Rotating planets in Newtonian gravity

Variational techniques have been used in applications of hydrodynamics in special cases but an action that is general enough to deal with both potential flows and solid-body flows, such as cylindrical Couette flow and rotating planets, has been proposed only recently. This paper is one of a series that aims to test and develop the new Action Principle. We study a model of rotating planets, a compressible fluid in a stationary state of motion, under the influence of a fixed or mutual gravitational field. The main problem is to account for the shape and the velocity fields, given the size of the equatorial bulges, the angular velocity at equator and the density profiles. The theory is applied to the principal objects in the solar system, from Earth and Mars to Saturn with fine details of its hexagonal flow and to Haumea with its odd shape. With only 2 parameters the model gives a fair fit to the shapes and the angular velocity field near the surface. Planetary rings are an unforeseen, but a natural and inevitable feature of the dynamics; no cataclysmic event need be invoked to justify them. The simple solutions that have been studied so far are most suitable for the hard planets, and for them the predicted density profiles are reasonable. The effect of precession was not taken into account, nor were entropic forces, so far. There has not yet been a systematic search for truly realistic solutions. The intention is to test the versatility of the action principle; the indications are are very encouraging.