In this note the new Lagrangian method is applied to the orbits of ECE2 relativity in two and three dimensions, and to the relativistic H atom with fine structure. From work such as UFT328 it is known that ECE2 relativity produces precessing orbits. That paper was based on simultaneous solution of the ECE2 lagrangian and hamiltonian. The method in this note is based on the relativistic lagrangian (1) for plane polar coordinates, and (7) for spherical polar coordinates. It produces precession by using Maxima to solve the relevant Euler Lagrange equations. We have shown over the past decade that Einsteinain general relativity is completely riddled with errors, so this method produces precession without using the Einstein field equation at all. The precession can be compared with data from astronomy for any object m orbiting any object M with an assumed radial gravitational potential. If this assumption is lifted, the orbits become very interesting as shown in UFT371. The relativistic H atom is computed with the same lagrangian method and the result can be compared with the analytical result (42) for the relativistic H atom. Eq. (42) gives fine structure. The Schroedinger equation for H does not give fine structure as is well known.