The unrestrained brachistochrone

Abstract

The unrestrained brachistochrone problem is to find the path of a frictionless track between two horizontally separated points along which a block with an initial velocity will travel in the shortest time. The track lies in a uniform gravitational field and must be horizontal at its beginning and its end. Furthermore, the block must slide along the track like a block on an inclined plane and it must remain in contact with the track at all times. Although the problem is well posed, the nature of the unusual constraints makes it quite difficult to solve using standard variational techniques, such as those of Euler and Lagrange. We present a solution that avoids explicit reliance on such techniques.

Minimum descent time along a set of connected inclined planes

“…, plotted as the dashed line in figure 4. This limiting value is only 5% larger than that found in equation (13).…”

“…ramps AB 1 4 1 2 and hence by the same logic as equation (13) one obtains Figure 5. The three-ramp geometry that results in fastest transit time from initial point A at ( ) 0, 0 to final point C at ( ) L, 0 .…”

“…It is true that a 60°ramp is steep and a ball on it would probably not roll without slipping, and might even lose contact and bounce at the transition points between the inclines and the table. One could thus impose additional constraints on the problem, in the spirit of the unrestrained brachistochrone problem proposed by Stork [13], such as that the ball has to start on horizontal starting and ending platforms and that there is a maximum slope of the track. Such constraints might again be more easily investigated theoretically and experimentally using connected inclined planes.…”

“…The values for the triple connected ramps in figures 1 and 3 are graphed as the square symbols, whereas the round symbols plot the cycloid values from equation(12)after numerically solving for θ max from the ratio of equations (3) and (4). The solid line marks the cycloidal limiting value p » 1.772 as  ¥ L H from equation(13), whereas the dashed line indicates the ramp limit of »…”

Time-optimal control of rolling bodies

“…Interesting variants include the constrained brachistochrone problem (Dreyfus, 1962), the unrestrained brachistochrone problem (Stork & Yang, 1988;Stork, Yang, & Stover, 1986), the unrestrained rolling brachistochrone problem , the brachistochrone with Coulomb friction (Hayen, 2005), the quantum brachistochrone problem (Carlini, Hosoya, Koike, & Okudaira, 2006), the relativistic brachistochrone problem (Goldstein & Bender, 1986), the giant brachistochrone problem (Parnovsky, 1998) and the brachistochrone problem with drag (Vratanar & Saje, 1998). The fundamental and evergreen nature of this problem and its variants is widely recognised.…”