Hamiltonian theory of the crossing of the 2Q_{x}-2Q_{y}=0 nonlinear coupling resonance

Abstract

In a recent paper, the adiabatic theory of Hamiltonian systems was successfully applied to study the crossing of the linear coupling resonance, Q_{x}-Q_{y}=0. A detailed explanation of the well-known phenomena that occur during the resonance-crossing process, such as emittance exchange and its dependence on the adiabaticity of the process, was obtained. In this paper, we consider the crossing of the resonance of nonlinear coupling 2Q_{x}-2Q_{y}=0 using the same theoretical framework. We perform the analysis using a Hamiltonian model in which the nonlinear coupling resonance is excited and the corresponding dynamics is studied in detail, in particular looking at the phase-space topology and its evolution, in view of characterizing the emittance exchange phenomena. The theoretical results are then tested using a symplectic map. Thanks to this approach, scaling laws of general interest for applications are derived.