Non-linear beam-beam effects in present and future storage rings; generalized Birkhoff method on conservative mappings

A relevant application of the recent developments in nonlinear dynamic theory lies in providing the description and consequent calculation of the interactions among charged particles that belong either to a given beam or to different beams in collision with each other in storage rings. This paper focuses on the weak-strong version of beam-beam interactions and further limits itself to only those interactions which occur in proton-proton and proton-antiproton rings. The motion of a test particle around its main reference orbit is analyzed by conservative mapping following from periodic kicks induced by the forces of beam-beam collisions which solicit otherwise regular betatron oscillations. Also considered is the effective spin polarization of the beams after long residence in the machine in the presence of depolarizing forces. Both fixed-angle and head-on collisions are treated. The Birkhoff procedure applied in a generalized form to the mapping consists of searching for a formal simplectic transformation which reduces the mapping to normal form or to the form of integrable mapping