Action-Angle Variables for Betatron Oscillations

In Chap. 3 we showed that choosing the action-angle canonical variables in a one-dimensional Hamiltonian system dramatically simplifies the dynamics: the action remains constant and the angle increases linearly with time. With minor modifications, the same transformation can be applied to the Hamiltonian ( 6.14) that describes betatron oscillations in an accelerator. This yields an invariant of the motion and is also a useful starting point for analyzing more complicated dynamics.