Betatron Oscillations

Let N be the number of periods per revolution, and
be the length of one period.
Since the y equation is linear, the solutions at two points s and s 0 are linearly related. If
Y is the column vector we can write
4.2
⎡ y ⎤
⎢
⎣ y ′ ⎥
⎦
Y()= s M s| s 0
( )Ys 0
( )
L= C N
( ) M( s 2
s 1 )
where M s 2
s 1 is a 2x2 matrix . The determinant of is unity, since the
*
Wronskian determinant is constant because the coefficient of y' is zero in 4.1. We note
that the transformations M form a group, since the identity Μ( s|s)= I exists, the
inverse M −1 ( s | s 0 )= M( s 0
| s) exists because det Μ ≠ 0 , and M( s| s 0 )= M( s| s 1 )M ( s 1
| s 0 ).
Some useful examples of M are when K is constant. For K=0 (drift),
4.3
M( ⎡
s| s 0 )= 1 s− s ⎤
0
⎢
⎣ 0 1 ⎥
⎦
19 Jun 2007 Accelerators: Theory and
Applications
*see notes
4