Second-order sufficiency criteria and local convexity for equivalent problems in the calculus of variations (Q912380)

Two classical theorems regarding sufficient conditions for optimality in the calculus of variations receive new proofs in this paper. The author shows that if the strengthened forms of the classical Euler-Lagrange, Weierstrass, Legendre, and Jacobi conditions hold along a certain arc x(t), then one can construct a variational problem equivalent to the original one which is locally convex near x(t). The classical sufficiency theorems follow from this, because in convex problems the first-order necessary conditions are also sufficient.