Exact Calculation of Nonlinear Orbit Properties of a Synchrotron

The trajectory of a particle in an arbitrary guide field can be computed as accurately as desired by numerical integration. This "measured" trajectory can then be fit with the phase-amplitude relation describing betatron oscillations, suitably generalized to account for nonlinear effects. Information about closed on- and off-momentum orbits, as well as the frequencies of small oscillations about them, can be obtained from such fits. Thus one can compute chromaticities exactly, among other things.