Brachistochrone for a Rigid Body Sliding down a Curveстатья
Информация о цитировании статьи получена из
Web of Science,
Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 26 сентября 2016 г.
Аннотация:Extremality conditions in the brachistochrone problem for a perfectly rigid body sliding without friction down an unknown (to be determined) curve in a vertical plane are found. In this case, the body has to track the tangent to the trajectory. According to the principle of constraint release, the reaction of the support creates a torque used as control. For dimensionless Appell’s equations with a single similarity coefficient, the standard problem of the fastest descent from a given initial point to a given terminal point assuming that the initial velocity is zero is formulated. The Okhotsimskii–Pontryagin method is used to analyze the differential of the objective function. Necessary optimality conditions are found, and a formula for the optimal control that does not involve adjoint variables is derived from them. Properties of the optimal trajectories are investigated analytically both in the general case and for the limiting (zero and infinite) values of the similarity coefficient. It is found that cycloid-shaped brachistochrones occur as the similarity coefficient tends to infinity. For some values of the similarity coefficient, numerical results are presented that demonstrate the shape of the corresponding brachistochrones and the optimal time of motion. The results are compared with those obtained by solving the classical brachistochrone problem.
DOI: 10.1134/S1064230713040084