CURVE OF DESCENT OF A MATERIAL POINT IN THE SHORTEST TIME ON A TRANSCENDENTAL SURFACE IN A UNIFORM VERTICAL GRAVITATIONAL FIELD
Keywords:Variational problem, Brachystochronous motion, Transcendental surface, Time functional, Cycloid, Euler–Lagrange equation, Response time
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).
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, “Brachistochrone for a rolling cylinder”, Mechanics of Solids, vol. 45, no. 1, pp. 27–33, 2010. doi: 10.3103/s002565441001005x
V.P. Legeza and S.G. Savchuk, “Brachistochronous motion of the material point on an inclined plane a uniform gravitational field”, KPI Science News, no. 2, pp. 15–23, 2019. doi: 10.20535/kpi-sn.2019.2.167495
V.P. Legeza and O.V. Atamaniuk, “The Brachistochronic movement of a material point in the horizontal vector field of a mobile fluid”, KPI Science News, no. 3, pp. 44–51, 2019, doi: 10.20535/kpi-sn.2019.3.175735
W. Dunham, Journey Through Genius. New York: Penguin Books, 1991, 304 p.
L.P. Eltsgolts, Differential Equations and Variational Calculus. Moscow, SU: Nauka, 1974, 432 p.
H. Erlichson, “Johann Bernoulli’s brachistochrone solution using fermat’s principle of least time”, Eur. J. Phys., vol. 20, no. 5, pp. 299–304, 1999. doi: 10.1088/0143-0807/20/5/301
N. Ashby et al., “Brachistochrone with Coulomb friction”, Am. J. Phys., vol. 43, no. 10, pp. 902–906, 1975. doi: 10.1119/1.9976
A.M.A. van der Heijden and J.D. Diepstraten, “On the brachistochrone with dry friction”, Int. J. Non-Linear Mechanics, vol. 10, no. 2, pp. 97–112, 1975. doi: 10.1016/0020-7462(75)90017-7
S. Lipp, “Brachistochrone with Coulomb friction”, SIAM J. Control Optim., vol. 35, no. 2, pp. 562–584, 1997. doi: 10.1137/S0363012995287957
V. Covic and M. Veskovic, “Brachistochrone on a surface with Coulomb friction”, Int. J. Non-Linear Mechanics, vol. 43, no. 5, pp. 437–450, 2008. doi: 10.1016/j.ijnonlinmec.2008.02.004
J.C. Hayen, “Brachistochrone with Coulomb friction”, Int. J. Non-Linear Mechanics, vol. 40, no. 8, pp. 1057–1075, 2005. doi: 10.1016/j.ijnonlinmec.2005.02.004
B. Vratanar and M. Saje, “On analytical solution of the brachistochrone problem in a non-conservative field”, Int. J. NonLinear Mechanics, vol. 33, no. 3, pp. 489–505, 1998. doi: 10.1016/S0020-7462(97)00026-7
H.A. Yamani and A.A. Mulhem, “A cylindrical variation on the brachistochrone problem”, Am. J. Phys., vol. 56, no. 5, pp. 467–469, 1988. doi: 10.1119/1.15755
D. Palmieri, “The brachistochrone problem, a new twist to an old problem”, Undergraduate Honors Thesis, Millersville University of PA, 1996.
P.K. Aravind, “Simplified approach to brachistochrone problem”, Am. J. Phys., vol. 49, no. 9, pp. 884–886, 1981. doi: 10.1119/1.12389
H.H. Denman, “Remarks on brachistochrone-tautochrone problem”, Am. J. Phys., vol. 53, no. 3, pp. 224–227, 1985. doi: 10.1119/1.14125
G. Venezian, “Terrestrial brachistochrone”, Am. J. Phys., vol. 34, no. 8, p. 701, 1966. doi: 10.1119/1.1973207
A.S. Parnovsky, “Some generalisations of the brachistochrone problem”, Acta Physica Polonica, Suppl. A 93, pp. 5–55, 1998.
G. Tee, “Isochrones and brachistochrones”, Neural, Parallel Sci. Comput., vol. 7, pp. 311–342, 1999.
H.F. Goldstein and C.M. Bender, “Relativistic brachistochrone”, J. Math. Phys., vol. 27, no. 2, pp. 507–511, 1986.
G.M. Scarpello and D. Ritelli, “Relativistic brachistochrone under electric or gravitational uniform field”, Z. Angew. Math. Mech., vol. 86, no. 9, pp. 736–743, 2006. doi: 10.1002/zamm.200510279
J. Gemmer et al., “Generalizations of the brachistochrone problem”, Pi Mu Epsilon J., vol. 13, no. 4, pp. 207–218, 2011.
S. Mertens and S. Mingramm, “Brachistochrones with loose ends”, Eur. J. Phys., vol. 29, pp. 1191–1199, 2008. doi: 10.1088/0143-0807/29/6/008
E. Rodgers, “Brachistochrone and tautochrone curves for rolling bodies”, Am. J. Phys., vol. 14, pp. 249–252, 1946. doi: 10.1119/1.1990827
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
S.S. Gurram et al., “On the brachistochrone of a fluid-filled cylinder”, J. Fluid Mech., vol. 865, pp. 775–789, 2019. doi: 10.1017/jfm.2019.70
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