Hydrogen Atom Set up by Electron Capture Describing a Continuous Path

We present a formulation in which an electron can follow a continuous path in the process of formation of a hydrogen atom. We use a field Lagrangean foundation for chaotic, non-conservative systems to shape the electron motion. We have found electron transition paths that vary from an infinity distance to the finite radii concerning the proton-nucleus, which correspond to the stable orbital states. To find these states corresponds to the problem of finding the possible stable orbital attractors in the chaotic dynamical motion problem. In this view, although there is a general collapsing tendency of the electron-proton system, the Hamiltonian of the system exhibits some metastable states in which the electron can stay for some time, which is not infinite, due to the chaotic perturbations that the electron is undergone in the formation of the atom. The Hamiltonian varies continuously along the electron path, from zero at the infinity to the negative value corresponding to some of the metastable orbits.