I agree that it is seemingly planar in beta, but beta as a function of theta and phi. This gives Eq. (25), and the orbits that you graphed for UFT269. These are perfectly valid, showing precessing orbits in the function phi. This is the key difference from the two dimensional analysis in plane polar coordinates r and phi alone. That gives a non precessing orbit. The animation shows that the orbit is an intricate function of time, and is three dimensional. The three dimensional analysis reduces to the usual two dimensional analysis if A(t), the constant of integration in Eq. (23), is zero, or time independent. The Euler Lagrange equation (7) gives the same result as the fundamantal calculation of angular momentum from first principles in UFT269, so it valid. The Euler Lagrange equation (8) is also a standard equation. I checked this with a literature search. The Euler Lagrange equation (6) is valid because the lagrangian can be defined as in Eq. (1). The analysis is completely self consistent, cross checked with lagrangian and first principles. The integration constant A(t) in Eq. (23) must be time dependent, so it plays a role in evolution. It must also be theta independent. So A(t) can be regarded in analogy with relaxation theory and correlation function or diffusion theory. Initially it may be very large in the primaeval three dimensional orbit, but it falls to zero over billions of years, resulting in a planar orbit. The cosmic backgropund in your suggestion can be developed with relaxation theory, the background being the stochastic force that appears in Langevin or memory function theory.
In a message dated 01/09/2014 12:12:11 GMT Daylight Time,
My view is the following: Obviously it comes out from this note that an orbit with a central potential of type k/r is planar as also comes out from Euler-Lagrange theory. If such an orbit is considered in 3D, the plane of the orbit will be placed in an arbitrary angle compared to the XY plane. Nevertheless this remains a planar orbit. The beta angle obviously is the transformed phi angle of the XY plane which depends from phi and theta since the transformation needs two coordinates. So it would be illustrative to show that eq.(17) of the note is just this coordinate transformation, probably identical with the tetrad.
The Lagrange equations (7) and (8) seem to be unnecessary because Lagrangian theory requires coordinates without constraints as you know. The constraints have been applied to transform theta and phi to beta.
The time-dependent integration constant remains interesting. The meaning is in my opinion that an external acceleration is applied. This leads to a deviation from the k/r potential and, consequently, to non-planar orbits.
EMyrone@aol.com hat am 1. September 2014 um 11:25 geschrieben:
All planar orbits precess by observation, so they must be limits of three dimensional orbits. The Kepler Hooke Newton non precessing ellipse is obtained if and only if condition (17) of note 270(3) is true. The reason for a planar non precessing ellipse can be expressed mathematically , as in note 270(3), but there must still be a deeper reason due to physics, and that must be the result of evolution. To work out the evolution is a highly non trivial n mass problem needing supercomputers and additional assumptions. In the two mass problem the orbit is in general three dimensional, this is proven by the fact that all observable orbits precess. It is now known that the inverse square law produces this precession if the solution is correctly worked out with spherical rather than cylindrical polar coordinates. In Marion and Thornton for example, the entire orbital theory is worked out with cylindrical polar coordinates. The force law for the type of three dimensional orbits in galaxies is completely different from inverse square. In the roughly two dimensional whirlpool galaxies the orbit is a hyperbolic spiral which can be worked out with ECE theory, so one can start with that two dimensional spiral and make it three dimensional by changing phi to beta. That ought to produce some more very interesting graphics and animations. The Cartan torsion of the spherical polar coordinates can be worked out easily, and that alone is sufficient to show that this is a torsion based theory. So all orbits are in one sense due to torsion.