content_image/442839.jpg content_image/442840.jpg content_image/442841.jpg content_image/442837.jpg content_image/442842.jpg content_image/442843.jpg content_image/442854.jpg content_image/442857.jpg content_image/442856.jpg content_image/442855.jpg
where ∂ ′ i = ∂ ∂⃗ r $_{′}$. After a straightforward calculation, the linear correction term as function of η is obtained as 

For the quadratic correction terms in equation (7.15), we have that 

After a direct calculation, the second-order correction term is obtained as 

For the cubic correction terms in equation (7.15), we can 

<img src='content_image/442839.jpg'>

<img src='content_image/442840.jpg'>

<img src='content_image/442841.jpg'>

write 

which allows to conclude that G ( n ) = 0 for n ≥ 3. Hence, the gauge-invariant first-order Lagrangian is written as 

where the symplectic potential is 

<img src='content_image/442837.jpg'>

<img src='content_image/442842.jpg'>

<img src='content_image/442843.jpg'>

To complete the gauge-invariant reformulation of the model, we will compute the infinitesimal gauge transfor- mation. In agreement with the method, the zero-mode ˜ ν (0) is the generator of infinitesimal gauge transforma- tions ( δ O = ε ˜ ν $^{(0)}$). Then, 

where ε ( ⃗ r, t ) is an infinitesimal time-dependent param- eter. In fact, under the infinitesimal transformations above, the invariant Hamiltonian ( ˜ V $^{(0)}$) changes as 

Considering the following transformations, 

then the Lagrangian density, equation (7.21), and the Hamiltonian, equation (7.22), become 

<img src='content_image/442854.jpg'>

These expressions are similar the original expressions for the Lagrangian in equation (7.1), and the Hamiltonian in equation (7.2), respectively. 

<img src='content_image/442857.jpg'>

<img src='content_image/442856.jpg'>

<img src='content_image/442855.jpg'>

Thus, at this point, it is important to point out that exist a hidden symmetry into the rotational fluid model. Note that, in Section II, it was not possible to realize this symmetry. Another interesting feature discovered here is that for the set of the differential equations obtained we have no other solutions, i.e. , the model has, in fact, only one hidden symmetry. Based on the investigation done by some of us in Ref. [41], where the extra global symme- tries proposed in [42] are lifted to the local status , we can conclude that these extra global symmetries do not exist in the rotational fluid model. Some other considerations will be depicted in the next section. 

From the Dirac point of view, a system classified as a gauge invariant theory is one that has only first-class constraints. When a theory has second-class constraints, the gauge invariance can be recovered by converting the second-class constraints into first-class constraints. In the literature there is a great variety of techniques, with pros and cons, to promote this kind of conversion. 

In this work we are concerned not only with the gauge invariance of second-class systems but also in obtaining a theory dual equivalent to the original one. We be- lieve that dual embedding formalism is the most ade- quate technique because it is not affected by ambiguity problems related to the introduction of the WZ variables [13]. Besides, as demonstrated by some of us, this method has the advantage that a convenient choice of a conve- nient zero-mode can lead to a theory dual equivalent to 

## VIII. FINAL DISCUSSIONS 

