The methods of shuttle-sums and of shuttle-integrals in ion optics

A new recursive technique which is called the shuttle-sums or a generalized analogue of the Gaussian brackets, is proposed and developed. A special form of this technique is continuous generalized analogue to the Gaussian brackets which is called the shuttle-integrals. The shuttle-sums are used for analytical and numerical investigations of ion optical systems. The shuttle-integrals are used to calculate the matrizant (or the transfer matrix function) of the nonlinear equations of beam motion for an arbitrary coefficient matrix. In this method there is rigorous conservation of the phase volume of the beam at each stage of the calculations.