Revisiting Sommerfeld's Atomic Model using Euler-Lagrange Dynamics

The purpose of this work is to present the atomic model proposed by Sommerfeld. We outline the classical calculation of the elliptical orbit and then apply the Wilson-Sommerfeld quantization rules to obtain expressions for the quantization of energy and orbit semi-axes. We then apply relativistic theory to solve the problem of degenerate orbits. Thus, we see that the Sommerfeld atom is a very rich topic, involving the integration of different subjects ranging from Lagrangian mechanics to relativity, as well as plane geometry and differential equations.