content_image/413349.jpg content_image/413350.jpg content_image/413348.jpg content_image/413347.jpg content_image/413346.jpg content_image/413345.jpg content_image/413354.jpg content_image/413353.jpg content_image/413352.jpg content_image/413351.jpg
## II. THE DUAL EMBEDDING FORMALISM 

The formalism is developed based on the symplectic framework [23, 24], that is a modern way to handle with constrained systems. The basic object behind this for- malism is the symplectic matrix, i. e., if this matrix is singular, the model presents a symmetry, if it is not sin- gular, the Dirac brackets can be obtained. In this way, we propose to render nonsingular symplectic matrix to a singular one. It will be carried out introducing arbitrary functions that depend on the original and WZ variables into the first-order Lagrangian. To appreciate this point, a brief review of the symplectic formalism will be fur- nished and, after that, general ideas of the symplectic gauge-invariant formalism will be presented. This for- malism, differently from the BFFT and other iterative constraint conversion methods, does not require special modifications into the procedures to convert Abelian or non-Abelian set of second-class constraints into first-class ones. 

This technique follows the Faddeev-Shatashivilli’s sug- gestion [9] and is set up on a contemporary framework to handle constrained models, namely, the symplectic for- malism [23]. In the following lines, as we said above, we will try to keep this paper self-sustained reviewing the main steps of the dual embedding formalism. We will follow closely the ideas contained in [25]. 

Let us consider a general noninvariant mechanical model whose dynamics is governed by a Lagrangian L ( a$_{i}$, ˙ a$_{i}$, t ), (with i = 1 , 2 , . . . , N ), where a$_{i}$ and ˙ a$_{i}$ are the space and velocity variables, respectively. Notice that this model does not result in a loss of generality nor physical content. Following the symplectic method the zeroth-iterative first-order Lagrangian one-form is writ- ten as 

and the symplectic variables are 

<img src='content_image/413349.jpg'>

where A (0) θ are the canonical momenta and V (0) is the symplectic potential. From the Euler-Lagrange equations of motion, the symplectic tensor is obtained as 

If the two-form 

is singular, the symplectic matrix (2.3) has a zero-mode ( ν $^{(0)}$) that generates a new constraint when contracted with the gradient of the symplectic potential, 

<img src='content_image/413350.jpg'>

<img src='content_image/413348.jpg'>

<img src='content_image/413347.jpg'>

<img src='content_image/413346.jpg'>

This constraint is introduced into the zeroth-iterative Lagrangian one-form equation (2.1) through a Lagrange multiplier η , generating the next one 

with γ = 1 , 2 , . . . , (2 N + 1) and 

As a consequence, the first-iterative symplectic tensor is computed as 

If this tensor is nonsingular, the iterative process stops and the Dirac brackets among the phase space variables are obtained from the inverse matrix ( f (1) γβ ) − 1 and, con- sequently, the Hamiltonian equation of motion can be formulated and solved, as discussed in [26]. It is well known that a physical system can be described at least classically in terms of a symplectic manifold M . From a physical point of view, M is the phase space of the system while a nondegenerate closed 2-form f can be identified as being the Poisson bracket. The dynamics of the system is determined just specifying a real-valued function (Hamiltonian) H on the phase space, i.e. , one of these real-valued function solves the Hamiltonian equa- tion, namely, 

and the classical dynamical trajectories of the system in the phase space are obtained. It is important to mention that if f is nondegenerate, the equation (2.8) has a very unique solution. The nondegeneracy of f means that the linear map ♭ : T M → T $^{∗}$M defined by ♭ ( X ) := ♭ ( X ) f is an isomorphism. Due to this, the equation (2.8) is solved uniquely for any Hamiltonian ( X = ♭ − $^{1}$( dH )). On the other hand, the tensor has a zero-mode and a new con- straint arises, indicating that the iterative process goes on until the symplectic matrix becomes nonsingular or singular. If this matrix is nonsingular, the Dirac brack- ets will be determined naturally. 

In [26], the authors consider in detail the case when f is degenerate. The main idea of this embedding formalism is to introduce extra fields into the model in order to obstruct the solutions of the Hamiltonian equations of motion. We introduce two arbitrary functions that hinge on the original phase space and on the WZ variables, namely, Ψ( a$_{i}$, p$_{i}$ ) and G ( a$_{i}$, p$_{i}$, η ), into the first-order one- form Lagrangian as follows 

<img src='content_image/413345.jpg'>

<img src='content_image/413354.jpg'>

<img src='content_image/413353.jpg'>

<img src='content_image/413352.jpg'>

<img src='content_image/413351.jpg'>

