# BRAHISTOCHRONOUS MOTION OF THE MATERIAL POINT ON AN INCLINED PLANE IN A UNIFORM GRAVITATIONAL FIELD

## DOI:

https://doi.org/10.20535/kpi-sn.2019.2.167495## Keywords:

Variational problem, Brachistochrone, Cycloid, Euler equations, Time functional, Transition time## Abstract

**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.

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