Too Complex for Ron

375(1): Equivalence of Cartesian and Plane Polar Force Laws

April 8, 2017

This is a note on the equivalence of the Cartesian force laws (4) and (5) and the plane polar force laws (15) and (16). This means that numerical integration in Cartesian coordinates can be applied to several problems, as in Horst’s section 3 of UFT374 just sent over.”

It is well known that complex numbers can very conveniently model two-dimensional rotation. One keeps the Cartesian framework, and simply makes the real and imaginary components time-dependent. The centrifugal and Coriolis fictitious forces then fall out immediately, and this method also has the advantage of showing how intimately they are related. This is rarely stressed in elementary textbooks.  Looking more closely, one finds that there is a myriad of possible fictitious forces. Only the first two have generally agreed names (centrifugal, Coriolis). The ‘next one up’ is variously called ‘jerk’, ‘transverse’, ‘Euler’, etc., and explains why fairground rides have to be carefully designed: even if curved and straight lengths of track are made to meet at exactly the same height and angle, there would still be a bone-jarring jolt at the join if the ride-designer were as ignorant of basic physics as is Ron. Is that relevant here? Yes, there is a jerk term in the Kepler orbit. Now that Ron knows about it, can he ‘twist torsion’ enough to model it?


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