Lagrange versus symplectic algorithm for constrained systems

The systematization of the purely Lagrangian approach to constrained systems in the form of an algorithm involves the iterative construction of a generalized Hessian matrix W taking a rectangular form. This Hessian will exhibit as many left zero modes as there are Lagrangian constraints in the theory. We apply this approach to a general Lagrangian in the first-order formulation and show how the seemingly overdetermined set of equations is solved for the velocities by suitably extending W to a rectangular matrix. As a byproduct we thereby demonstrate the equivalence of the Lagrangian approach to the traditional Dirac approach. By making use of this equivalence we show that a recently proposed symplectic algorithm does not necessarily reproduce the full constraint structure of the traditional Dirac algorithm.