Fwd: PS:Re: Consistency of calculations in notes 413(5) and 414(1) with Lagrangian

Conditions for a decreasing orbit

Many thanks, this is very interesting work! ECE2 is a universal theory of gravitation and so it applies to all orbits. My guess is that there is no orbit in the universe that does not obey fundamental geometry (ECE2 theory is fundamental geometry as you know). I plan to do some more calculations using the relativistic hamiltonian, the results should be the same as from the relativistic kinematics and the relativistic lagrangian. That would be a triple cross check. Of particular interest are decreasing and retrograde orbits. EGR cannot produce retrograde orbits at all.

Consistency of calculations in notes 413(5) and 414(1) with Lagrangian

I changed the initial conditions and used a negative omega_1. Then a decrease of orbit is visible. As I supposed, the angular velocity must go down by the frame rotation. Then a point is reached where the angular momentum is no more conserved and the radius shrinks.
Will check this further and work out.

Horst

Am 07.09.2018 um 17:46 schrieb Myron Evans:

Agreed. Your Larange integrator is a powerful and very important method of going from the equations of motion to the orbit.
Consistency of calculations in notes 413(5) and 414(1) with Lagrangian

Ok, we can use the frame (r, phi’). Then the transformation

phi’ = phi + omega_1 t

transforms from the oberserver space to the local frame of the orbiting mass. This should be kept in mind.

Horst

Am 07.09.2018 um 15:11 schrieb Myron Evans:

PS:Re: Consistency of calculations in notes 413(5) and 414(1) with Lagrangian

The integration of the two simultaneous equations using your Lagrange integrator is the most important advance. I went through all the calculations again and summarized them in Note 414(4). Precise agreement s found between the kinematic and lagrangian methods, so I propose basing UFT414 on Note 414(4) ff. The calculations are carried out in frame ( r , phi’ ). This note also clears up some difficulties we had in previous uses of the Lagrange theory. The Lagrange variables are r and phi’. Note 414(4) is valid in any coordinate system such as Cartesian, as in some of your previous numerical work. The kinematic method is particularly clear. The transformation from frame (r ,phi’) back to the observer frame (r , phi) defines the spin connection.

PS:Re: Consistency of calculations in notes 413(5) and 414(1) with Lagrangian
PS: attached is the protocol of the Lagrange calculations.

Horst

Am 07.09.2018 um 11:11 schrieb Horst Eckardt:

In the notes you wrote the terms for L and Omega_r (eqs. 4 and 7 in 414(1)) with overall positive signs. However the kinetic energy for the Lagrangian variable phi’ is defined by
T=

(where phi corresponds to phi’). Therefore there are negative terms in the constant of motion:

.
Here is dphi’/dt = omega (or more correctly omega’).
From the Lagrange equation for r (eq.1, to be written with phi’) follows

.

There is a sign change in front of the factor 2 omega.
Concering radius shrinking form L=const, I guess that omega counteracts the term t*d omega_1/dt as long as possible, keeping the radius constant. I will check this by examples.
It seems that in the notes the variables phi and phi’ have accidentally be interchanged. The relevant transformation is:

phi = phi’ – omega_1*t.

Another point: I had erroneausly added the spin connection term to the full equation for r dot dot. According to the above derivation, using the spin connection is only a formal re-writing. The equations remain the same. I will do further checks and write up a preliminary version of UFT 413,3 hopefully before my holiday next week.

Horst


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