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

Abstract

Data from orbits of a symplectic integrator can be interpolated so as to construct an approximation to the generating function of a Poincare 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 refined, approaching the exact generator of the Poincare map as defined by the symplectic integrator, in some parallelepiped of phase space centered at the origin. (orig.)SIGLEAvailable from TIB Hannover: RA 2999(97-163) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDepartment of Energy, Washington, DC (United States)DEGerman