[en] A proposal of using extrapolation algorithms as an analytical - numerical tool for the evaluation of the beam-beam effects in storage rings is formulated. These algorithms already known to experts of Applied Mathematics are described in some detail. Physical models for the beam-beam interaction effects on betatron motion and polarization of a test particle are discussed and put in such a form to allow the application of the extrapolation algorithms

[en] This chapter proposes an improvement in the linearization of non-linear fundamental equations on the basis of a perturbative assumption. Discusses laser fusion formalism, and models of the linearized theory and parametric instabilities. This method contemplates the linearized iterative solution as an intermediate step before reaching an outcome that makes sense physically even if strong non-linearity is present. The subsequent stage, that would provide the solution for the initial non-linear system, is reached by the use of an algorithm that uses the terms of the formal series as ingredients

[en] The general expression of the Stern-Gerlach (SG) force is deduced for a relativistic charged spin-1/2 particle which travels inside a time varying magnetic field. This result was obtained either by means of two Lorentz boosts or starting from Dirac's equation. Then, the utilization of this interaction for attaining the spin states separation is reconsidered in a new example using a new radio-frequency arrangement. On the basis of the previous estimates, we feel ready to propose the time varying SG interaction as a method for attaining a spin state separation of an unpolarized beam of, say (anti)protons, since the energy of particles with opposite spin orientations will differ and beams in the two states can be separated. In a first stage of the study of a sensible practical design, we intend to proceed with numerical simulations. As a first step, we intend to verify the correctness of Eqs.(42) and (43) setting once β_ph = 2 and then β_ph = 3, in a cavity where the field line pattern can be realistically controlled. Beyond the verification of the present theory, there is also the aim of studying the effects generated by the spin precession inside the cavity, that we did not yet address in this note. Next, we shall consider a spin splitter scheme based on the lattice of an existing or planned (anti)proton ring endowed with an array of splitting cavities. The principal aim of the latter implementations is to check the mixing effect of the longitudinal phase-plane filamentation, i.e. the actual foe which could frustrate the entire spin splitting process.

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[en] In this paper we point out possible method of polarizing a beam of spin (1/2) charged particles by using repeatedly the Stern-Gerlach forces for both cases of transversal and longitudinal polarizations in a storage ring. Realistic estimates of the method are considered

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[en] The use of Pade approximant methods in the study of the evolution of non-integrable systems of classical mechanics is proposed, with particular attention to the accelerator physics problem of the beam-beam interaction in storage rings. An extrapolation technique is devised in order to deduce the behaviour at asymptotic times of the betatron oscillation of a test particle in the weak beam - strong beam approximation. The methods are tested numerically on simple one- and two-dimensional cases. (author)

[en] The use of converging analytical approximations (such as Pade approximants and related algorithms) is pointed out to be quite relevant for the treatment of strongly interacting matter-radiation waves, as for very dense plasmas driven by a powerful laser. Such an approach might provide us with predictions (for instance dispersion relations or normal modes) that differ quite substantially from the ones given by perturbation theory