| The nonlinear dynamics of particles in modern accelerators are governed by the fields of the guiding and focusing superconducting magnets. While for the linear design of the lattice it is sufficient to treat the fields by a rather coarse approximation, a thorough analysis of the magnetic fields is necessary in order to study the nonlinear effects. As an example, the Large Hadron Collider (LHC) is studied, using detailed field data for the High Gradient Quadrupoles in the interaction region. The influence of the resulting nonlinearities on the dynamics is analysed via high-order maps determined with Differential Algebraic (DA) techniques and the code COSY INFINITY. Normal form methods are utilised to determine amplitude dependent tune shifts as well as resonance strengths. It is shown that the end effects change the nonlinear characteristics of the lattice significantly.;This made a rigorous treatment of the fields desirable, which is presented in the second part. Using DA-techniques the Taylor expansion of the field is calculated. Based on this expansion two analytical algorithms to determine the multipole content of the field are developed. |
