Hamiltonian analysis of interacting fluids

Ideal fluid dynamics is studied as a relativistic field theory with particular importance on its hamiltonian structure. The Schwinger condition, whose integrated version yields the stress tensor conservation, is explicitly verified both in equal-time and light-cone coordinate systems. We also consider the hamiltonian formulation of fluids interacting with an external gauge field. The complementary roles of the canonical(Noether) stress tensor and the symmetric one obtained by metric variation are discussed. Finally, a non-relativistic reduction of the system in light-cone coordinates has been carried out which reproduces results found earlier in the literature.


Introduction
The recent idea of fluid-gravity correspondence [1] has brought, in the forefront, the theoretical study of fluid dynamics from the high energy and gravitational physics perspective. The basic premise is that relativistic or non-relativistic fluid dynamics can reproduce the low energy behavior of systems in local thermal equilibrium in a universal way. Indeed, this is an offshoot of the AdS/CFT correspondence [2] that paves the way for studying strongly coupled systems from their weakly coupled analogues in one dimension higher. Generically one exploits AdS/CFT correspondence to study strongly correlated condensed matter systems as boundary conformal theories from results obtained in weakly coupled classical gravity theories in one higher dimension. However, the mutual exchange of ideas can work bothways in fluid-gravity correspondence: fluid systems can yield results relevant in eg. black hole physics, Hawking radiation [3] while gravitational physics can provide new ideas in the context of viscous fluids, turbulence, to name a few. All these considerations require a systematic study of the fluid system as a field theory in the Euler scheme, which is essentially a hamiltonian framework. Surprisingly, apart from the treatment presented in [4], there are not many articles devoted to a systematic hamiltonian analysis of even the relativistic ideal fluid, let alone on the non-ideal fluid.
In the present paper we have performed a thorough hamiltonian analysis of the relativistic fluid both in equal-time and light-cone variables. It is imperative to introduce Clebsch variables [5] in a first order formulation of the fluid action and the resulting constraint system is analyzed in Dirac's framework [6]. The relevant constraints are identified and the system is found to be second class. The modified symplectic structure is obtained by replacing the Poisson brackets by Dirac brackets. It may be recalled that, for a consistent formulation of relativistic field theory the validity of the Schwinger condition is essential. We have proved this condition explicitly, once again both in equal-time and light-cone formulations. As a spin-off from this condition, we have shown the conservation of the stress tensor.
Another area of topical interest is the non-relativistic reduction of relativistic fluid systems where light-cone analysis plays a pivotal role [7,8]. Quite interestingly, it has been demonstrated in [7] that rewriting the conservation relation of relativistic energy-momentum tensor in light-cone variables, and compactifying a spatial light-cone coordinate, one can map the relativistic fluid dynamics to its non-relativistic counterpart in one dimension lower. This naturally enforces a nontrivial map between relativistic and non-relativistic variables. This approach is essentially algebraic in nature. In this paper we have, on the other hand, developed a dynamical approach in the lightcone formulation, to achieve the non-relativistic reduction. Using the current conservation and relativistic Euler equation, we have successfully carried out the reduction and reproduced the map between the relativistic and non-relativistic fluid variables found earlier [7].
The paper is organized as follows: In Section 2 the relativistic fluid model in terms of Clebsch variables is introduced in equal-time coordinate system. The sysmplectic structure is derived in the hamiltonian formalism and the Schwinger condition is verified. Section 3 deals with the fluid in light-cone coordinate system. The dynamics and light-cone Schwinger condition are discussed. Section 4 is devoted to the non-relativistic reduction of the fluid in the light-cone formalism. The paper ends with our conclusions in Section 5.
2 Relativistic fluid mechanics in equal-time coordinates: In the Eulerian description of relativistic fluid the dynamical variables are the matter density j 0 and the currents j i , i = 1, 2, 3 that satisfy the conservation law, A manifestly Lorentz covariant lagrangian description may be adopted by introducing a generalized scalar potential function f ( j µ j µ ), where, the auxiliary degrees of freedom a µ are defined in terms of three scalar Clebsch variables [5] θ, α, β, We take (2) as the lagrangian density of an ideal relativistic fluid [4].
The expanded form of the lagrangian (2) with j µ j µ = n 2 , is In the above we have defined ρ = j 0 . Our prescription is the following: the variables associated with time derivatives like ρ, α, β, θ are treated as dynamical whereas j i are regarded as auxiliary variables. From the lagrangian (4), equations obtained by varying β, α, ρ and j µ are respectively 1 , θ + αβ + ρ n f ′ (n) = 0.
Note that variation of θ reproduces the covariant current conservation law (1).
