Agreed, in one of our previous papers it was shown that the Lagrange derivative is an example of the covariant derivative of Cartan. You showed that the spin connection of the Lagrange derivative is the Jacobian. So this shows that:
g = Dv / Dt
where D denotes the Cartan covariant derivative. The reduction to the static frame in which
g =dv / dt
occurs when the spin connection vanishes. I will think about the direct transformation you suggest below, it is directly related to the spin connection.
To: EMyrone@aol.com
Sent: 30/10/2016 09:53:25 GMT Standard Time
Subj: Re: Discussion of Note 360(5)OK, in the moving frame, r is perpendicular to v, we have to two cartesian coordinate systems, one for the fixed frame and one for the moving frame. It could be interesting to describe the direct transformation between both, maybe this will become quite complicated.
Horst
Am 29.10.2016 um 08:41 schrieb EMyrone:
Many thanks, this point is answered in UFT359 by using moving frames of reference. You provided very helpful graphics for these moving frames in UFT359, Section 3, by expressing X and Y in terms of r and theta. For a circular orbit, the frame of reference is X = r cos theta, Y = r sin theta, but for the ellipse and any other orbit it is different. The Lagrange or convective derivative is the derivative in the moving frame, and the latter is defined by observation for each orbit. Therefore v of Eq. (2) is equated with the observed v, and this defines X and Y. In the moving frame so defined, v is perpendicular to r. In the static frame X = r cos theta, and Y = r sin theta, r is not perpendicular to v for the conic sections. However, in the moving frame, r is perpendicular to v because v is proportional to – X i + Y j and r is proportional to X i + Y j. The definitions of X and Y are however different in the moving frame. The use of the convective derivative in this radically new way is a direct consequence of UFT349 ff., already very popular papers.
To: Emyrone
Sent: 28/10/2016 20:15:32 GMT Daylight Time
Subj: note 360(5)As far as I understand the notes, the convective derivative leads to the
general lawbold g = – v^4 / (MG) bold e_r.
I am not totally sure about this becaus eq.(2) in note 360(5) describes
directions of v which are perpendicular tobold r = (X, Y).
Such a velocity direction is only valid for circular orbits. In elliptic
orbits bold r is not perpendicular to bold v.Horst