DOI: https://doi.org/10.20535/kpi-sn.2019.3.175735

### THE BRACHISTOCHRONIC MOVEMENT OF A MATERIAL POINT IN THE HORIZONTAL VECTOR FIELD OF A MOBILE FLUID

Viktor P. Legeza, Oleksiy V. Atamaniuk

#### Abstract

Background. Since the brachistochronic motion of a material point in a flat vector field of a mobile fluid was not previously considered, the formulated variational problem of searching for extremal trajectories in such a formulation is new and relevant.

Objective. The aim of the study is to obtain the algebraic equations of extremal trajectories of motion, along which the material point moves from a given starting point to a given finish point in the shortest possible time.

Methods. The solution of the problem was carried out using classical methods of the calculus of variations (to obtain a differential equation for the motion of a material point), as well as using Taylor series (for approximate integration of the resulting differential equation). For a given variant of the boundary conditions, approximate algebraic equations of extremals of the motion of a material point were established in the form of segments of power series. A comparative analysis of the time of movement was carried out both along extreme trajectories and along an alternative shortest path – along a straight line, which connects two given points of start and finish.

Results. It is shown that the considered variational problem has two different solutions, which differ only in sign. At the same time, only one solution provides the minimum time for the movement of a material point between the given start and finish points. Studies have also found that the extremal trajectory of the brachistochronic movement of a point is not straight and has an oscillatory character.

Conclusions. The proposed approach allows plotting in advance such a logistical route of a material point (motorboat) in a flat vector field of a mobile fluid between the given start and finish points, which ensures the minimum travel time between them. In this case, the extremal trajectory will not necessarily be the shortest line that connects the start and finish points.

#### Keywords

Variational problem; Brachistochronic motion; Vector field of a mobile fluid; Time functional; Euler equation; Boundary conditions; Taylor series; Extremal trajectory

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