Many thanks for the computation, the figures do not appear, but I am sure they are most interesting. This method is similar to 369(1). The coordinate r is the coordinate of the point of the gyro in the lab frame (X, Y, Z), defined by Eq. (1) as usual. The translational kinetic energy of the gyro is (1/2) m r dot dot r dot. The potential energy from the general definition (3) is given by Eq. (6). It is a lagrange variable because the relevant Euler Lagrange (15) equation gives the self consistent result (19). Similarly r is a Lagrange variable in orbital theory. It is always possible to make the construction r = h sub 1 cos theta where h sub 1 is a constant. This construction converts the tautology (19) into Eq. (20), introducing an interaction between translation and rotation. If this construction is not used, the translational and rotational motions are independent.
Sent: 10/02/2017 00:01:14 GMT Standard Time
Subj: Re: 370(2): Gyro Subjected to the General External Torque
I do not fully understand the meaning of the coordinate bold r. It seems to represent a time-dependent external force. then r, r dot and r dotdot are predefined, and r is not a Lagrange coordinate.
It seems that L_phi and L_psi are no constants of motion any more, because additional angular terms are introduced into the rigid body equations for phi and psi.
Why do you introduce h_1 to which bold r is a prejection? Is the external force working in parallel to h_1?
I have preliminary calculated the Lagrange equations of the angular variables in the attachment, pp. 1-3.
Am 09.02.2017 um 12:27 schrieb EMyrone:
This note deals with a force applied directly upwards to the point of the gyro, as described in Fig. (1). Subject to checking by computer algebra the lagrangian analysis gives the four simultaneous equations (20) to (23), in the lagrangian variables r, theta, phi, and chi. The interesting result is found that the applied force lifts the gyro and also changes its nutations and precessions. The applied force can be generated by an applied torque as in the Laithwaite and Shipov experiments. The dynamics are far too complicated to solve by hand but can be solved by computer to give a large amount of information. Eq. (10) is a simple transformation from frame (1, 2, 3) to frame (X, Y, Z).