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

Symplectic integrator plays a pivotal role in the long-term tracking of charged particles within accelerators. To get symplectic maps in accurate simulation 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, the 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 the boundary surface's field data due to uniqueness theorem. Finding generating functions to meet the Hamilton-Jacobi equation via a perturbative ansatz, we derive symplectic maps that are independent of the expansion order. This method yields a direct mapping from initial to final coordinates in a large step, bypassing the transition of intermediate coordinates. Our developed particle-tracking algorithm translates the field harmonics into beam trajectories, offering a high-resolution integration method in accelerator simulations.