Viktor P. Legeza, Svitlana G. Savchuk


Background. The variational problem, which is posed and solved in this work, is a natural generalization of the classical problem of I. Bernoulli about the search for brachistochrones in a vertical plane. In the proposed formulation, it is new and relevant from a practical point of view in such areas as engineering, transport and logistics, sports events, etc.

Objective. The aim of the paper is to find such a curve on an inclined plane, moving along which, without an initial velocity in a uniform gravitational field, from one given point of the plane to another, the material point will make such a transition in the shortest time.

Methods. To achieve this goal, the classical methods of the calculus of variations were used, namely, the Euler equation.

Results. A time functional is constructed, using which the differential equation of the spatial brachistochrone, which lies on an inclined plane, is analytically derived. After its integration in a closed form, an algebraic brachistochrone equation is obtained. The results of the study are illustrated graphically. At the starting point M of the brachistochrone, the direction of the initial velocity of the material point is established. A comparative analysis of the transition time for the optimal brachistochrone curve and two alternative paths of motion of the material point is carried out.

Conclusions. It is proved that the projection of the brachistochrone on the plane  is not a cycloid. It is shown that the vector of the initial velocity of the material point at the starting point M of the brachistochrone is perpendicular to the x-axis. It was established that the minimum time of transition depends on the parameter a of the inclined plane, the energy dissipation coefficient k, and also on the coordinates of the starting M and finishing N points through which the brachistochrone passes.


Variational problem; Brachistochrone; Cycloid; Euler equations; Time functional; Transition time


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