  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.
