These are sections 1 and 2, and the new equation of precessing orbits in a plane is Eq. (56). The final paper corrects some minor errors in Note 402(7). Computer algebra can be used to find whether Eq. (56) or the simpler approximation eq. (60) have analytical solutions.

## Comparing the ECE2 Precessing Orbit with Astronomical Data

The origin of any orbit has been traced to vacuum fluctuations which define the spin connection of ECE2 relativity in for example Eq.(16) of Note 402(4), and in UFT401 it has been shown by Horst Eckardt that there can be retrograde as well as forward precession, depending on the sign of the spin connection. This goes far in advance of the obsolete Einstein theory, which cannot produce retrograde precession, and is completely riddled with errors of all kinds (it has been refuted in almost a hundred different ways in the UFT series and Stephen Crothers has refuted it in many more ways). So if clean data on orbital precession can be found, the orbit can be explained by changing the mean square vacuum fluctuation so that the precession at the perihelion is explained exactly. In the solar system, orbital precession is badly affected by the influence of other planets as is well known. Myles Mathis (online) has argued convincingly that the astronomers do not account for this influence correctly. They use Newtonian theory for the planetary influence on the precession of a given planet, but use Einsteinian theory for the tiny part that is claimed to be due to EGR. However in some systems the precession is that of one object around another, and presumably is accurately known. The analytical orbit can be found from the relativistic Binet equation as in Note 402(7). From Eq. (35) of this note it is clear that the orbit depends on the orbital velocity contained in the Lorentz factor. So if the orbital velocity is used as an input parameter the orbit will be changed. So I will proceed to write up Sections 1 and 2 of UFT402. It would be very interesting to produce graphics in Section three ilustrating how the mean square fluctuation of the vacuum affects the precession at the perihelion, and how the orbital velocity affects the precession at the perihelion.

## The Relativistic Binet Equation of Orbits

This given by Eq. (25) using the results of Note 402(5). It is the same result as derived in earlier UFT papers, giving a proof of overall self consistency. It is developed to give the precise orbital equation (36), which is approximated by Eq. (42), a second order non linear differential equation. It is known from numerical integration in UFT401 that the starting equation, the relativistic Newton equation, gives precessing ellipses. Eq. (36) gives the precise analytical result for the same orbit, so Eq. (36) must give precession.

## Minor Erratum in UFT401

In eqs. 7 and 19 – 21, m should be included. This is a minor typo that does not affect the paper. It is already being intensely studied.

## 402(3): The Generalized Momentum

The latest note gives a rigorously self consistent solution with conservation of relativistic angular momentum. Marion and Thornton define a beta = v / c which is used throughout chapter fourteen of teh third edition. I can give a few examples of how this beta should be used. The Lorentz boost in different directions is also discussed by Carroll and Ryder, and also in "The Enbigmatic Photon" (online in the Omnia Opera). In their lagrangian in Eq. (14.111) there are components u sub i and also beta. I can write a note to clarify how beta should be used.

Date: Thu, Feb 22, 2018 at 8:38 AM

Subject: Re: 402(3): The Generalized Momentum

To: Myron Evans <myronevans123>

There seems to be an intricate point with the gamma factor: according to eq.(11) of the note, gamma contains the velocity component v_i for each generalized coordinate q_i. This is different from using the modulus of v in all component equations.

Horst

Am 19.02.2018 um 15:44 schrieb Myron Evans:

This Eq. (6.151) of Marion and Thornton, third edition. the term as introduced in Kelvin and Tait, "Natural Philosophy" (1867). It is the origin of the Lagrange equations of motion, and also the origin of the relativistic lagrangian. The relativistic momentum as used by Einstein is derived from the conservation of momentum. Horst has found that the relativistic Newtonian force, the time derivative of the relativistic momentum, gives retrograde precession, a major discovery because EGR fails to give retrograde precession.

## The Rigorously Self Consistent ECE2 Covariant Force Equation

This is equation (3),valid for any coordinate system. In the plane polar system it is resolved into the relativistic Leibniz orbital equation (6) and the conservation of relativistic angular momentum, Eq. (7). Eqs. (6) and (7), when solved simultaneously, give precessing orbits. Forward and retrograde precessions can both be obtained, depending on the sign of the spin connection as just shown by Horst. This note is the result of extensive discussions between co author Horst Eckardt and myself. Provided that Eq. (3) is resolved into Eqs. (6) nad (7), everything is rigorously self consistent, leading to an entirely new understanding of all kinds of precession. This shows the value of these discussions, or dialogue, which have been going on for over twelve years continuously. This is one of the most rigorous dialogues in the history of physics. The AIAS / UPITEC staff has also contributed importantly. The theory is part of a generally covariant unified field theory with finite torsion and curvature.

## 402(4): The Vacuum Fluctuations for a Precessing Planar Orbit

This Section will be the start of a new phase of ECE theory in which cosmology is based on the same vacuum fluctuation theory as Lamb shift theory. It is obviously important to try to get precise agreement with experimental data, and this method can gradually be extended to other phenomena. Resonance can also be worked in to the theory, there is very intense current interest in UFT399.

