Exact Numerical Calculation of Chromaticity in Small Rings

A Newton's searcch method is presented which efficiently finds closed orbits in a ring whose lattice contains both linear elements, such as drifts and quadrupoles, and nonlinear elements such as dipoles with a small radius of curvature, sextupoles, etc. The method simultaneously determines the tune of the closed orbit. By observing how the location of the closed orbt depends on the total momentum, the η and η' functions are determined exactly (including nonlinear terms in δ). By observing how the tunes of the closed orbit depend on the total momentum, the chromaticities are determnined exactly. The essential tool employed is the simultaneous integration of the equations of motion for a trajectory along with the variational equations for neighboring trajectories.