Many thanks for this checking work. I rechecked my hand calculation for the phi component as attached and it is correct. It is the same as Eq. (7) of Problem 21-25 of “Vector Analysis Problem Solver”. VAPS has a notation in which phi and theta are interchanged. These checks give complete international confidence in our work. The spin connection is a three dimensional rotation matrix, which describes the rotation of the axes in the spherical polar system. There are three dimensional centrifugal and Coriolis accelerations and so on. Knowing the spin connection, the field equations can be worked out from the Cartan torsion and Cartan curvature of the spherical polar coordinate system. I will also give the convective derivative in spherical polars in one of the next notes. The spin connection matrix is probably related to the product of rotation generators of theta and phi when these are defined as Euler angles.
In a message dated 31/01/2017 20:55:10 GMT Standard Time, writes:
Checking eq.(28) by Maxima, the third component gives a different result. I computed
a – partial v / partial t – Omega*v
which should give zero. Then I tried to solve the linear equation system for Omega. There are 6 equations from antisymmetry and 3 eqautions from eq.(20). The system is not solveable. Is the acceleration a defined correctly?)
Am 31.01.2017 um 15:37 schrieb EMyrone:
This is given by the 3 x 3 matrix in Eq. (28), it is an antisymmetric rotation matrix in three dimensions. It can be used to describe a three dimensional orbit, and in general can be used for any problem in dynamics. This proves for the first time that the spherical polar coordinate system is a special case of Cartan geometry. Note that theta and phi are the same as the theta and phi Euler angles. Any curvilinear coordinate system is a special case of Cartan geometry.