These are important results in my opinion, the vector formats of the Euler Lagrange method are elegant. It is possible to use an Euler Lagrange equation with derivatives with respect to a four vector, for example in quantum field theory, the equation at the foot of page 96, second edition, of Ryder’s “Quantum Field Theory”, second edition. It contains partial lagrangian / partial A sub mu on the left hand side and partial lagrangian / partial A dot sub mu on the right hand side. See also page 108, where the lagrangian for Yang Mills theory is given by Eq. (3.132), and the following equation is an Euler Lagrange equation with derivatives with respect to a four vector W sup i sub mu and a four vector W dot sup i sub mu. Also, in Atkins, “Molecular Quantum Mechanics”, second edition, 1982, page 454, second equation from foot of page, a derivative partial lagrangian / partial r dot bold appears. The very interesting thing is that the vector formalism gives different results from the scalar formalism. This was revealed by computer algebra, it is impossible to see it analytically. There is a deep consequence to this in mathematics, as well as physics.

To: EMyrone@aol.com

Sent: 09/05/2017 15:27:46 GMT Daylight Time

Subj: Re: Forward and Retrograde Precessions from ECE2 RelativityTo my understanding the vector Equation (10) is only a short notation for the Euler-Lagrange equations (2,3). A derivative according to a vector is not defined. The problem of different results seems to result from eq.(14). the term

d/dt ( partial L/partial r dot )

is not identical to

gamma^3 m r dotdot.

The definition of the relativistic momentum seeems not to be compatible with the Lagrangian formulation. So we have to choose what produces the right experimental results as you stated in the note.

BTW, in (4,5) there should stand gamma instead of gamma^2 because you used the derivative according to t, not tau.

All four possible equations can be written in vector form, see attachment.Horst

Am 09.05.2017 um 12:18 schrieb EMyrone:

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 propoer Lagrange variables are chosen to reproduce the expreimental 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.