## Forward and Retrograde Precessions from ECE2 Relativity

This note checks Horst’s calculations received this morning and shows that a planar orbit in ECE2 relativity can always have a forward and retrograde precession, coming from the same relativistic lagrangian. This can always be analysed with the scalar Euler Lagrange equations (2) and (3), or with the vector Euler Lagrange equation (10). The relation between the scalar and vector Euler Lagrange equations is given by Eqs. (35) to (38). The scalar Euler Lagrange equations are the two components in a plane of the vector Euler Lagrange equation. The theoretical results for the S2 star system are in the middle of the experimental range. In the non relativistic limit both scalar and vector Euler Lagrange equations give the same result, the static ellipse. Note carefully that the Lagrangian is mathematically the same as that of special relativity. So we make the fundamentally important discovery that special relativity gives forward and retrograde precessions of any planar orbit. Now we can add the field equations of ECE2 in order to get more information. These are astounding results, they can be fitted to experimental data by using the spin connection to fix initial conditions as in the last note. Then the theory can be applied to the solar system, where all precessions appear to be forward precessions. These are entirely new concepts in astronomy and cosmology. There is freedom of choice of proper Lagrange variables in Euler Lagrange theory, so the proper Lagrange variables are chosen to reproduce the experimental data. In the S2 star there appears to be retrograde precession, so we choose the vector Euler Lagrange equation as the physical equation. In the solar system all precessions appear to be forward precessions, so we chose the scalar Euler Lagrange equations. Prior to the emergence of ECE2 theory it was thought that special relativity does not give precession at all. ECE2 covariance can be thought of as special relativity in a space with finite torsion and curvature. The Euler Lagrange equations are the result of the Hamilton Principle of Least Action.

a377thpapernotes4.pdf