From symplectic integrator to poincaré map: spline expansion of a map generator in Cartesian coordinates Original Research Article

Data from orbits of a symplectic integration can be interpolated so as to construct an approximation to the generating function of a Poincaré 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 approximation has been constructed previously for full-turn maps of large particle accelerators, and a large saving in time (for instance 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 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 refined, approaching the generator of the Poincaré map as defined by the symplectic integrator, in some parallelepiped of phase space centered at the origin.