This would be very interesting. The purpose of this note is to define precisely the torque associated with a potential energy of mgh cos theta. Having understood this, the effect of the Shipov torque can be calculated. I agree about the generalized Lagrange method, one can see it at work in a book such as Ryder, “Quantum Field Theory”, and in our UFT paper on the Lagrangian method of deriving the Lorentz force equation. The great advantage we have now is the ability of Maxima to solve very complicated simultaneous differential equations. So we can experiment with many situations in dynamics.
Sent: 15/02/2017 14:13:11 GMT Standard Time
Subj: Re: 370(7): Motion of the Gyroscope, General Theory
What we need for the Shipov experiment is a torque around the Z axis, i.e. the force F of
bold Tq = bold r x bold F
has a component only in the phi_1 direction of the lab frame. I looked up the Lagrange method, there are “generalized forces” that can be used instead of the potentials (for torques we would need a vector potential which is not integrated into the Lagrange method). The result will be that the equation for phi_1 has an additive constant in the simplest case. This gives a modification of nutation/precession but no coupling with the r component.
Something like a Lorentz force, however, would change the situation.
Am 15.02.2017 um 13:39 schrieb EMyrone:
This general theory is developed in terms of Euler angles and spherical polar coordinates and can be used for any situation. It is exemplified by the motion of a gyro with one point fixed. In general the complete information about a gyroscope should be obtained by using both the Euler angles and the spherical polar coordinate system, related by Eq. (8). The latter can be used as a check on the correctness of the numerical solution. The torque and force calculations are exemplified by working out the lab frame torque and force which give the potential energy U = mghcos theta of the gyroscope with one point fixed. In the spherical polar system the torque is Eq. (29), adn in the Cartesian system the lab frame torque is Eq. (34). The lab frame force is the force of gravitation on the centre of mass of the gyroscope, Eq. (36), in the -k direction. The torque balance equation is Eq. (38) which shows that the torque due to gravitation is balanced by the torques of the Euler equations. This is the reason why the gyro does not fall over. We are now ready to evaluate the Laithwaite experiment and Shipov experiment by adding a torque in the lab frame. Any kind of torque can be used in Eq. (38).