THE USE OF VARIATIONAL TECHNIQUES IN THE OPTIMIZATION OF FLIGHT TRAJECTORIES

General equations which describe optimum aircraft flight tra­ jectories are obtained in this paper by the use of the methods of the calculus of variations. These general equations are specialized for minimum time, maximum range, and minimum fuel consumption flight paths and are presented in a form which lend themselves to analog computer solution. A few brachistochronic flight trajectories, obtained from the analog computer, are illustrated. The approach and approximations used to obtain optimum flight trajectory equations and the analog computer method of solution repre­ sent the distinctive features of this investigation. An energy equa­ tion, with arc length as the independent variable, is employed as the dynamical constraint equation in place of the equations of motion customarily used. The optimizing Euler Lagrange equations are devel­ oped on a basis which includes the effects of thrust, parasite drag, gravity, mass flow, and wind on an aircraft. Thrust is assumed to be a specified function of altitude and velocity. By neglecting induced drag and limitations on lift, the optimizing equations are signifi­ cantly simplified.. Analog computer solutions to brachistochronic problems indicate that the shape of an optimum flight path is a strong function of normal forces applied to the aircraft, but only weakly dependent on the ap­ plied tangential forces. i