Problem of the calculus of variations with a quadratic functional and higher-order derivatives

This paper presents a complete study of the problem of the calculus of variations with higher-order derivatives and a quadratic functional.The main result is the theorem about quadratic functionals.We answer the following question: when does the feasible extrema provide an absolute minimum in the given problem with a quadratic functional?This problem was previously investigated by V. Alekseev, E. Galeev, V. Tikhomirov in [1].Points c) and d) of the above theorem were firstly formulated in [1] without proof.In this article we prove these points and add two more cases a) and b).Why are these two new points significant?What does happen if necessary condition is satisfied (the Jacobi condition), but sufficient condition is not (the strong Jacobi condition)?Usually, in such cases we use definition of absolute minimum and then we prove that feasible extrema provide absolute minimum or not by using sequences of feasible functions.But now, using the result of this paper we can answer this question just by checking two conditions.To prove the above theorem, we modified and applied methods used by E. Galeev in [2] to prove similar result in a case of the classic problem of the calculus of variations.