betatron phase.
2.4.2 Principal trajectories
The solutions of Hill’s equation can be found in the form of
principal trajectories. These are two particular solutions of the
homogeneous Hill’s equation
y
'' + k(s)y = 0
which satisfy the following initial conditions
C (s 0 ) = 1; C
' (s 0 ) = 0; cosine−like solution