This is very interesting as usual! This equation is in fact the same as the lagrangian result, Marion and Thornton, third edition, equation (7.21). I will prove that today and it can be checked by computer. This is a nice result that shows that the entire kinematic analysis is self consistent from beginning to end. When v is parallel to r the object m falls directly into M and there is no spin connection. When v is not parallel to r it is in an orbit and as soon as v is not parallel to r there is a spin connection and an angular momentum. If a stone is dropped it falls with v parallel to r, but if thrown sideways starts with v perpendicular to r and if it does not have sufficient velocity, falls into a trajectory with v becoming parallel to r. If thrown fast enough it goes into orbit. A goal of this co authored paper, UFT236, should be to explain the velocity curve of a spiral galaxy, i.e v becomes constant with increasing r. Computer algebra is ideal for this. The angular momentum L, if defined in k, is parallel to the spin connection vector omega, which is also the angular velocity vector. Coriolis and centrifugal forces are of course real and measurable, and are due to the spin connection. This new analysis is fully and correctly relativistic with none of the errors of Einsteinian analysis. All orbits are due to the Cartan spin connection, which defines the Cartan torsion and curvature. Finally the Frenet analysis is a special case of the Cartan analysis. The Frenet frame is the Cartan tangent frame.

To: EMyrone@aol.com

Sent: 30/01/2013 23:12:47 GMT Standard Time

Subj: Re: 236(3): Some results for accelerationThese are some results for the acceleration eq.(67). Could be checked with results of previous papers. One has to observe that the inverse function theta(r) has to be inserted into dr/dtheta in order to obtain the correct derivative

d/dr (dr/dtheta).

All results for the spirals contain the negative centrifugal term 1/r^3. For the hyperbolic spiral this comes out directly, which is identical to

omega x (omega x r),

indicating that spiral galaxies are dominated by the rotating frame alone. Newtonian attraction has completely disappeared.I still have to check the equations of the note.

Horst

Am 30.01.2013 12:41, schrieb EMyrone

This note gives a clear and easily understandable review of the way in which the ellipse is derived from the inverse square law incorrectly attributed to Isaac Newton, and in fact inferred by Robert Hooke as in John Aubrey “Brief Lives” (an online classic of literature). Newton takes the credit for fluxions, and the first part of this note is his original method in modern vector notation (see “Vector Analysis Problem Solver”). It is checked that the Hooke / Newton equation (23) is the same as eq. (28), derived in recent papers from pure kinematics. It is shown very clearly that the Hooke Newton method is exactly what it was intended to be, a proof that a particular force law and equation of motion gives a ellipse. It does not explain why the ellipse exists, while ECE theory clearly explains the ellipse as the motion of axes, i.e. of space itself, with a spin connection now known to be the angular velocity of the axes. This is a correctly relativistic explanation given by a generally covariant unified field theory (ECE theory). The correctly relativistic equation of all planar orbits is eq. (64). It is shown finally that this reduces to Hooke Newton for an elliptical orbit, QED or quod erat demonstrandum. This is Latin for “that which was intended to show”. It would be very interesting to apply computer algebra now to find the acceleration for any orbit from eq. (67). This also replaces the Einstein theory of orbits and is a simple and powerful new equation. It is also shown why Hooke Newton is a very successful empirical description of space and aerospace dynamics. The reason is that it defines the ellipse in the form of E = T + V (eq. (75) of the attached note). In other words it just rewrites the ellipse in terms of E, T and V, with the concept of force defines as minus the derivative of potential energy with respect to r. It is very important to understand that the new eq. (67) applies to any planar orbit of any kind observable by astronomy, and it is correctly relativistic as well. Hooke Newton fails spectacularly outside the solar system, or vanishingly tiny corner of the universe.