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


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.

Author Biographies

Viktor P. Legeza, Igor Sikorsky Kyiv Polytechnic Institute

Віктор Петрович Легеза

Oleksiy V. Atamaniuk, Igor Sikorsky Kyiv Polytechnic Institute

Олексій Віталійович Атаманюк


S.O. Gladkov and S.B. Bogdanova, “Analytical and numerical solution of the problem on brachistochrones in some general cases”, in Itogi Nauki i Tehniki. Sovremennaja Matematika i Ee Prilozhenija. Tematicheskie Obzory, vol. 145. Moscow, SU: VINITI, 2018, pp. 114–122.

A.V. Zarodnyuk and O.Yu. Cherkasov, “Qualitative analysis of optimal trajectories of the point mass motion in a resisting medium and the brachistochrone problem”, J. Comput. Syst. Sci. Int., vol. 54, no. 1, pp. 39–47, 2015. doi: 10.1134/S106423071501013X

A.V. Zarodnyuk and O.Yu. Cherkasov, “A qualitative analysis of the brachistochrone problem with dry friction and maximizing the horizontal range”, Moscow Univ. Mech. Bull., vol. 71, no. 4, pp. 93–97, 2016. doi: 10.3103/S002713301604004X

A.V. Zarodnyuk and O.Yu. Cherkasov, “On the maximization of the horizontal range and the brachistochrone with an accelerating force and viscous friction”, J. Comput. Syst. Sci. Int., vol. 56, no. 4, pp. 553–560, 2017. doi: 10.1134/S1064230717040177

A.V. Zarodnyuk and O.Yu. Cherkasov, “Support reaction in the brachistochrone problem in a resistant medium”, in Dyna­mical Systems in Applications, Springer Proceedings in Mathematics & Statistics, Łódź, Poland, Dec. 11–14, 2017, vol. 249, pp. 451–460. doi: 10.1007/978-3-319-96601-4_40

A.S. Vondrukhov and Yu.F. Golubev, “Brachistochrone with an accelerating force”, J. Comput. Syst. Sci. Int., vol. 53, no. 6, pp. 824–838, 2014, doi: 10.1134/S1064230714060124

A.S. Vondrukhov and Yu.F. Golubev, “Optimal trajectories in the brachistochrone problem with an accelerating force”, J. Comput. Syst. Sci. Int., vol. 54, no. 4, pp. 514–524, 2015. doi: 10.1134/S1064230715040139

A.S. Vondrukhov and Y.F. Golubev, “Optimal trajectories in brachistochrone problem with Coulomb friction”, J. Comput. Syst. Sci. Int., vol. 55, no. 3, pp. 341–348, 2016. doi: 10.1134/S1064230716030163

A.S. Sumbatov, “The problem on a brachistochrone (classification of generalizations and some recent results)”, Proc. MIPT. Ser. Mechanics, vol. 9, no. 3, pp. 66–75, 2017.

R.T. Boute, “The brachistochrone problem solved geometrically: A very elementary approach”, Mathematics Magazine, vol. 85, no. 3, pp. 193–199, 2012. 10.4169/math.mag.85.3.193

H.W. Broer, “Bernoulli’s light ray solution of the brachistochrone problem through Hamilton's eyes”, Int. J. Bifurcation and Chaos, vol. 24, no. 8, 2014. doi: 10.1142/S0218127414400094

O. Jeremic et al., “On the brachistochrone of a variable mass particle in general force fields”, Math. Comp. Model., vol. 54, pp. 2900–2912, 2011. doi: 10.1016/j.mcm.2011.07.011

V.P. Legeza, “Cycloidal pendulum with a rolling cylinder”, Mechanics of Solids, vol. 47, no. 4, pp. 380–384, 2012. doi: 10.3103/S0025654412040024

V.P. Legeza “Efficiency of a vibroprotection system with an isochronous roller damper”, Mechanics of Solids, vol. 48, no. 2, pp. 168–177, 2013. doi: 10.3103/S0025654413010088

M. Levi, Classical Mechanics with Calculus of Variations and Optimal Control. AMS, Providence: Pennsylvania State Univer­sity, 2014, 299 p.

Y. Nishiyama, “The brachistochronic curve: The problem of quickest descent”, Int. J. Pure Appl. Math., vol. 82, no. 3, pp. 409–419, 2013.

A. Obradovic et al., “The brachistochronic motion of a vertical disk rolling on a horizontal plane without slip”, Theor. Appl. Mech., vol. 44, no. 2, pp. 237–254, 2017. doi: 10.2298/tam171002015o

R. Radulovic et al., “Analysis the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint”, FME Trans., vol. 42, no. 4, pp. 290–296, 2014. doi: 10.5937/fmet1404290r

R. Radulovic et al., “The brachistochronic motion of a wheeled vehicle”, Nonlinear Dynamics, vol. 87, no. 1, pp. 191–205, 2017. doi: 10.1007/s11071-016-3035-3

S. Salinic et al., “Brachistochrone with limited reaction of constraint in an arbitrary force field”, Nonlinear Dynamics, vol. 69, no. 1, pp. 211–222, 2012. doi: 10.1007/s11071-011-0258-1

A.S. Sumbatov, “Brachistochrone with Coulomb friction as the solution of an isoperimetrical variational problem”, Int. J. Non-Linear Mech., vol. 88, pp. 135–141, 2017. doi: 10.1016/j.ijnonlinmec.2016.11.002.