Principal quadratic functionals

Abstract

This paper is a continuation of a study undertaken earlier by Marston Morse and the writer [ML, 1](2). While the present paper is essentially selfcontained, some familiarity on the part of the reader with the earlier paper, particularly with the importance of the singularity function in studying the minimizing of singular quadratic functionals when evaluated along A-admissible curves, is highly desirable. Classical Sturm separation and comparison theorems are used freely and usually without reference throughout the paper. The integrals employed in the analysis are Lebesgue integrals and their extensions. The functional J with which this paper has to do is termed a principal quadratic functional (?2). In general it is singular at x = 0. It should be understood, however, that it is so defined as to include many nonsingular functionals as well. The function p(x) which appears in J is assumed to be of one sign near x = 0. It will be seen that the Euler equation of J thus includes, for example, all the classical singular second-order linear ordinary differential equations. The variation of J is studied both under fixed end conditions and when the y-axis is regarded as a kind of singular end curve with the second end point fixed. The concept of the focal point of the y-axis is introduced. Its theory characterizes precisely the variation of J when F-admissible curves are the comparison curves. It is also found that the focal point of the y-axis rather unexpectedly plays a role in the study of the fixed end point problem (Theorem 4.1).