CORRECT LOCAL DESCRIPTION FOR TRACKING IN RINGS

We discuss the absolute minimum knowledge required to design a symplec tic integrator for rings. We include details about: (1) symplectic int egrators for compound bends, (2) coordinate patches for bends with arb itrary entry and exit angles, (3) general fringe field effects, (4) ar bitrary magnet displacements, and (5) radiation effects in electron ri ngs. Of course, we expect the usual automatic differentiation to be im plemented in such integrators and to be linked with a perturbation the ory package capable of analyzing symplectic and non-symplectic maps fo r the beam envelope analysis (calculation of equilibrium second-order moments).