SLAC, Menlo Park, California
UNM, Albuquerque, New Mexico
The goal is to construct a symplectic evolution map for a large section of an accelerator, say a full turn of a large ring or a long wiggler. We start with an accurate tracking algorithm for single particles, which is allowed to be slightly non-symplectic. By tracking many particles for a distance S one acquires sufficient data to construct the mixed-variable generator of a symplectic map for evolution over S. Two ways to find the generator are considered: (i) Find its gradient from tracking data, then the generator itself as a line integral *. (ii) Compute Hamilton's principal function on many orbits. The generator is given finally as an interpolatory C2 function, say through B-splines or Shepard's meshless interpolation. A test of method (i) is given in a hard example: a full turn map for an electron ring with strong sextupoles. The method succeeds where Taylor maps fail, but there are technical difficulties near the dynamic aperture due to oddly shaped interpolation domains. Method (ii) looks more promising in strongly nonlinear cases. We also explore explicit maps from direct fits of tracking data, with symplecticity imposed on local interpolating functions.
