Brachistochrone problem with linear and quadratic drag and accelerating force

Abstract

Motion of the material point in vertical plane is considered under assumption, that gravitational field and atmosphere are homogeneous. The problem is to determine the shape of the trajectory,ensuring the maximum horizontal distance from initial position for fixed time interval. Problem formulated above is close to the famous brachistochrone problem with viscous drag. Maximum Principle is applied to reduce optimal problem to the boundary-value problem for the system of two nonlinear differential equations. The qualitative analysis of the resulting system makes it possible to reveal previously unknown features of optimal trajectories for a general model of the resistance force. This analysis is illustrated by a numerical simulation. The particular case of the accelerating viscous force also analyzed.