We present a geometric method in Calculus of Variations that, in the presence of a constraint given by partial differential equations, allows to determine, for any Lagrangian density, a 'constrained' Cartan form which satisfies an analogous of the classical Lepage congruences. The problematic of defining a second variation from the Lagrangian density can be solved with this new object, from which we derive Euler-Lagrange and Cartan equations for constrained critical sections, a Hessian bilinear form and a notion of Jacobi vector fields. We compare our results with those arising in the classical Lagrange multiplier approach, showing that certain special care should be taken when working with this classical framework. An example of immersions of surfaces in the euclidean space with prescribed area element illustrates the theory.