Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics
aa r X i v : . [ phy s i c s . p l a s m - ph ] A p r Under consideration for publication in J. Plasma Phys. Lagrangian and Dirac constraints for the idealincompressible fluid and magnetohydrodynamics
P. J. Morrison † , T. Andreussi, and F. Pegoraro Department of Physics and Institute for Fusion Studies, The University of Texas at Austin, Austin, TX78712-1060, USA SITAEL S.p.A., Pisa, 56121, Italy Dipartimento di Fisica E. Fermi, Pisa, 56127, Italy(Received xx; revised xx; accepted xx)
The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called La-grangian variable description using his method of multipliers in the Lagrangian (variational)formulation. An alternative is the imposition of incompressibility in the Eulerian variable de-scription by a generalization of Dirac’s constraint method using noncanonical Poisson brackets.Here it is shown how to impose the incompressibility constraint using Dirac’s method in termsof both the canonical Poisson brackets in the Lagrangian variable description and the noncanon-ical Poisson brackets in the Eulerian description, allowing for the advection of density. Bothcases give dynamics of infinite-dimensional geodesic flow on the group of volume preservingdi ff eomorphisms and explicit expressions for this dynamics in terms of the constraints andoriginal variables is given. Because Lagrangian and Eulerian conservation laws are not identical,comparison of the various methods is made.
1. Introduction
Background
Sometimes constraints are maintained e ff ortlessly, an example being ∇ · B = ∇ · v = ff erent methods for imposing the compress-ibility constraint in ideal (dissipation-free) fluid mechanics and its extension to magnetohydro-dynamics (MHD), classical field theories intended for classical purposes. This endeavor is richerthan might be expected because the di ff erent methods of constraint can be applied to the fluidin either the Lagrangian (material) description or the Eulerian (spatial) description, and the con-straint methods have di ff erent manifestations in the Lagrangian (action principle) and Hamilto-nian formulations. Although Lagrange’s multiplier is widely appreciated, it is not so well knownthat he used it long ago for imposing the incompressibility constraint for a fluid in the Lagrangianvariable description (Lagrange 1788). More recently, Dirac’s method was applied for imposingincompressibility within the Eulerian variable description, first in Nguyen & Turski (1999, 2001)and followed up in several works (Morrison et al. et al. et al. † Email address for correspondence: [email protected]
P. J. Morrison, T. Andreussi, and F. Pegoraro t preserves volume, and he did thisby the method of Lagrange multipliers. It is worth noting that Lagrange knew the Lagrangemultiplier turns out to be the pressure, but he had trouble solving for it. Lagrange’s calculationwas placed in a geometrical setting by Arnold (1966) (see also Appendix 2 of Arnold (1978)),where the constrained maps from the initial conditions were first referred to as volume preserv-ing di ff eomorphisms in this context. Given this background, in our investigation of the threedichotomies described above we emphasize geodesic flow.For later use we record here the incompressible Euler equations for the case with constantdensity and the case where density is advected. The equations of motion, allowing for densityadvection, are given by ∂ v ∂ t = − v · ∇ v − ρ ∇ p , (1.1) ∇ · v = , (1.2) ∂ρ∂ t = − v · ∇ ρ , (1.3)where v ( x , t ) is the velocity field, ρ ( x , t ) is the mass density, p ( x , t ) is the pressure, and x ∈ D ,the region occupied by the fluid. These equations are generally subject to the free-slip boundarycondition n · v | ∂ D =
0, where n is normal to the boundary of D . The pressure field that enforcesthe constraint (1.2) is obtained by setting ∂ ( ∇ · v ) ∂ t =
