Symplectic integrator for $s$-dependent static magnetic fields based on mixed-variable generating functions

TL;DR

This paper introduces a symplectic integration method using mixed-variable generating functions to accurately model charged particle trajectories in complex static magnetic fields without relying on paraxial approximation.

Contribution

It presents a novel technique for constructing exact symplectic transfer maps in three-dimensional static magnetic fields using mixed-variable generating functions.

Findings

Successfully applied to a quadrupole magnet with octupole component

Ensures exact symplecticity in transfer maps

Does not require paraxial approximation

Abstract

We describe a technique for constructing a symplectic transfer map for a charged particle moving through an accelerator component with arbitrary three-dimensional static magnetic field. The transfer map is constructed by symplectic integration; by representing the map at each step of the integration by a mixed-variable generating function, exact symplecticity is ensured. By using an appropriate integration algorithm, there is no necessity to make the paraxial approximation. The technique is illustrated by application (in one degree of freedom) to a quadrupole magnet with strong octupole component and fringe field.