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 the 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.