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.

a370thpapernotes5.pdf

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