Flight Mechanics and Variational Problems of a Linear Type

Abstract

SUMMARY A general analysis is presented of variational problems of linear type. The characteristic of these problems is that the functional forms to be extremized, the differential equations of constraint, and/or the eventual isoperimetric conditions only in­ volve the first power of the derivatives of the unknown functions. For the above problems the Legendre-Clebsch condition fails to yield any information on the minimal or maximal nature of the extremal arc; moreover, the Weierstrassian function is zero along a major portion of the Eulerian solution. By the use of a new technique, based on Green's theorem, the afore-mentioned difficulties can be overcome. Rigorous sufficiency proofs are developed for linear problems of both the simplest type and the isoperimetric type. Several physical problems, associated with the mechanics of flight of either a constant mass or a variable mass vehicle, are treated within the general frame of problems of linear type.