The unexpected flight of the electron in a classical hydrogen-like atom

We study the behaviour of a particle in an attractive Coulomb potential according to the nonrelativistic approximation of the Lorentz–Dirac equation that is the equation describing the motion of a charged point particle in classical electrodynamics when radiation reaction is taken into account. We prove that for any choice of the initial data the particle always eventually escapes to infinity. Its acceleration either vanishes asymptotically, corresponding to a scattering process, or increases exponentially fast with time, with a so-called ‘runaway’ behaviour. No bound states of any type are possible.