Symplectic tracking through straight three dimensional fields by a method of generating functions

For simulating single-particle trajectories, the derivation of final coordinates from known initial coordinates through arbitrary electromagnetic fields is of key interest in accelerator physics. We address this task in the case of straight stationary magnetic fields, using generating functions via a perturbative ansatz for the corresponding Hamilton-Jacobi equation. Such an approach is always symplectic, independent of the expansion order. We set up the Hamiltonian by static fields, represented by Fourier series, and outline this approach for the correct and complete set of 3D-multipole fields. Different types of multipoles can be treated with the same formalism, combining them with a specific table of Fourier coefficients characterizing their fields. The resulting particle-tracking routine maps the multipole in a single step. Results are compared with analytical estimations and high-resolution integration methods.