Perfect fluid coupled to a solenoidal field which enjoys the l-conformal Galilei symmetry

A non-relativistic (Galilei-invariant) model of a perfect fluid coupled to a solenoidal field in arbitrary spatial dimension is considered. It contains an arbitrary parameter $\kappa$ and in the particular case of $\kappa=1$ it describes a perfect fluid coupled to a magnetic field. For a special value of $\kappa$, the theory admits the Schrodinger symmetry group which is consistent with the magnetic case in two spatial dimensions only. Generalization to the case of the l-conformal Galilei group for an arbitrary half-integer parameter l is constructed.


Introduction
Current interest in fluid dynamics with conformal symmetry is caused by efforts to understand the hydrodynamic limit of the AdS/CFT-correspondence (see review [1] and references therein).In this limit, fluid mechanics can provide an effective description of a strongly coupled quantum field theory.From the symmetry standpoint, the main object here is the conventional conformal algebra.
Application of the non-relativistic version of AdS/CFT-correspondence [2][3][4] to strongly coupled condensed matter systems generated a great deal of interest in non-relativistic conformal algebras.There are various conformal extensions of the Galilei algebra, the most general of which is the so-called ℓ-conformal Galilei algebra [5,6].A peculiar features of this algebra is that temporal and spatial coordinates scale differently under the dilatation: t ′ = λt, x ′ i = λ ℓ x i .In condensed matter physics, z = 1 ℓ is known as the dynamical critical exponent.The algebra is finite-dimensional provided ℓ is a (half)integer number.The cases ℓ = 1  2 and ℓ = 1 are referred to as the Schrödinger algebra [7] and the conformal Galilei algebra [8], respectively.In recent years, dynamical realizations of the ℓ-conformal Galilei algebras attracted considerable attention (see e.g.[9]- [16] and references therein).
As is known, the non-relativistic perfect fluid dynamics enjoys the Schrödinger symmetry provided a specific equation of state is chosen [17,18].Quite recently, generalized perfect fluid equations were formulated which enjoy the ℓ-conformal Galilei symmetry [19] (see also [20,21]).In both cases, the fluid is considered as a closed system in the absence of external forces.It is natural to wonder whether the formalism in [19] can be extended so as to accommodate external forces.Of particular interest is the magnetic force.The goal of this work is to study the case involving a solenoidal field.To the best of our knowledge, this issue has not yet been discussed in literature.
The paper is organized as follows.In the next section, the equations of motion, the energymomentum tensor and the Hamiltonian formulation are constructed for a non-relativistic perfect fluid coupled to a solenoidal field B i (t, x) in three spatial dimensions.They contain an arbitrary parameter κ.In particular case of κ = 1 they describe the perfect magnetohydrodynamics. Symmetries of the dynamical system under consideration are established and the corresponding conserved charges are constructed.The Galilei group is present for an arbitrary value of κ, while additional conformal symmetries enlarging it to the Schrödinger group can be realized for a special choise of κ only.These results are generalized to the case of an arbitrary number of spatial dimensions in Sect.3, where the model of the perfect fluid coupled to antisymmetric solenoidal field F ij (t, x) is considered.A generalization corresponding to the ℓ-conformal Galilei algebra for an arbitrary half-integer value of the parameter ℓ is constructed as well.In the concluding Sect. 4 we summarize our developments.

Perfect fluid coupled to a vector field B i
In non-relativistic space-time parameterized by the coordinates (t, x i ), i = 1, ..., d, a fluid is characterized by the density ρ(t, x) and the velocity vector field υ i (t, x).Let us consider a model of the perfect fluid coupled to a vector field B i (t, x) defined by the following equations1 where p(t, x) is the pressure obeying the equation of state p = p(ρ) and κ is an arbitrary parameter.For κ = 1 and d = 3 the above equations describe the perfect fluid coupled to the magnetic field B i (see for example [22]), which include the continuity equation (2.1), the Euler equation in magnetic field (2.2), the low-frequency version of the Maxwell equations for a medium of infinite conductivity (2.3), (2.4).The last equation (2.4) implies the vector field is solenoidal.
The equations above admit the Hamiltonian formulation which relies upon the Hamiltonian and the Poisson brackets in which derivatives on the right-hand sides are evaluated with respect to x i .It is assumed that the potential V (ρ) entering (2.6) is related to the pressure via the Legendre transformation Note that, in order to verify the Jacobi identity, properties of delta-functions should be used and total spatial derivative terms must be disregarded.Within the context of magnetohydrodynamics (κ = 1), the brackets (2.7) were formulated in [23].
The model under consideration enjoys the Galilei symmetry for an arbitrary value of the parameter κ.The conserved energy coincides with the Hamiltonian H, while the conserved momentum, the Galilei boost generator and the angular momentum generator read (2.9) They form the centrally extended Galilei algebra under the Poisson bracket where the central charge M = dxρ links to the mass of the fluid.For a special choice of κ and V , one can construct additional conserved quantities.The conventional way to prove this is to construct the non-relativistic energy-momentum tensor T µν , µ = 0, i associated with the equations (2.1-2.4).The following energy density and flux satisfy Similarly, the momentum density and stress tensor Note that T i0 = T 0i because the theory is not Lorentz-invariant but T ij = T ji as the theory is invariant under spatial rotations.If additional conformal symmetry exist [18].In our case this condition requires  and allows one to build conserved charges associated with the dilatation and special conformal transformation Together with H they form so(2, 1) subalgebra under the Poisson bracket The remaining structure relations read (2.10) and (2.19) describe the Schrödinger algebra [7].Thus, we have formulated the perfect fluid equations coupled to the solenoidal vector field B i , which enjoy the Schrödinger symmetry, and constructed the corresponding Hamiltonian formulation.The potential V = 1  2 pd together with the relation (2.8) give the equation of state p = νρ 1+ 2 d (ν is some constant), which is the same as in the absence of external forces [17].
3 Perfect fluid coupled to an antisymmetric field F ij

