On the analytical solution of the brachistochrone problem in a non-conservative field
We present a new approach to obtain an analytical solution of the brachistochrone problem in a non-conservative velocity-dependent frictional resistance field. Geometrical and energy constraints are incorporated into a time functional through Lagrangian multipliers, and the Euler-Lagrange equations are derived in a natural coordinate system. 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, all function-variables are eliminated and the remaining two unknown parameters are eventually determined from the two nonlinear equations. 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 \(k\to \infty\) is also dealt with.
scientific article; zbMATH DE number 1174786