Equivalence between Dirac and symplectic approaches of quantization

In this work we show that the symplectic matrix can be set up directly from the Hamiltonian, without having to construct a first-degree Lagrangian formulation first. Also we show that the Dirac matrix of constraints can be constructed directly from a Lagrangian. Moreover, when the Hamiltonian is not quadratic in the momenta, a first-degree Lagrangian can be constructed anyway using a linearization prescription based on the Routh function. From all this we conclude that the difference between the Dirac and the Symplectic methods is restricted to the choice of the matrix to be used to quantize the theory (of constraints or symplectic) instead of to the choice of the framework (Hamiltonian or first-degree Lagrangian) as is frequently believed. (author)