On Jacobi’s condition for the simplest problem of calculus of variations with mixed boundary conditions

The purpose of this paper is an extension of Jacobi’s criteria for positive definiteness of second variation of the simplest problems of calculus of variations subject to mixed boundary conditions. Both non constrained and isoperimetric problems are discussed. The main result is that if we stipulate conditions (21) and (22) then Jacobi’s condition remains valid also for the mixed boundary conditions.