Symplectic Tracking Using Point Magnets and a Reference Orbit Made of Circular Arcs and Straight Lines

Abstract

In order to study long term stability, it appears desirable that the particle tracking be symplectic. One way to achieve symplectic tracking is to replace the magnets by a series of point magnets and drift spaces. This approach is modified here by using a reference orbit that is made up of arcs of circles and straight lines which join smoothly with each other. This makes the symplecticity more evident, and simplifies in some way the particle tracking, as the coordinate system based on this reference orbit is not changing discontinuously between elements. It also allows the use of transfer matrices to find the linear orbit parameters. For this choice of reference orbit, the required results are obtained to track particles, which are the transfer functions, the transfer matrices and the transfer time, for the different elements present in the accelerator. It is shown that, in the absence of longitudinal magnetic fields theses results provide a symplectic, second order integrator. Existing tracking programs that use a reference orbit, made up of arcs of circles and straight lines, can be modified, using the results given here to do symplectic tracking with point magnets. The results have been used to modify the ORBIT tracking program. The ORBIT program will now, by changing an indicator, either track using the usual large accelerator approximation for the transfer functions or do symplectic tracking with point magnets, and will use the same reference orbit in both cases.