A Lie transform perturbation scheme for Hamiltonian averaging in self consistent systems

A periodic focusing system is reduced to an equivalent continuous focusing one for a beam with space charge by averaging over the lattice oscillations. The Lie transform perturbation method is used to canonically transform the laboratory phase space variables to slowly oscillating variables. A similar averaging over the lattice period was performed by R.C. Davidson, H. Qin and PJ. Channell [Phys Rev ST 2, 074401 (1999)] using the Poincare-Von Ziepel perturbation method. The Lie transform method offers certain advantages in that it retains the original form of the Hamiltonian before beginning the process of canonical transformation to a slowly oscillating coordinate frame. On the other hand, the Poincare-Von Ziepel method requires one to make a Taylor expansion of the Hamiltonian in terms of the as yet undetermined expansion terms of the transformed phase space variables. The Lie transform method avoids such a Taylor expansion and so the formulation is less tedious. It will be demonstrated that performing the reverse transformation to the original phase space variables is also straight forward in the Lie transform method.