The method of formal series: applications to nonlinear beam dynamics and invariants of motion

A novel technique to determine invariant curves in nonlinear beam dynamics based on the method of formal series has been developed. It is first shown how the solution of the Hamilton equations of motion describing nonlinear betatron oscillations in the presence of a single sextupole can be represented in a nonperturbative form. Further, the solution of the Hamilton-Jacobi equation is obtained in a closed symbolic form as a ratio of two series in the perturbation parameter (and the nonlinear action invariant), rather than a conventional power series according to canonical perturbation theory. It is well behaved even for large values of the perturbation parameter close to strong structural resonances. The relationship between existing invariant curves and the so-called scattering orbits in classical scattering theory has been revealed.