Non-linear single-particle beam dynamics in circular machines

The studies of non-linear single-particle beam dynamics are devoted to two main fields: large superconducting hadron colliders, dominated by unintentional imperfections of the magnetic field-shape, and compact low-emittance lepton accelerators, governed by chromatic aberration correctors. The non-linear fields are at the origin of two effects: the betatron tunes change with the amplitude and the momentum of the circulating particles, and, for certain combinations of the horizontal, vertical, and synchrotron tunes, non-linear resonances are excited. These phenomena have a profound influence on the stability of the particle motion. Analytical, numerical and experimental approaches have been used to evaluate and possibly compensate mechanisms which lead to particle losses. (author) 36 refs