368(6): Final Version of Note 368(5)

This is the final version for checking with computer algebra. This is the motion of a symmetric top with one point fixed, first worked out by Lagrange, who was a student of Euler. In general the motion must be found by solving Eqs. (5), (6) and (7) for theta(t), phi(t) and psi(t), the time dependent Euler angles. This is a difficult problem even for contemporary supercomputers, so Lagrange worked it out by approximation in “Mecanique Analytique” (1811 and 1815). Eq. (5) must be solved for time dependent theta(t). There is no way in which the gyroscope can lift itself off the ground in this type of problem, because its point is defined as the origin of both (X, Y, Z) and (1, 2. 3). In order to consider the possibility of lifting off the ground, the more general Eqs. (11) and (12) must be used, in which the centre of mass of the gyroscope is allowed to move. The first steps towards this end were made in UFT367 and will be developed further in the next note. It may be possible to get some insight into Eqs. (5) to (7) with Maxima, as suggested in the note.


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