Date: Tue, Feb 20, 2018 at 3:43 PM

Subject: Re: 402(4): The Vacuum Fluctuations for a Precessing Planar Orbit

To: Myron Evans <myronevans123>

This looks similar as I am preparing for UFT 401. Will finish the section the next days.

Horst

Am 20.02.2018 um 11:11 schrieb Myron Evans:

The mean square vacuum fluctuation responsible for a precessing planar orbit is given by Eq. (16), in which the orbital velocity v for small precessions is given by the Newtonian (17). The angular frequency of fluctuations for any planar orbit is given by Eq. (19). These results are a straightforward result of the relativistic Newton equation (1) used with the ECE2 force equation (3). The orbital precession is obtained as in UFT377 by numerical integration of Eq. (1) with given initial conditions. This theory can be used with any coordinates system. This calculation is to second order in the tensorial Taylor series as in UFT401. The aim is to use this theory to obtain complete agreement with experimental data.

## 402(4): The Vacuum Fluctuations for a Precessing Planar Orbit

The mean square vacuum fluctuation responsible for a precessing planar orbit is given by Eq. (16), in which the orbital velocity v for small precessions is given by the Newtonian (17). The angular frequency of fluctuations for any planar orbit is given by Eq. (19). These results are a straightforward result of the relativistic Newton equation (1) used with the ECE2 force equation (3). The orbital precession is obtained as in UFT377 by numerical integration of Eq. (1) with given initial conditions. This theory can be used with any coordinates system. This calculation is to second order in the tensorial Taylor series as in UFT401. The aim is to use this theory to obtain complete agreement with experimental data.

## 402(3): The Generalized Momentum

This Eq. (6.151) of Marion and Thornton, third edition. the term as introduced in Kelvin and Tait, "Natural Philosophy" (1867). It is the origin of the Lagrange equations of motion, and also the origin of the relativistic lagrangian. The relativistic momentum as used by Einstein is derived from the conservation of momentum. Horst has found that the relativistic Newtonian force, the time derivative of the relativistic momentum, gives retrograde precession, a major discovery because EGR fails to give retrograde precession.

## Self Consistency problem with relativistic Newtonian force

As in Marion and Thornton pp. 213 ff. of the third edition the Lagrange equations can be derived from the Newtonian p = partial T / partial r dot (Eq. 6.102) so they are not independent of this equation. This is also eq. (14.107) of Marion and Thornton and they show that it leads to the relativistic lagrangian. So this equation is not independent. Finally the relativistic momentum is given by p rel sub X = partial lagrangian / partial X dot and the Y component, as in Eqs. (15) to (19). The origin of relativistic momentum is discussed in Section 14.7 of Marion and Thornton. It is conservation of linear momentum. Finally the equation p sub i = partial lagrangian / partial q dot sub in generalized coordinates is eq. (6.151) of Marion and Thornton. From eq. (6.151), the Lagrange equations follow immediately by differentiating both sides, to give Eq. (6.152). This is elegant, and the easiest way of seeing that Eq. (2) of Note 402(2) is a well known equation. It simply merges Eqs. (6.151) and (6.152) but uses the slightly unfamiliar differentiation with respect to a vector. So p = partial lagrangian / partial q dot is an equation from which the Euler Lagrange equations can be derived. They can also be derived from Hamilton’s Principle of Least Action as is well known. The equivalence of the Lagrange and Newton equations for rectangular coordinates is discussed in Section 6.6. The Lagrange and Hamilton equations are more general than the Newton equations if generalized coordinates are used. I know that we have showed M and T can go wildly wrong, when dealing with general relativity, but the foregoing basics are well established. It is very good to discuss these basics. Does anyone else have a comment? This kind of free and collegial discussion is essential, it may sometimes lead to new discoveries. We have made more discoveries in the past decade than needles on a hedgehog.

Date: Mon, Feb 19, 2018 at 12:17 PM

Subject: Self Consistency problem with relativistic Newtonian force

To: Myron Evans <myronevans123>

To my understanding the resulting inconsistency is quite clear. Lagrange theory gives relativistic equations of motion from the relativistic Lagrangian. If the equations of motion are derived from another equation directly, in this case

F = m d(p_rel)/dt = gamma^3 m dv/dt,

this is an INDEPENDENT approach, and it cannot be expected that both methods give the same result a priori. For the results to be identical, it must be

p_rel = partial L / partial bold r dot,

and this is obviously not the case. Nevertheless the Lagrangian gives the relativistic angular momentum as a constant of motion. But there is no prescription that the above first equation contains the same p_rel as obtained formally from the Lagrangian. To my knowledge the approach

p_rel = gamma m v

comes from generalization of relativistic dynamics based on the Lorentz transform which only holds for constant relative motion. Perhaps this is the problem.

Horst

Am 18.02.2018 um 13:21 schrieb Myron Evans:

There is freedom of choice of proper Lagrange variables, but the review just sent over seems to be the only way to achieve complete and rigorous self consistency, and in this sense the method is unique. The formal Euler Lagrange equation using a proper Lagrange variable vector r is rigorously correct but if and only if it is correctly interpreted and correctly expressed in any given coordinate system. The formal equation to my mind is elegant and economical.