Construction of Nonlinear Symplectic Six-Dimensional Thin - Lens Maps by Exponentiation

The aim of this paper is to construct six dimensional symplectic thin–lens transport maps for the tracking program SIXTRACK [2], continuing an earlier report [1] by using another method which consistes in applying Lie series and exponentiation as described by W. Gröbner [3] and for canonical systems by A.J. Dragt [4]. As in Ref. [1] we firstly use an approximate Hamiltonian obtained by a series expansion of the square root { 1− [px + H · z] 2 + [pz − H · x]2 [1 + f(pσ)] 2 }1/2 up to first order in terms of the quantity [px + H · z]2 + [pz − H · x]2 [1 + f(pσ)] 2 . Furthermore, nonlinear crossing terms due to the curvature in bending magnets are neglected. An improved Hamiltonian, excluding solenoids, is introduced in Appendix A by using the unexpanded square root mentioned above, but neglecting again nonlinear crossing terms in bending magnets. It is shown that the thin lens maps remain unchanged and that the corrections due to the new Hamiltonian are fully absorbed into the drift spaces. Finally a symplectic treatment of the crossing terms appearing in bending magnets is presented in Appendix B, taking into account only the lowest order. The equations derived are valid for arbitrary particle velocity, i.e. below and above transition energy and shall be incorporated into the tracking code SIXTRACK [2]. ∗CERN, SL-Division, Geneva, Switzerland 1