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 pathalong 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.

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).