Paul Bunyan's brachistochrone and tautochrone
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In this paper we concern ourselves with modified versions of the traditional brachistochrone and tautochrone problems. In the modified version of each problem the constant gravity model is replaced with an attractive inverse square law, consequently we name these the 1/r2 brachistochrone and 1/r2 tautochrone problems. With regard to the 1/r2 brachistochrone problem, we show that the shape of the minimizing curve is formally constructed from an infinite series of elliptic integrals, and we use a numerical optimal control technique to generate the trajectories. The 1/r2 tautochrone problem is solved using fractional calculus techniques and we show that the solution satisfies Lagrange's rule for tautochronous curves.
