January 17, 2017
These are described in the attached note, the motion is the same as the spherical orbit motion graphed in UFT270, with some additional features.The equations of motion (15) to (17) can be integrated to give the trajectories beta(r), phi(r) and theta(r). The gyroscope is characterized by the relation between beta and phi in Eq. (20) and between theta and phi in Eq. (22). The lagrangian method is used as in UFT270 and its notes. As mentioned by Michael Jackson recently, this is the first complete solution of the classical gyroscope problem in about four hundred years. The same is true for the solution of spherical orbit motion in UFT270. Some of the very interesting graphics in UFT270 could be extended and developed for UFT367 and UFT368. The next stage is to introduce a lifting force as in the Braithwaite and Shipov experiments. Much of this theory became the basis for quantum mechanics, there is a photograph on the net of Heisenberg and Pauli looking intently at a simple gyroscope (spinning top). I think that they were both students of Sommerfeld.”
The highlighted statement is of course rubbish as the gyroscope was not invented until some 200 years ago; far too many people (including those who should know better) confuse the gyroscope with the ‘spinning-top in a frame’. Ancient peoples played with peg-tops and whipping-tops, but those cannot be placed on a little tower (friction makes them jump off). It is a sad commentary on so-called homo sapiens that it took millennia to think of making that small modification. As for rotational dynamics in general, everything needed was written down by Euler in the 18th century. But his equations are rather like Einstein’s gravitational equation: easy to state, but hard to solve (apart from a few simplified situations). Laithwaite claimed that, because of this, ‘antigravitational’ properties lay hidden within Euler’s equations. But there cannot be any, because the equations are predicated on Newton’s third law. So then the nutters have to attack the third law … and then they have to attack the first law … of thermodynamics.