Elimination of Jacobi equation in extremal variational problems (Q2896654)

The classical scheme of a study of local extrema for the one-dimensional Euler-Lagrange functional includes the check of the strengthened Legendre condition and a condition for the solution of the Jacobi equation. The author presents conditions on the integrand in the variational functional, under which there is no need to use the Jacobi equation. There are two cases: in one of them, there is a restriction on the length of the interval of integration; the second case is free from such restrictions. For another approach to the problem see \textit{G. S. Licea} [Journal of Applied Mathematics \& Bioinformatics, vol.1, no.1,2011, 1-20].