Let us now develop a hamiltonian formulation. Being first order in time derivatives the system is a constraint system and has a non-trivial symplectic structure, that can be identified with the Dirac brackets of the variables in a hamiltonian formalism [6]. The first step is to define the conjugate momenta for the dynamical variables, which are They yield four primary constraints Using canonical Poisson brackets of the generic form 2 {q(x), π q (y)} = δ(x − y), we can easily show that the constraint algebra does not close indicating that they form a set of four second class constraints [6]. In a generic system with n second class constraints Ω i , i = 1, 2, ..n, the modified symplectic structure (or Dirac brackets) are defined in the following way, where {Ω i , Ω j } is the invertible constraint matrix. From now on we will only use Dirac brackets but for notational simplicity we will refer to them as {, } instead of {, } * . The non-vanishing Dirac brackets are explicitly listed below Incidentally (12) gives rise to two independent canonical pairs (ρ, θ) and (α, ρβ). The canonical hamiltonian density for the fluid corresponding to (4) is, Using the Dirac brackets (12) the hamiltonian equation of motion for ρ is and we findρ yielding the current conservation law (or in fluid dynamics terminology the continuity equation), obtained earlier (1). In the same way we can find equations of motion for α, β, from which we recover and These equations are the same as the Euler-Lagrange equations of motion (5,6). Finally, fromθ we This is same as (7) and equivalent to the time component of (8).
In our case, the space components of (8) just correspond to the equation for the nondynamical variable j i .
The cornerstone of consistently formulating a relativistic field theory is the verification of the Schwinger condition. For this purpose, it is first essential to define an appropriate stress tensor.
The symmetric and covariant energy-momentum tensor obtained from L is [4] From (4) and (8) the above expression for the stress tensor can be written as, Next we calculate the hamiltonian density from Θ µν , Using (8) to replace j i , we recover the canonical form of the hamiltonian obtained earlier (2), This matching of the expression for the hamiltonian is indeed reassuring. Furthermore, the covariant form of Θ µν in (21) is advantageous since it is manifestly symmetric.
The prerequisite for a consistent relativistic field theory is the validity of the Schwinger condition which is given by, 3 Using the expression of Θ 00 obtained above we need to compute, A straightforward calculation using the Dirac brackets (12) leads to the result, The terms in the parenthesis on the right hand side are just the Θ 0i components of (21). Hence the Schwinger condition (25) is reproduced. This is the first instance it has been explicitly computed in relativistic fluid dynamics and constitutes one of our major results.
Integrating (25) over y and after rearranging terms, one obtains which is nothing but the (ν = 0) time component of the covariant energy momentum conservation equation, In an identical manner we can show the validity of the ν = i component of (29). Interestingly enough, this will hold provided the following relation is satisfied, This is the relativistic generalization of the Euler equation as noted by [4]. Although in nonrelativistic fluid mechanics, Euler equation is frequently used, quite surprisingly the relativistic Euler equation is not very familiar. However, in our subsequent work in the context of nonrelativistic reduction (see section(4)) it will play an essential role.
3 Relativistic fluid mechanics in light-cone (null plane) coordinates: In this section we study the fluid mechanics in light-cone coordinates. The motivation will become clearer in the next section when we discuss the non-relativistic reduction of the fluid model. We define the light-cone coordinates as in [9], {x + , x − ,x} wherex ≡ x 1 , x 2 and x ± = 1 √ 2 (x 0 ± x 3 ). The nonvanishing metric components are g +− = g −+ = 1, g ii = −1, i, j = 1, 2. The fluid lagrangian in this coordinate system is, where, in the last step, we have exploited the definition of a µ (3). Note that x + plays the role of time and the dynamical variables are identified following our previous prescription, that is variables involved in x + -derivatives only are considered as dynamical. In the present setup the degrees of freedom are j − , θ, α, β. The momentum is defined as for a generic φ and ∂ + ≡ ∂ t . The first order model (3) produces the constraints, where π − is the momenta conjugate to j − . Constraint analysis once again provides the Dirac . It is worthwhile to point out that the above bracket structure in light cone coordinates is same as the one derived earlier in (12) in equal time coordinate system. This is simply because the lagrangian (4) was also first order. 4 The hamiltonian density is given by from which, using (3) and (33), the hamiltonian of the fluid is, Before proceeding further we need to check the overall consistency of the light-cone framework mainly because of the unconventional mapping of the variables that we have prescribed.
Let us start by comparing the lagrangian and hamiltonian equations of motion. First comes the continuity equation. From the lagrangian (3) by varying θ we obtain, which is the continuity equation in light-cone coordinates. On the other hand, in hamiltonian framework, we have which reproduces (36). It is interesting to observe that the spatial part is now broken up into two sectors x − andx that are qualitatively somewhat distinct.
Let us rederive the light-cone version of the rest of the lagrangian variational equations (5-7).
The hamiltonian equation, can be rearranged to yield (5) while reproduces (6). In a similar way ∂ + θ obtained below is the light-cone version of (7).
Let us now derive the Schwinger condition in these coordinates. Once again we start from (21).
The Θ +− component reads, We identify this with the canonical hamiltonian density (H) defined in (35). This may be easily seen by replacing a − using (8).
Finally we come to the verification of the Schwinger condition in light-cone coordinates. This requires us to compute, After some algebra we end up with, On further simplification we obtain, and thereby recover the cherished form of the Schwinger condition in light-cone coordinates, We emphasize that this is a completely new result in the context of light-cone formulation of fluid.