0, which implies ∆ ρ p : = ∇ · ρ ∇ p ! = −∇ · ( v · ∇ v ) . (1.4)For reasonable assumptions on ρ and boundary conditions, (1.4) is a well-posed elliptic equation(see e.g. Evans 2010), so we can write p = − ∆ − ρ ∇ · ( v · ∇ v ) . (1.5)For the case where ρ is constant we have the usual Green’s function expression p ( x , t ) = − ρ Z d x ′ G ( x , x ′ ) ∇ ′ · ( v ′ · ∇ ′ v ′ ) , (1.6)where G is consistent with Neumann boundary conditions (Orszag et al. v ′ = v ( x ′ , t ).Insertion of (1.6) into (1.1) gives ∂ v ∂ t = − v · ∇ v + ∇ Z d x ′ G ( x , x ′ ) ∇ ′ · ( v ′ · ∇ ′ v ′ ) , (1.7)which is a closed system for v ( x , t ).For MHD, equation (1.1) has the additional term ( ∇ × B ) × B /ρ added to the righthand side.Consequently for this model, the source of (1.5) is modified by the addition of this term to − v ·∇ v . agrangian and Dirac constraints for fluid and MHD Overview
Section 2 contains material that serves as a guide for navigating the more complicated calcu-lations to follow. We first consider the various approaches to constraints in the finite-dimensionalcontext in Sections 2.1–2.3. Section 2.1 briefly covers conventional material about holonomicconstraints by Lagrange multipliers – here the reader is reminded how the free particle withholonomic constraints amounts to geodesic flow. Section 2.2 begins with the phase space actionprinciple, whence the Dirac bracket for constraints is obtained by Lagrange’s multiplier method,but with phase space constraints as opposed to the usual holonomic configuration space con-straints used in conjunction with Hamilton’s principle of mechanics, as described in Section 2.1.Next, in Section 2.3, we compare the results of Sections 2.1 and 2.3 and show how conventionalholonomic constraints can be enforced by Dirac’s method. Contrary to Lagrange’s method,here we obtain explicit expressions, ones that do not appear in conventional treatments, for theChristo ff el symbol and the normal force entirely in terms of the original Euclidean coordinatesand constraints. Section 2 is completed with Section 2.4, where the previous ideas are revisitedin the d + ff erent manifestationsof constraints in our dichotomies.Section 4 begins with Section 4.1 that reviews Lagrange’s original calculations. Because theincompressibility constraint he imposes is holonomic and there are no additional forces, his equa-tions describe infinite-dimensional geodesics flow on volume preserving maps. The remainingportion of this section contains the most substantial calculations of the paper. In Section 4.2 forthe first time Dirac’s theory is applied to enforce incompressibility in the Lagrangian variabledescription. This results in a new Dirac bracket that generates volume preserving flows. Asin Section 2.3.1, which serves as a guide, the equations of motion generated by the bracketare explicit and contain only the constraints and original variables. Next, in Section 4.3, areduction from Lagrangian to Eulerian variables is made, resulting in a new Eulerian variablePoisson bracket that allows for density advection while preserving incompressibility. This wasan heretofore unsolved problem. Section 4.4 ties together the results of Sections 4.2, 4.3, and3.4. Here both the Eulerian and Lagrangian Dirac constraint theories are compared after theyare evaluated on their respective constraints, simplifying their equations of motion. BecauseLagrangian and Eulerian conservation laws are not identical, we see that there are di ff erences.Section 4 concludes in Section 4.5 with a discussion of the full algebra of invariants, that of theten parameter Galilean group, for both the Lagrangian and Eulerian descriptions. In addition theCasimir invariants of the theories are discussed.The paper concludes with Section 5, where we briefly summarize our results and speculateabout future possibilities. P. J. Morrison, T. Andreussi, and F. Pegoraro
