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

Symplectic integrators play a pivotal role in the long-term tracking of charged particles within accelerators. To obtain symplectic maps in accurate simulations of single-particle trajectories, two key components are addressed: precise analytical expressions for arbitrary electromagnetic fields and a robust treatment of the equations of motion. In a source-free region, electromagnetic fields can be decomposed into harmonic functions, applicable to both scalar and vector potentials, encompassing both straight and curved reference trajectories. These harmonics are constructed according to boundary surface field data due to the uniqueness theorem. By finding generating functions to satisfy the Hamilton-Jacobi equation via a perturbative ansatz, we derive maps that are symplectic at any order of the expansion. This method yields a direct mapping from initial to final coordinates, bypassing the tracking of intermediate coordinates. Our developed particle-tracking algorithm translates the field harmonics into beam trajectories, offering a high-resolution integration method in accelerator simulations.