Lawrence Berkeley National Laboratory

Linear equations of motion are derived that describe small-amplitude centroid oscillations induced by displacement and rotational misalignments of the focusing solenoids in the transport lattice, dipole steering elements, and initial centroid offset errors. These equations are analyzed in a local rotating Larmor frame to derive complex-variable "alignment functions" and "bending functions" that efficiently describe the characteristics of the centroid oscillations induced by mechanical misalignments of the solenoids and dipole steering elements. The alignment and bending functions depend only on properties of the ideal lattice in the absence of errors and steering and have associated expansion amplitudes set by the misalignments and steering fields. Applications of this formulation are presented for statistical analysis of centroid deviations, calculation of actual lattice misalignments from centroid measurements, and optimal beam steering.