# CURVE OF DESCENT OF A MATERIAL POINT IN THE SHORTEST TIME ON A TRANSCENDENTAL SURFACE IN A UNIFORM VERTICAL GRAVITATIONAL FIELD

## DOI:

https://doi.org/10.20535/kpi-sn.2019.5-6.188298## Keywords:

Variational problem, Brachystochronous motion, Transcendental surface, Time functional, Cycloid, Euler–Lagrange equation, Response time## Abstract

**Background.** The article deals with the original variational problem of the brachystochronous motion of a material point on a cycloidal surface between two given points in a vertically homogeneous gravitational field. The novelty and relevance of the work is explained by the choice of the transcendental surface, since earlier the motion of a material point was considered on algebraic surfaces of the second order.

**Objective.** Find a curve on the transcendental surface, moving from one set point (starting point) to another set point (finish point) on this surface without friction a material point will make such a transition in a minimum time. The transcendental surface has a guide curve of the cycloid lying in one of the coordinate planes, and its generatrix are perpendicular to that plane.

**Methods.** To achieve this goal, we used the classical methods of variational calculus (Euler–Lagrange equation), as well as the classical method of integrating ordinary differential equations in a closed form (Bernoulli method).

**Results.** A time functional was constructed, using which the differential equations of the spatial brachystrochron, which lies on the transcendental surface, are analytically deduced. After integration in a closed form, algebraic equations of the spatial brachystrochron in parametric form are obtained. The results of the study are illustrated graphically: the projections of the trajectory of the brachystrochron on the coordinate planes *OXY*** **and *OXZ*. The slope angles of the optimal trajectory at the start point are determined. A comparative analysis of the time of action in the process of motion of a material point along two trajectories is carried out: along the obtained brachystrochron and along the alternative trajectory.

**Conclusions.**The proposed approach allows to pre-plot such a logistic route of a material point on a given transcendental surface between two fixed points, which will provide a minimum travel time between them in a uniform vertical gravity field. In this case, an extreme trajectory will not necessarily be the shortest line on the surface that connects the two predetermined points (start and finish).

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