Discussion of Note 362(1)

Many thanks. Cartan’s differential geometry is very powerful and more general than the Lagrange derivative (1778 – 1783). This is a very interesting problem because the Lagrange or convectirve or material derivative reduces to the Coriolis accelerations in Eq. (3) if and only if the plane polar system is used, and if and only if v = v(t). Otherwise new accelerations appear as in UFT361. These are represented in the spin connection matrix (5). The plane polar coordinate system reduces this to Eq. (6), which is the spin connection for Eq. (3), the usual textbook result for the Coriolis accelerations of 1835. The latter appear in a rotating frame of reference as you know. The rotation of the frame is presented in the spin connection matrix by the second matrix, an antisymmetric two by two rotation generator matrix containing the angular velocity of the frame on the off diagonals. There is a very interesting contradiction in the usual textbook approach because the plane polar coordinates imply partial r / partial theta = 0. This is true if and only if the orbit is a circle. If the orbit is an ellipse or conic section or a precessing ellipse then partial r / partial theta is not zero as you know. So the next note will develop this problem in the elliptical polar coordinate system. I intend to derive the Lagrange derivative and the Coriolis accelerations in the elliptical polar coordinate system. This procedure may well result in new orbital forces which can in principle be observable in astronomy. It is already known that the Cartan covariant derivative in general leads to the ECE2 field equations, which are true for all coordinate systems. So to calculate the most general orbit the ECE2 field equations must be solved. Essentially, UFT361 and 362 are simplifications of that procedure.

To: Emyrone@aol.com
Sent: 15/11/2016 10:25:22 GMT Standard Time
Subj: note 361(2)

It seems that eq.(2) is identical to eq.(1) if we develop the total
derivative in the usual way:

d bold v / dt = partial bold v / partial t + partial bold v
/ partial X * partial X/ partial t + …

This gives exactly eqs. (3) and (4). Obviously the Cartan spin
connection (5) here is not the most general one because the first lower
index is always zero. That means, Cartan geometry is more general than
ordinary differential calculus represented by (2).

Horst

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