ON THE ANALYTICAL SOLUTION OF THE BRACHISTOCHRONE PROBLEM IN A NONCONSERVATIVE FIELD

A new approach to obtain an analytical solution of the brachistochrone problem in a non-conservative velocity-dependent frictional resistanc e field is presented. Geometrical and energy constraints are incorpora ted into a time functional through Lagrangian multipliers and the Eule r-Lagrange equations in a natural coordinate system are derived. The n ovelty of the present approach is a parametrization of the Euler-Lagra nge equations by the slope angle of the trajectory. By exploiting a sp ecial structure of the governing equations of the problem, all functio n-variables are eliminated and the remaining two unknown parameters ar e eventually determined from the two non-linear equations. This approa ch offers several advantages over the well-known solution by Bolza, an d establishes an analogy of the brachistochrone problem with other mec hanical problems, in particular with a bending of a planar beam. The s olution of the classical (Bernoulli's) brachistochrone problem is deri ved in explicit, yet alternative formulae. A numerical example assumin g the linear resistance law (Newtonian fluid) is presented, and the in fluence of the coefficient of viscous friction, k, on the brachistochr one motion is analyzed. The limiting case k --> infinity is also dealt with. (C) 1997 Elsevier Science Ltd.