Journal of Applied Mathematics and Mechanics (about journal) Journal of Applied
Mathematics and Mechanics

Russian Academy of Sciences
in January 1936
(Translated from 1958)
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ISSN 0021-8928
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IssuesArchive of Issues2009-4pp.371-378

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L.D. Akulenko, "The brachistochrone problem for a disc," J. Appl. Math. Mech. 73 (4), 371-378 (2009)
Year 2009 Volume 73 Issue 4 Pages 371-378
Title The brachistochrone problem for a disc
Author(s) L.D. Akulenko (Moscow, Russia,
Abstract The motion of a vertical disc along a curve under the influence of gravity is investigated. On the assumption of regular rolling without slip and separation of contact points, the problem of plotting the curve of most rapid motion of the disc centre from the origin of coordinates to an arbitrary fixed point of the lower half-plane is solved. As usual, the velocity at the initial instant of time is zero, and at the final instant of time it is not fixed. In explicit parametric form, the classical brachistochrone for contact points of the disc is plotted and investigated. The response time, trajectory and kinematic and dynamic characteristics of motion are calculated analytically. Previously unknown qualitative properties of regular rolling are established. In particular, it is shown that the disc centre moves along a cycloid connecting specified points. The envelopes of the boundary points of the disc, produced as its centre moves along the cycloid, are brachistochrones. The feasibility of mechanical coupling of the disc and the curve by reaction forces at the contact point (the normal pressure and dry friction) is investigated.
Received 09 October 2008
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