Integrating over y we recover or equivalently the energy conservation condition since this is the ν = − component of the covariant conservation law ∂ µ Θ µν = 0. Note that this computation can be repeated for ν = +, i but infact that is unnecessary since the covariant conservation law follows directly from the lagrangian (12) and we have checked individually that the hamiltonian equations of motion in light-cone coordinates match correctly with their lagrangian counterpart. Finally, as discussed at the end of Section 2, the light-cone version of the relativistic Euler equation will also appear in the present setup.
To the best of our knowledge, in our work, for the first time, the light-cone analysis of relativistic fluid model has been carried through where the specific identification of the physical degrees of freedom with the Clebsch variables have been spelt out. A specific use of the light-cone formulation will be discussed in the next section.

Non-relativistic light-cone reduction
In recent times non-relativistic conformal theories have become an active area of research in the context of AdS/CFT correspondence [10]. This is mainly due to the possibility of verifying AdS/CFT correspondence experimentally in physically realizable non-relativistic systems, such as in cold atom [11]. Originally AdS/CFT provided a mapping between relativistic theories. Moreover its analogues in non-relativistic scenario have also been explored [10]. Indeed, there are several ways of invoking the non-relativistic limit and we particularly follow the limiting procedure in light-cone framework, advocated in [10] and applied in [7] in the fluid dynamical context. This prescription is to reduce the relativistic conformal symmetry to Galilean symmetry just by compactifying the x − coordinate. This will induce the selection of a preferred light-cone direction [7]. Operationally this means that we will simply omit all x − dependence reducing ∂ − terms to zero.
Following the prescription of dropping x − -dependence, we write the lagrangian and hamiltonian density of the fluid in non-relativistic regime as, We are going to derive the non-relativistic fluid equations following our earlier prescription. Subsequently, we will compare the relations with the usual non-relativistic fluid equations and finally we will provide a map between the non-relativistic and the relativistic fluid variables.
Continuity equation, written in this reduced coordinate system is Note that the variables now depend only on x + ,x. The continuity equation in terms of nonrelativistic degrees of freedom reads,ρ Comparing (50) and (51) we identify j − = ρ and j i = ρv i .
As advertised in Section 2, we now exploit the relativistic Euler equation (30) [4], Henceforth we impose a restriction that the fluid is pressureless, which basically implies a free system [4]. Comparing (21) with the conventional ideal fluid energy momentum tensor, the expression for (relativistic) pressure turns out to be, The subscript rel indicates relativistic variables. Hence, the pressureless condition implies, with k a numerical constant. The relativistic Euler equation (30) for a pressureless fluid simplifies to, For ν = i we have, Comparing this with the non-relativistic Euler equation in terms of the usual fluid variables, we note that for with u + a constant, the two equations (56) and (57) will become identical. The ν = − component of (55) will be trivially satisfied for a constant u + = u − . Now, for ν = +, (55) reduces to which can be identified with the Euler equation provided we define u + in the following way: This identification brings in two new constants γ and δ. Let us now exploit the relativistic identity, to determine the constants. Substituting the expressions for u µ we arrive at Thus with a one-parameter (u + ) arbitrariness the mapping becomes, We have successfully given a complete mapping between the relativistic and the non-relativistic variables as promised. This mapping agrees with the one suggested by [7] once the zero-pressure condition is imposed. Hence, as promised, we have provided a complete dynamical framework to study non-relativistic fluid in light-cone coordinates.

Conclusion and future prospects
Fluid dynamics has generally been considered as an applied science but there has been a paradigm shift in modern physics perspective where deep theoretical aspects of the theory are being studied in the context of Fluid/Gravity correspondence, conformal symmetry of non-relativistic fluid dynamics, etc.A stepping stone in this direction would be to study fluid dynamics from a modern field theory point of view. This has been the motivation of the present paper.We have discussed in detail ideal fluid dynamic models, principally using a hamiltonian formalism. The Clebsch parametrization plays an essential role in our framework where the fluid turns out to be a second class constraint system. Next we reconsider the fluid model in light-cone coordinate system, qualitatively different from the equal-time coordinate system considered earlier. In both equal-time as well as light-cone formulation we have shown the validity of the Schwinger condition, a hallmark of any realtivistic field theory. The Schwinger condition involves the computation of the algebra of the stress tensor components. Since the fluid is a constrained system, it is essential to use Dirac brackets to calculate this algebra.
Next, we have discussed the non-relativistic reduction of the free relativistic fluid. A dynamical approach was adopted in the light-cone formulation. The fundamental relativistic fluid equationsthe conservation law and the Euler's equation-were reduced to their non-relativistic versions by compactifying a light-cone coordinate [7,10]. This naturally led to a mapping among the variables in the two sectors that reproduced earlier findings [7] based on an algebraic treatment.
There are diverse channels along which further work can be pursued. We have studied a pressureless fluid but it will be interesting to see the effect of relaxing this condition. Effectively this implies the introduction of interactions. It will be worthwhile to generalize our analysis for viscous fluids.