Fast Symplectic Mapping, Quasi-invariants, and Long-term Stability in the Lhc

A systematic program to explore stability of orbits in hadron storage rings is based on the following steps: (a) beginning with a symplectic tracking code, construct the mixed-variable generator of the full-turn map in a Fourier-spline basis; (b) use the resulting fast mapping to follow long orbits and estimate the long-term dynamic aperture; (c) contruct quasi-invariants and examine their variation in time to set long-term bounds on the motion for any initial condition in a specified region. First results from an application of the program to the Large Hadron Collider (LHC) are reported. Maps can be constructed in a few hours and evaluated at a speed 60 times greater than that of one-turn tracking, on a workstation computer. Orbits of 107 turns take 3.6 hours. The value of a “stroboscopic” view of the synchro-betatron motion is emphasized. On a Poincaré section at multiples of the synchrotron period, one can study resonances and invariant surfaces in two dimensions, thereby taking advantage of techniques that have proved effective in treating pure betatron motion.