Many thanks, this solves the problem, and I attach a corrected and final version of Note 369(4), now fully checked by computer algebra. The spin connection remains the same. I agree that the nine elements of the spin connection cannot be found a priori, but one can use the fact that the matrx must be antisymmetric (a three dimensional rotation generator). What we can do is to compute the nutations and precessions, to see if the cyclic patterns have any resemblence to the Milankowich theory. I agree that we are pioneering an entirely new method. I have not studied the Milankowich theory very much, but know of its existence from googling it up.
In a message dated 02/02/2017 11:26:13 GMT Standard Time, writes:
I checked the definition of acceleration from VAPS. In note 369(4), eq. (24), a factor of sin(theta) was missing for the first term at rhs. Now all works well. The general solution for Omega from eqs.(25-27) including the antisymmetry conditions is undertedermined, there is one free parameter %r1, as the calculation shows. If this parameter is set appropriately, the solution (29) comes out. This is probably the most simple solution.
I did not know the theory that climate changes are caused by the Milankowich cycles. These could have been calculated 30 years ago since numerical methods for solving diff. equations are available for what you and me are pioneers.
Am 02.02.2017 um 09:48 schrieb EMyrone:
I have checked a sub phi (attached Note 369(4)) a few times and it is OK. To make sure, the last four lines of this note can be run through Maxima. In Note 369(5) a new general law is derived for the Cartan covariant derivative in any coordinate system. I am very interested in the ability of Maxima to solve simultaneous differential equations and also in the resulting graphics. Various external torques or forces can be applied to the gyroscope. Your method seems OK, and the simple method of UFT368 produces the Laithwaite result. Laithwaite was subjected to a grave miscarriage of justice. As you know, the earth is a symmetric top and its gyroscopic motions are important in climatology. So the new code can be applied to predict Milankowich cycles for example. We could apply various gravitational forces of the sun , moon and planets. This would be very interesting for UFT369. Having derived the spin connection in spherical polars, the Cartan torsion and curvature can be calculated, so the gravitational fields can be calculated. So this method and previous methods (UFT119) can be tied together. The big advance made in UFT368 is that the nutations and precessions can all be graphed. These are the causes of the climate changes of the Milankowich cycles. It will be interesting to try to see whether the climate warms or cools due to the nutations and precessions of the earth (a giant gyroscope). Your graphics are as usual of immediate interest. The canonical or general method of dealing with torques is given by the Euler equations in frame (1, 2, 3) of the principal moments of inertia. These can be transformed to frame (X, Y, Z) using a transformation matrix made up of the three rotation generator matrices defined by Euler. This is horrendously complicated but complication is no longer any problem. I was delighted as a young post graduate when I found that no matter how complicated my equations, the computer could deal with them (see early Omnia Opera papers). The Elliott 4130 at that time had 48 kilobytes of memory for the entire College and used cards and paper tape. Every time I used it I had to drive about three miles up and down the hill to the main campus. The Euler Langevin equations for example are appalling to solve by hand, but the computer could deal with them and compare them with far infra red data. In fact, they could be solved with your new code.