A new notion of conjugacy for isoperimetric problems
The main purpose of this paper is to introduce, in the setting of fixed-endpoint problems of the calculus of variations, a new set of points \(R(x)\) such that the emptiness of \(R(x)\) is equivalent to the following property of the trajectory \(x\): the second variation of the problem at \(x\) is nonnegative along the admissible variations. The set \(R(x)\) generalizes the usual notion of conjugate points introduced by Loewen and Zheng in 1994 and by Zeidan in 1996. In contrast to the classical conjugate point theory which concerns the solution of a two point boundary value problem for a linear ordinary differential equation, the notion proposed in this paper consists in finding a trajectory with special properties which assure the possibility making a quadratic form negative.
scientific article; zbMATH DE number 2135337