Brachistochrone with friction

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Abstract
The brachistochrone problem is solved under the action of dry and viscous friction. For this purpose, this problem is represented as a problem of choosing a time optimal normal component (control) of the reaction of the support curve, whose shape is to be found. The Okhotsimsky-Pontryagin method of investigation of the functional differential is applied. Necessary optimality conditions that give the solution of the classical brachistochrone problem without friction and provide the corresponding optimal curves with friction are found. Parametric formulas for brachistochrones under the action of dry and viscous frictions are found analytically. Their properties are investigated. Attainability domains are determined. For certain values of the friction coefficients, the results of calculations are presented, which show the shape of the found brachistochrones and the optimal motion time.

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