Abstract:
Systems consisting of two interacting magnetic moments, or spins, in an applied magnetic field play a role in the study of quantum chaos where they are used as a classical analogue of a quantum spin system. The equations of motion for such systems are particularly amenable to solution via a geometric integrator constructed by splitting the Hamiltonian into integrable pieces. Such methods have the advantage of restricting numerical solutions to the correct manifold and respecting various geometric properties of the system. Some of these properties, such as approximate conservation of total energy, are important when producing Poincaré surfaces of section (PSSs) which are used to study the onset of chaos in the classical system. The choice of coordinate system for the equations of motion is also important with some choices leading to coordinate singularities. We compare PSSs obtained via a generalized 'leapfrog' integrator with those from a classical 'black-box' Runge–Kutta integrator. The effects of the coordinate singularity on the accuracy and the energy of the solution is also investigated.