The Calculus of Variations in Applied Aerodynamics and Flight Mechanics

Abstract

The calculus of variations has become increasingly popular in applied aerodynamics through the study of the optimum shapes of aircraft and missile components and flight mechanics through the study of the optimum trajectories of aircraft, missiles, and spaceships. The most general problems of the calculus of variations in one independent variable are the problems of Bolza, Mayer, and Lagrange. This chapter discusses the indirect methods of the calculus of variations with particular emphasis on the problems of Bolza, Mayer, and Lagrange. After the Euler-Lagrange equations, the corner conditions, the transversality condition, the Legendre-Clebsch condition, and the Weierstrass condition are reviewed, and the difficulties inherent to the solution of the mixed boundary-value problem are discussed in the chapter. The Legendre–Clebsch condition is a consequence of the Weierstrass condition. The use of variational techniques in applied aerodynamics is also discussed in the chapter, with particular regard to the determination of the geometry of the slender body of revolution having minimum pressure drag in Newtonian flow, as well as the two-dimensional wing having minimum pressure drag in linearized supersonic flow. The use of variational techniques in flight mechanics is reviewed with particular regard to the determination of the optimum two-dimensional paths for a rocket vehicle flying in either a vacuum or a resisting medium.