  By using the supersymmetric version of the Faddeev-Jackiw symplectic
formalism, a family of first-order constrained Lagrangians for the t-J model is
found. In this approach the Hubbard ${\hat X}$-operators are used as field
variables. In this framework, we first study the spinless fermion model which
satisfies the graded algebra spl(1,1). Later on, in order to satisfy the
Hubbard ${\hat X}$-operators commutation rules satisfiying the graded algebra
spl(2,1), the number and kind of constraints that must be included in a
classical first-order Lagrangian formalism for the t-J model are found. This
model is also analyzed in the context of the path- integral formalism, and so
the correlation generating functional and the effective Lagrangian are
constructed.
