Morse and Leighton (Singular quadratic functionals. Transactions of the American Mathematical Society, volume 40 (1936) pp. 252-286) gave a systematic approach to the problem of minimizing a singular quadratic functional for one dependent variable considering integrands of the type f (x, y, y') = r(x) y'2 + 2q (x) y y' + p (x) y2 where r, q, and p are single-valued continuous functions of the real variable x on the interval (0, d)(2) and r is positive. They defined first conjugate point of the singular point x = 0. They defined minimum limit of a functional and determined conditions under which [o, b] would afford such a limit to a functional among several classes of comparison curves.
Proceedings of the Iowa Academy of Science
©1952 Iowa Academy of Science, Inc.
Chellevold, John O.
"Conjugate Points of Singular Quadratic Functionals for N Dependent Variables,"
Proceedings of the Iowa Academy of Science, 59(1), 331-337.
Available at: https://scholarworks.uni.edu/pias/vol59/iss1/40