362(1): Spin Connection of the Plane Polar System

This note shows that the Cartan spin connection of the plane polar system is Eq. (6), and that this is a special case of the more general spin connection (5) given by the convective derivative calculated in the plane polar system in which v is a velocity field v(t, r(t), theta(t)). In the usual dynamics v is a function only of t. Most generally, the orbit is described by the general spin connection, and use of the Cartan derivative gives the ECE2 gravitational field equations which therefore give the most general orbit, and which have just summarized in chapter three of “ECE2: The Second Paradigm Shift”. These are also the field equations of electrodynamics and fluid dynamics. In the classical approach an orbit is a function superimposed on a coordinate system. The Cartan equations are relations between components, and this leads to general relativity, in which the orbit is defined by the general spin connection. In a sense the plane polar coordinate system is an example of general relativity, because the spin connection is not zero and the axes move. When the static Cartesian axes are used there is no spin connection present, but that means that the Coriolis forces are not defined.


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