A generalized condition of Clebsch under weak normality assumptions

Abstract For unconstrained problems in the calculus of variations, the condition of Legendre is a second order necessary condition for optimality which has played a crucial role in the theory. In the presence of equality constraints, it is commonly called the condition of Clebsch. In this paper, this condition is generalized for problems with equality and inequality constraints under normality assumptions weaker than the standard assumptions found in the literature.