From symplectic integrator to Poincare map: spline expansion of a map generator in Cartesian coordinates

Data from orbits of a symplectic integration can be interpolated so as to c onstruct an approximation to the generating function of a Poincare map. The time required to compute an orbit of the symplectic map induced by the gen erator can be much less than the time to follow the same orbit by symplecti c integration. The approximation has been constructed previously for full-t urn maps of large particle accelerators, and a large saving in time (for in stance a factor of 60) has been demonstrated. A shortcoming of our work to date arises from the use of canonical polar coordinates, which preclude map construction in small regions of phase space near coordinate singularities . Here, we show that Cartesian coordinates can be used, thus avoiding singu larities. The generator is represented in a basis of tensor product B-splin es. Under weak conditions, the spline expansion converges uniformly as the mesh is refined, approaching the generator of the Poincare map as defined b y the symplectic integrator, in some parallelepiped of phase space centered at the origin. (C) 1999 Elsevier Science B.V. and IMACS. All rights reserv ed.