Approach to combined-function magnets via symplectic slicing

In this article we describe how to obtain symplectic “slice” maps for combined-function magnets, by using a method of generating functions. A feature of this method is that one can use an unexpanded and unsplit Hamiltonian. From such a slice map we obtain a first-order map which is symplectic at the closed orbit. We also obtain a symplectic kick map. Both results were implemented into the widely used program MAD-X to regain, in particular, the twiss parameters for the sliced model of the Proton Synchrotron at CERN. In addition, we obtain recursion equations for symplectic maps of general time-dependent Hamiltonians, which might be useful even beyond the scope of accelerator physics.

Figure 1

Relative errors for various re-entries for 4 slices of a single CFM with quadrupole and sextupole component against the Lie-transformation coefficients. Those errors which are zero are not shown.

Figure 2

Counter-check with convergence behavior of the relative errors for the re-coefficients of Fig. 1.

Figure 3

Convergence behavior of the horizontal- and vertical chromaticities in the PS against the PTC results (red).

Figure 4

Horizontal and vertical phase space with the corrected (symplectic) code. The units are [m] and [rad] respectively.