Muonic helium atom as a classical three-body problem

We study the classical problem of the muonic helium atom, a helium atom with one of its electrons replaced by a muon. First, we establish the connection of the model with the one-dimensional frozen planetary approximation of the helium atom and find that there is classically stable motion for the configuration alpha-particle-electron-muon, and no stable motion for the alpha-particle-muon-electron configuration. After that, we introduce the restricted muonic helium problem, a model for the movement of an electron in the potential of the alpha particle/muon pair moving in a circular orbit. In this model, the equilibrium points, their associated Lyapunov families of periodic orbits, and their stability parameters were studied. The most interesting feature is the stability of the halo orbits, for a range of energy values. The vicinity of the alpha particle is also studied, examining Poincare sections for increasing energy values showing an increase of the chaotic motion.