60
W. Herr and E. Forest
3.5.3 Action-Angle Variables
More appropriate for studies of beam dynamics is the use of Action-Angle variables.
Once the particles “travel” on a circle, the motion is better described by the
canonical variables action J x and angle x :
2J
ψ
with the definitions and the choice is (3.14):
x =
√
2J x β x cos(( x )
p x = −
2J x
β x
(sin(( x ) + α x cos(( x ))
J x =
1
2 (γ x x 2 + 2α x xp x + β x p 2
x )
(3.15)
– the angular position along the ring becomes the independent variable!
– The trajectory of a particle is now independent of the position s!
– The constant radius of the circle
√
2J defines the action J (invariant of motion)
3.5.4 Beam Emittance
A sad and dismal story in accelerator physics is the definition of the emittance.
Most foolish in this context is to relate emittance to single particles. This is true in
particular when we have a beam line which is not periodic. In that case the CourantSnyder parameters can be determined from the beam. These parameters are related
to the moments of the beam, e.g. the beam size is directly related to the second order
moment < x 2 >. Using the expression above for the action and angle, we can write
for this expression:
< x
2 > = < 2J x β x · cos
2 (( x ) > = 2β x < J x · cos
2 (( x ) > .
(3.16)