370(1): Orbital Motion of the Asymmetric Top

This note makes some remarks on UFT369 and sets up the usual lagrangian (2) used in a textbook such as Marion and Thornton. This lagrangian assumes that the translational and rotational kinetic energies are independent, so the problem reduces to solving independently the set of equations (8) to (10) and the set of equations (12) to (15). This means that the nutations and precessions of the symmetric top do not depend on its orbital position, e.g. they are the same at aphelion and perihelion. This theory is too simple to describe the Milankowitch cycles, so a more general five variable theory is needed. One way of developing this is to assume that the potential energy is in general a function of all five variables, and is no longer central, i.e. is no longer a function only of r. The mathematics of this note are examples of Cartan geometry, and insight which allows a great deal of development in future. Euler was the first to devise the idea of principal moments of inertia in 1750, and the idea is used in far infra red spectroscopy – rotational spectra of molecules.


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