A Method to Render Second Order Beam Optics Programs Symplectic

Abstract

Violations of Liouville's theorem have been observed in various tracking studies employing programs based on a second order matrix ray trace. This is because the truncated Taylor's series transformation used by the codes in question is not canonical. We describe a simple method, based on a generating function, that brings the ray trace to canonical form. This method utilizes the second order matrix elements provided by existing beam optics programs, and may be implemented by the use of a few FORTRAN subroutines. As an example, the program DIMAT has been modified to accept the symplectric ray trace; results obtained using both symplectic and nonsymplectic transformations are compared.