Construction of symplectic maps for nonlinear motion of particles in accelerators

Abstract

We explore an algorithm for the construction of symplectic maps to describe nonlinear particle motion in circular accelerators. We emphasize maps for motion over one or a few full turns, which may provide an economical way of studying long-term stability in large machines such as the Superconducting Super Collider (SSC). The map is defined implicitly by a mixed-variable generating function, represented as a Fourier series in betatron angle variables, with coefficients given as B-spline functions of action variables and the total energy. Despite the implicit definition, iteration of the map proves to be a fast process. The method is illustrated with a realistic model of the SSC. We report extensive tests of accuracy and iteration time in various regions of phase space, and demonstrate the results by using single-turn maps to follow trajectories symplectically for ${10}^{7}$ turns on a workstation computer. The same method may be used to construct the Poincar\'e map of Hamiltonian systems in other fields of physics.