We begin with methods that provide highly accurate representations of fields, a pre-requisite for a detailed analysis of nonlinear motion, and discuss surface-based methods that assure accurate field representation far from midplane or axis. We then discuss methods to enforce the symplectic symmetry of the motion by minimal deformations in a suitable metric of Hamiltonian spaces, which is important in the attempt of long-term tracking of the dynamics. We then develop an analytical mechanism most closely related to the conventional aberration correction in conventional systems, the method of normal forms. As a result, quantitative measures of those linearities that matter are obtained in the form of resonance strengths and tune shifts. Residual effects can be described quantitatively using normal form defect methods, which allow rigorous predictions of long-term stability.
For practical calculations, the above methods can be expressed seamlessly in terms of differential algebraic structures on Taylor representations, either in a truncated representation in the conventional DA framework, or under consideration of remainder error terms in a fully rigorous setting.
M. Berz, K. Makino, Nuclear Instruments and Methods A645 (2011) 175-181
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