Evaluation of the Dynamic Aperture of the CERN-LHC using Fast Indications of Long Term Stability

The dynamic of the particles in the CERN-LHC is largely dominated by the unavoidable magnetic errors of the superconducting magnets used to guide the particles along the 27km long circumference. Due to that, the volume of the phase space wherestable motion occurs shrinks, hence reducing dramatically the performance of the accelerator. We willfirst introduce the concept of thedynamic aperture, reviwing theoretical and computational issues. Different methods for a fast and accurate estimate of this quantity are discussed and compared. After that, the issueof finding effective fast indicators of the long term stability of the particle motion is discussed in detail. Dynamical quantities, like the variation with time of the instantaneous betatron frequency (tune) and the maximal value of the Lyapunov exponent are presented and used to detect the onset of chaotic motion. Their predictivity is extensively checked with the classical element-by-element computer tracking simulations over many turns. Advanced methods and large computational resources are required to achieve in a positive manner this ambitious goal.