Trajectory Paradox is a Boundary Layer!

Abstract

This study derives the asymptotic conditions under which the massless stretched string formulation is valid for the equivalent dynamics of the inertial string carrying a moving point mass. Consequently, the governing equation under such conditions can be written as a singularly perturbed partial differential equation when the inertia of the string is included. When the inertia of the string is ignored, the governing equation reduces to a non-homogeneous hypergeometric ordinary differential equation governing the transverse motion of the moving mass. This hypergeometric equation has a discontinuous solution at the terminating boundary known as the trajectory paradox. However, numerically solving the actual equation yields a continuous solution with a rapid variation near the boundary. The complete solution resembles the inner solution as well, whereas the hypergeometric ordinary differential equation leads to an outer solution, revealing that the trajectory paradox is a boundary layer.