The Schrödinger symmetry
Although the magnetic field (κ = 1, d = 3) does not fit the restriction (2.17), in an arbitrary spatial dimension, where one can introduce an antisymmetric tensor F ij = −F ji instead, the Schrödinger symmetry can be successfully accomodated.Below we discuss this issue in detail.
Consider a model governed by the equations which reproduce (2.1)-(2.4) in three spatial dimensions provided Identifying F ij with the spatial (magnetic) components of the electromagnetic field strength F µν , µ = 0, i one can take these equations to define a perfect fluid coupled to the magnetic field in arbitrary spatial dimension (κ = 1).To the best of our knowledge, Eqs.(3.1)- (3.2) have not yet been discussed in the literature.The second equation in (3.2) implies the field F ij is solenoidal and results in the useful identity The equations above can put into the Hamiltonian form provided one introduces the Hamiltonian and the Poisson brackets The non-relativistic energy-momentum tensor associated with the equations (3.1) and (3.2) reads It is conserved for an arbitrary value of the parameter κ Additional conformal symmetry entering the Schrödinger group arises provided 2T 00 = δ ij T ij , which imposes the same restriction (2.17) upon the potential V and determines the parameter κ as follows Note that for d = 3 one gets κ = 2, which correctly reproduces (2.17) in the previous section.At the same time, the case of a perfect fluid in a magnetic field (i.e.κ = 1) admits the Schrödinger symmetry for d = 2 only.Conserved charges corresponding to the Schrödinger group look like in the previous section (2.9), (2.18) with the energy H defined by the Hamiltonian (3.4).

The ℓ-conformal Galilei symmetry
The ℓ-conformal Galilei algebra [6] is the most general non-relativistic conformal algebra which is characterized by an arbitrary (half)-integer parameter ℓ and for ℓ = 1 2 reproduces the Schrödinger algebra.In addition to time translation H, dilatation D, special conformal transformation K, spatial rotations3 , spatial translation i ] = kC and can be realized in a non-relativistic space-time (t, x i ) by the following operators [6] Consider the set of equations of motion which reproduces (3.1) and (3.2) for ℓ = 1 2 .In the absence of F ij = 0, they hold invariant under the action of the ℓ-conformal Galilei group [19].Below we will fix the value of κ , for which (3.10), (3.11) accomodate the ℓ-conformal Galilei symmetry.To do this, we use the Hamiltonian formulation and restrict ourselves to the case of a half-integer ℓ = n + 1 2 , n = 0, 1, 2, ....

Introducing auxiliary fields υ (k)
i , k = 0, 1, ..., 2n with υ (0) i = υ i , one can rewrite the second equation in (3.10) in the equivalent first order form (3.12) Then the Hamiltonian generates the dynamical equations as follows provided one introduces the Poisson brackets Here δ (k)(m) is the Kronecker symbol.In the absence of F ij = 0 the Hamiltonian and the brackets were formulated in a recent paper [21].Conserved energy is represented by the Hamiltonian (3.13), while conserved charges corresponding to the dilatation, special conformal transformations and vector generators read where υ are conserved for an arbitrary value of κ, whereas the conservation of D and K imposes the restriction .
Under the Poisson brackets the charges obey the algebra (3.9), which involves the central charge [24] {C

Conclusion
To summarize, in this work we have formulated the equations of motion of a non-relativistic perfect fluid coupled to a solenoidal field in arbitrary spatial dimension.The equations involve an arbitrary parameter κ and enjoy the Galilei symmetry.For κ = 1 and d = 3 they reduce a perfect fluid coupled to a magnetic field.It was demonstrated that the dynamical system admits the Schrödinger symmetry group for a specific value of κ, which links to spatial dimension.The corresponding Hamiltonian formulation was built and the full set of conserved charges, which obey the Schrödinger algebra under the Poisson bracket, was presented.
Equations describing a perfect fluid coupled to a solenoidal field admitting the ℓ-conformal Galilei symmetry were proposed as well.For an arbitrary half-integer ℓ, both the Hamiltonian formulation and the full set of conserved charges were constructed.It was shown that under the Poisson bracket the latter form the centrally extended ℓ-conformal Galilei algebra.