OK thanks again, I will write a note to show that Eq. (8) of the protocol with positive sign is the same as Eq. (7.38) of Marion and Thornton if the spin connection is zero. In this case its solution is Eq. (7.39), a conic section with half right magnitude (7.40a) and eccentricity (7.40b). The conic section is Eq.(7.41) The spin connection is very small, so an analytical approximation can proceed on that basis. A numerical integration is the best way forward.
Fwd: 403(5): Solution of the ECE2 Force Equation for Small Eccentricity (Planetary Systems)
To: Myron Evans <myronevans123>
Eq.(8) of the protocol is a kind of implicit solution of(7) because it is not u(phi) but an integral form of the inverse function phi(u). This is obviously what we need here.
The bracket in the upper line of Eq. (8) marks that there are two solutions, written as a vector in brackets. So both lines are a solution, differering only by the sign of the integral.
The integral can be solved analytically in case of two alternative simplifications:
1. The log function is omitted. Then the result does no more depend on omega_u. It is essentially a arcsin function, see eq. (9) of the augmented protocol.
2. The linear and quadratic terms of u are omitted. Then an error function expression is obtained ( see (10)).
If one of these simplifications can be justified, I can graph the result. Otherwise we could try a numerical solution of the integral.
Horst
Am 14.03.2018 um 10:56 schrieb Myron Evans:
Thanks again. This is very useful as usual. Eq. (8) of the protocol is the needed analytical solution and it is a precessing ellipse. As the spin connection goes to zero, the integral gives an ellipse as in Marion and Thornton chapter seven of the third edition. So the integral in Eq (8) of the protocol can be evaluated numerically or perhaps with an analytical approximation that does not restrict the range of phi.. It is similar to the integral of the Einstein theory, except that one term is replaced by a log term. It is known from the apsidal method for small eccentricities (note 403(6)) that the ellipse will be a precessing ellipse. So I think that Eq. (8) of the protocol should be used in UFT403 together with Note 403(6), and the rest of the notes mentioned but not used. I have one question, what is the bracket in the upper line of Eq. (8) of the protocol/ I assume that the lower line of Eq. (8) is the answer. From the apsidal method, any force that is not inverse square will give a precession. However, the force of the ECE2 equation is derived as part of a unified field theory rigorously based on geometry as you know. The major discovery is that precession is due to vacuum fluctuations An entirely new cosmology can be developed from the theory of vacuum fluctuations of the same type as used in Lamb shift theory. As shown in UFT399, infinite energy from the vacuum can also be based on vacuum fluctuations.
Date: Tue, Mar 13, 2018 at 6:43 PM
Subject: Re: 403(5): Solution of the ECE2 Force Equation for Small Eccentricity (Planetary Systems)
To: Myron Evans <myronevans123>This is which solutions Maxima gives for eqs. (1) and (2). It has to be assumed that omega_r is constant, otherwise there is no analytical solution. omega is written omega_u because the variable u=1/r has to be used in the equation solver. Then eq.(1) has an analytical solution, see eq. o6 in the protocol. By precondition, this solution is only valid in the range r ~ alpha, as far as I understand this. It can be seen that this solution comprises a homogeneous part (with constants %k1 and %k2) and an inhomogeneous part.
Eq. (2) of the note only gives an incomplete solution (with integrals) because u appears in the denominator:
r*omega_r = omega_u / u.
The approximatin of eq. (5) leads to the analytical solution o10, again with parts from the homogeneous and inhomogeneous diff. equation. This seems to differ from what the Worlfram solver puts out. However, o10 seems to be more or less identical with eq.(10) of the note.
Since we have r ~ alpha, this solution seems only to be valid in the range phi ~ pi/2. Using this (and setting %k1=0) in o10 gives the result o11 which depends on omega_u, in contrast to your eq.(14). Where did omega_r go in your calculation?Horst
Am 09.03.2018 um 13:39 schrieb Myron Evans:
This is given by Eq. (7) in the approximation of Eq. (5), and by Eq. (10) in the next approximation, the equation following equation (8). These equations can be used to describe any precessing orbit with precision and the ECE2 equation is clearly preferred to the Einstein equation because the latter has been refuted in nearly a hundred different ways in the UFT series and refuted in many other ways by many authors for many years. It is used only by dogmatic ostriches who hope that ECE2 will go away, but it’s here to stay.