Muonic helium atom as a classical three-body problem

We study the classical problem of the muonic helium atom, a helium atom wit h one of its electrons replaced by a muon. First, we establish the connecti on of the model with the one-dimensional frozen planetary approximation of the helium atom and find that there is classically stable motion for the co nfiguration alpha -particle-electron-muon, and no stable motion for the alp ha -particle-muon-electron configuration. After that, rye introduce the res tricted muonic helium problem, a model for the movement of an electron in t he 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 inte resting feature is the stability of the halo orbits, for a range of energy values. The vicinity of the Lu particle is also studied, examining Poincare sections for increasing energy values showing an increase of the chaotic m otion.