Dynamics of Coupled Motion

Abstract

Coupling between horizontal and vertical betatron oscillations plays an important role in beam dynamics. Since coupling is caused by linear as well as nonlinear fields, we observe the effects on beam characteristics in virtually any accelerator. In order to be able to introduce or eliminate coupling, whichever may be the case, in a controlled and predictable way, we need to understand the dynamics of coupling in more detail. In this chapter we will derive first the equations of motion for the two most general sources of coupling, the solenoid field and the field of a rotated quadrupole, solve the equations of motion and formulate modifications to beam dynamics parameters and functions of linear uncoupled motion.