Generalized Weierstrass Condition in the Classical Calculus of Variations

We study the extremals of problems of the classical calculus of variations. We obtain a new necessary condition for a strong minimum under the assumption that the Weierstrass and Legendre conditions degenerate, i.e., are satisfied as equalities, on the extremal under study; this condition is called a generalized Weierstrass condition. We also give a modified condition that is a necessary condition for a weak minimum. © 2019, Pleiades Publishing, Ltd.