Extremum conditions for functionals involving higher derivatives of several variable vector valued functions

TL;DR

This paper develops necessary and sufficient extremum conditions for variational problems involving higher derivatives of vector functions, providing explicit formulas that do not rely on conjugate points, thus advancing the theoretical framework of calculus of variations.

Contribution

It introduces a general formulation of first and second order extremum conditions for higher derivative variational problems, including explicit formulas for sufficiency without conjugate points.

Findings

Derived first order necessary conditions for constrained variational problems.

Established global second order necessary extremum conditions.

Presented explicit second order sufficient conditions avoiding conjugate points.

Abstract

This paper addresses both necessary and relevant sufficient extremum conditions for a variational problem defined by a smooth Lagrangian, involving higher derivatives of several variable vector valued functions. A general formulation of first order necessary extremum conditions for variational problems with (or without) constraints is given. Global Legendre second order necessary extremum conditions are provided as well as new general explicit formula for second order sufficient extremum condition which does not require the notion of conjugate points as in the Jacobi sufficient condition.