Discussion of Note 370(5)

The three angular velocities in frame (1, 2, 3) (or any frame) can be represented either by Euler angles or by spherical polar coordinates. The theta and phi are the same. There is a relation between chi and the theta and phi of the spherical polar system which I can work out in the next note.

To: EMyrone@aol.com
Sent: 12/02/2017 19:41:23 GMT Standard Time
Subj: PS: Re: 370(5): Freely Rotating Asymmetric Top in Terms of the Sphercal Polar Angles

PS: These are the equations for theta and phi of note 370(2) with all psi terms omitted. psi couples the rotation of the body with the theta-phi motion. Omitting the coupling should give different results.

Am 12.02.2017 um 20:09 schrieb Horst Eckardt:

How did you obtain eq.(9)? The Lagrange equation for phi (eq. 7) gives eq. o75 in my protocol. This is different from eq.(14). Eq. (17) is the same as in my calculation (o74). Please note that in the case I_1=I_2=I_3 (all moments of inertia equal) the rhight-hand sides of all equations are zero, i.e. we have a free motion with linear increase of theta and phi, see second solution in the protocol. Is this plausible?

Horst

Am 11.02.2017 um 14:15 schrieb EMyrone:

By use of the spin connection matrix in Eq. (1) it is shown that the free rotation of an asymmetric top can be expressed by the lagrangian (6) and two Lagrange variables theta and phi of the spherical polar coordinate system. So the motion is completely defined by simultaneous solution of Eqs. (14) and (17) with Maxima. This is a great simplification and advance over the use of the Euler angles. This motion is also the motion of a freely rotating gyroscope. Now we are ready to consider an additional force of any kind applied to the gyroscope, for example a gravitational force. This lagrangian can also be quantized using the Schroedinger equation to give the far infra red rotational spectrum of a freely rotating asymmetric top molecule. For free rotation the lagrangian is the same as the hamiltonian. In both types of problem the dynamics and quantized dynamics are governed by the spin connection of Cartan geometry, and become aspects of ECE theory. The trajectories theta(t) and phi(t) describe nuntations and precessions of the earth which can be related to the Milankovitch cycles.


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