Spline Expansion of a Map Generator in Cartesian Coordinates

Data from orbits of a symplectic integrator can be interpolated so as to construct an approximation to the generating function of a Poincar e map. The time required to compute an orbit of the symplectic map induced by the generator can be much less than the time to follow the same orbit by symplectic integration. The construction has been carried out previously for full-turn maps of large particle accelerators, and a big saving in time (for instance a factor of 60) has been demonstrated. A shortcoming of our work to date arose from the use of canonical polar coordinates, which precluded map construction in small regions of phase space near coordinate singularities. This paper shows that Cartesian coordinates can also be used, thus avoiding singularities. The generator is represented in a basis of tensor product B-splines. Under weak conditions the spline expansion converges uniformly as the mesh is re ned, approaching the exact generator of the Poincar e map as de ned by the symplectic integrator, in some parallelepiped of phase space centered at the origin. Invited paper at the NSF/CBMS Regional Conference on Numerical Analysis of Hamiltonian Systems, Colorado School of Mines, Golden, Colorado, June 2{6, 1997. Proceedings of the conference to be published in Applied Numerical Mathematics. Work supported in part by Department of Energy contract DE{AC03{76SF00515 and the Deutsches Elektronen-Synchrotron (DESY), Hamburg.