Checking Note 370(5)

Thanks again! I think that you used phi instead of phi dot in the protocol. The lagrangian (6) does not contain terms in phi. I think that the linear increase of phi and theta for a symmetric top is the right result because this note simply uses the spherical polar representation of the rotational kinetic energy of a freely rotating asymmetric top. I can see that the your solution produces oscillations in general.

Sent: 12/02/2017 19:09:42 GMT Standard Time
Subj: Re: 370(5): Freely Rotating Asymmetric Top in Terms of the Sphercal Polar Angles

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?


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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