Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism 20 this new term in the model. We will introduce a final result that is new in the literature, As was demonstrated in [41], the isentropic irrotational fluid model has a dynamically equivalent family of WZ gauge-invariant descriptions. Thus, in order to establish our ideas, we are going to investigate how the inclusion of a dissipative term affects the dynamics of the fluid model considering only one of these zero modes presented in [41]. Note that in the prescription of the symplec- tic formalism, for each zero-mode chosen we will have a gauge-invariant version for the model. In the next subsection, the scalar rotational fluid the- ory will be analyzed from the symplectic point of view [23], and the fundamental Dirac brackets of the fields will be computed. which lead to the invariant Hamiltonian. To demonstrate that the gauge invariant formulation of the non-Abelian Proca model is dynamically equivalent to the original noninvariant model, the symmetry is fixed by using the unitary gauge fixing procedure, δA a i = − ∂ x i ε $^{a}$, δπ a i = 0 , δA a 0 = 0 , (6.37) δθ a = ε $^{a}$, δπ a θ = 0 , which leads to the bracket below, A. Symplectic analysis of the rotational fluid ϕ a = θ a ≈ 0 , (6.38) { Ω a $_{1}$( x ) , ϕ $^{b}$( y ) } = − δ $^{ab}$δ $^{(3)}$( ⃗ x − ⃗ y ) . (6.39) In this section, we will analyze the rotational fluid dy- namic model from the dual embedding point of view. It is well known that systems that have vorticity and/or viscosity present Casimir invariants which ob- struct the construction of a canonical formalism for fluid, as demonstrated in [43]. However, this obstruction can be eliminated using the Clebsch parameters, as shown by Lin [44] and by two of us in [45]. In fact, with the introduction of the Clebsch parameters, it is possible to obtain a Lagrangian density for the rotational fluid with dissipation, in 3-dimensional, as being Due to this, a new Dirac brackets must be computed. The corresponding Dirac matrix for this set of constraints is Using the inverse of this matrix, the Dirac brackets among the physical phase space fields are computed, which is equal to the one calculated in the original de- scription given by equation (6.9). This result demon- strate that the symplectic formalism can be seen in fact as a dual mapping between the original theory and the final one. where the symplectic potential is The symplectic coordinates are ξ (0) = ( ρ, θ, α, β ) with the corresponding zeroth-iterative one-form canonical mo- menta given by L = − ρ ( ˙ θ + α ˙ β ) − V, (7.1) V = 1 2 (1 − k ) ρ ( ∂$_{i}$θ + α∂$_{i}$β )( ∂ $^{i}$θ + α∂ $^{i}$β ) + V ( ρ ) . (7.2) C = ( 0 − 1 1 0 ) δ $^{(3)}$( ⃗ x − ⃗ y ) . (6.40) A (0) ρ = 0 , A (0) θ = − ρ, (7.3) A (0) α = 0 , A (0) β = − αρ. (7.4) f (0) =    0 − δ ( ⃗ r − ⃗ r $^{′}$) 0 − α δ ( ⃗ r − ⃗ r $^{′}$) 0 0 0 0 0 0 − ρ α 0 ρ 0    , (7.5) The zeroth-iteration symplectic matrix, given by Recently, some of us have proposed a Wess-Zumino (WZ) gauge-invariant version for the isentropic irrota- tional fluid model [41]. In that work, we have demon- strated that the irrotational fluid model has a set of dy- namically equivalent WZ gauge invariant versions. Fur- ther, in that paper the extra global symmetries, namely, Galileo antiboost and time rescaling, first obtained in [42], were lifted to local symmetries. In this section, we propose an investigation of the sym- metries of the rotational fluid model, but now with an extra term, like kρ ( ∂$_{i}$θ + α∂$_{i}$β ) $^{2}$, where k is a constant, ρ ≡ ρ ( t, ⃗ r ) is the mass density and θ ≡ θ ( t, ⃗ r ) is the ve- locity potential. This term introduces a dissipative force into the model. The main motivation is that in the real world there is always dissipation, albeit sometimes ex- tremely small. Our purpose is, to verify if the extra sym- metries found in [41] are broken with the introduction of is a nonsingular matrix and, consequently, the model is not a gauge-invariant field theory. As settle by the sym- plectic formalism [23], the Dirac brackets of the phase VII. HIDDEN SYMMETRY IN THE ROTATIONAL FLUID MODEL