Analysis of phase space topologies for models of 4D betatronic motion in view of 4D beam splitting

The novel technique of multiturn extraction is used to ll the Super Proton Synchrotron (SPS) at CERN with a high-intensity proton beam delivered by the Proton Synchrotron (PS). This technique involves manipulating nonlinear fields of sextupoles and octupoles in the PS to trap the beam into stable islands in the transverse phase space. By varying the tunes slowly and crossing a resonance condition it is possible to trap particles by means of stable structures. This generates a number of well-dened beamlets, which may be extracted over a number of turns proportional to the order of the resonance that is crossed and its stability. In this report, the theoretical background of how to construct the Normal Form of a 4D map incorporating a sextupolar and an octupolar eld contribution that act on a particle in our beam close to a 4th order difference resonance is detailed. A Hamiltonian will be analysed after transforming from a 4D to a 2D system using a canonical transformation, which allows us to visualise the island structure that may allow for particle trapping.