Modern explorations regarding the search for a curve subject to a minimization principle and its inverse, namely the search for the minimization object based on a given curve, are made. By using the Udwadia–Kalaba theory, which subsumes the Coulomb friction force as a non-ideal constraint, we obtain the analytic expression of the equation of motion for a particle moving along the Brachistochrone cycloid curve under Coulomb friction. Then we perform the inverse problem for the moving particle. That is, while the Brachistochrone cycloid curve is given, we seek the corresponding minimization object. Both the situations with and without friction are addressed. Finally, we return to the search for a curve subject to a minimization principle to complete the loop. However, this time we presume the minimization object is the total travel time, which was addressed in the classical Brachistochrone problem (hence, frictionless), while recognizing the presence of Coulomb friction. All three analyses come to meet at the special case when there is no friction. Our research reveals profound insights that have not been reported previously. The loop analysis also suggests a new angle for the study of dynamic systems.
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