Geodesic
Qualitative analysis of the brachistochrone problem with a dry friction and maximization of horizontal distance
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2016), pp. 54-59

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The range maximization problem of a particle moving in a vertical plane under the action of gravity and dry friction and the corresponding brachistochrone problem are considered. The optimal control problem is reduced to a boundary value problem for a system of two nonlinear differential equations. A qualitative anaslysis of the trajectories of this system is carried out, their typical features are found and illustrated by numerical solving of the boundary value problem. It is shown that the normal component of the support reaction should be positive when moving along the optimal curve. The optimality of the found extremal trajectories is discussed. 
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     author = {A. V. Zarodnyuk and O. Yu. Cherkasov},
     title = {Qualitative analysis of the brachistochrone problem with a dry friction and maximization of horizontal distance},
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     pages = {54--59},
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A. V. Zarodnyuk; O. Yu. Cherkasov. Qualitative analysis of the brachistochrone problem with a dry friction and maximization of horizontal distance. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2016), pp. 54-59. http://geodesic.mathdoc.fr/item/VMUMM_2016_4_a8/