On the analytical solution of the brachistochrone problem in a non-conservative field

Abstract

A new approach to obtain an analytical solution of the brachistochrone problem in a non-conservative velocity-dependent frictional resistance field is presented. Geometrical and energy constraints are incorporated into a time functional through Lagrangian multipliers and the Euler-Lagrange equations in a natural coordinate system are derived. The novelty of the present approach is a parametrization of the Euler-Lagrange equations by the slope angle of the trajectory. By exploiting a special structure of the governing equations of the problem, all function-variables are eliminated and the remaining two unknown parameters are eventually determined from the two non-linear equations. This approach offers several advantages over the well-known solution by Bolza, and establishes an analogy of the brachistochrone problem with other mechanical problems, in particular with a bending of a planar beam. The solution of the classical (Bernoulli's) brachistochrone problem is derived in explicit, yet alternative formulae. A numerical example assuming the linear resistance law (Newtonian fluid) is presented, and the influence of the coefficient of viscous friction, k, on the brachistochrone motion is analyzed. The limiting case k → ∞ is also dealt with.