2. Constraint methods
Holonomic constraints by Lagrange’s multiplier method
Of interest are systems with Lagrangians of the form L ( ˙ q , q ) where the overdot denotes timedi ff erentiation and q = ( q , q , . . . , q N ). Because nonautonomous systems could be included byappending an additional degree of freedom, explicit time dependence is not included in L .Given the Lagrangian, the equations of motion are obtained according to Hamilton’s principleby variation of the action S [ q ] = Z t t dt L ( ˙ q , q ) ; (2.1)i.e. δ S [ q ; δ q ] : = dd ǫ S [ q + ǫδ q ] (cid:12)(cid:12)(cid:12)(cid:12) ǫ = = Z t t dt ddt ∂ L ∂ ˙ q i − ∂ L ∂ q i ! δ q i = Z t t dt δ S [ q ] δ q i ( t ) δ q i = , (2.2)for all variations δ q ( t ) satisfying δ q ( t ) = δ q ( t ) =
0, implies Lagrange’s equations of motion,i.e. δ S [ q ] δ q i ( t ) = ⇒ ddt ∂ L ∂ ˙ q i − ∂ L ∂ q i = , i = , , . . . , N . (2.3)Holonomic constraints are real-valued functions of the form C A ( q ), A = , , . . . , M , whichare desired to be constant on trajectories. Lagrange’s method for implementing such constraintsis to add them to the action and vary as follows: δ S λ : = δ ( S + λ A C A ) = , (2.4)yielding the equations of motion δ S λ [ q ] δ q i ( t ) = ⇒ ddt ∂ L ∂ ˙ q i − ∂ L ∂ q i = λ A ∂ C A ∂ q i , i = , , . . . , N , (2.5)with the forces of constraint residing on the right-hand side of (2.5). Observe in (2.4) and (2.5)repeated sum notation is implied for the index A . The N equations of (2.5) with the M numericalvalues of the constraints C A ( q ) = C A , determine the N + M unknowns { q i } and { λ A } . In practice,because solving for the Lagrange multipliers can be di ffi cult an alternative procedure, a exampleof which we describe in Section 2.1.1, is used.We will see in Section 4.1 that the field theoretic version of this method is how Lagrangeimplemented the incompressiblity constraint for fluid flow. For the purpose of illustration andin preparation for later development, we consider a finite-dimensional analog of Lagrange’streatment.2.1.1. Holonomic constraints and geodesic flow via Lagrange
Consider N noninteracting bodies each of mass m i in the Eucledian configuration space E N with cartesian coordinates q i = ( q xi , q yi , q zi ), where as in Section 2.1 i = , , . . . , N , but ourconfiguration space has dimension 3 N . The Lagrangian for this system is given by the usualkinetic energy, L = N X i = m i q i · ˙ q i , (2.6)with the usual “dot” product. The Euler-Lagrange equations for this system, equations (2.3), arethe uninteresting system of N free particles. As in Section 2.1 we constrain this system by addingconstraints C A ( q , q , . . . , q N ), where again A = , , . . . , M , leading to the equations m i ¨ q i = λ A ∂ C A ∂ q i . (2.7) agrangian and Dirac constraints for fluid and MHD N equations of (2.7) together with the M numerical values of theconstraints, in order to determine the unknowns q i and λ A , we recall the alternative procedure,which dates back to Lagrange (see e.g. Sec. IV of Lagrange (1788)) and has been taught tophysics students for generations (see e.g. Whittaker 1917; Corben & Stehle 1960). With thealternative procedure one introduces generalized coordinates that account for the constraints,yielding a smaller system on the constraint manifold, one with the Lagrangian L = g µν ( q ) ˙ q µ ˙ q ν , µ, ν = , . . . , N − M , (2.8)where g µν = N X i = m i ∂ q i ∂ q µ · ∂ q i ∂ q ν . (2.9)Then Lagrange’s equations (2.3) for the Lagrangian (2.8) are the usual equations for geodesicflow ¨ q µ + Γ µαβ ˙ q α ˙ q β = , (2.10)where the Christo ff el symbol is as usual Γ µαβ = g µν g να ∂ q β + g νβ ∂ q α − g αβ ∂ q ν ! . (2.11)If the constraints had time dependence, then the procedure would have produced the Coriolis andcentripetal forces, as is usually done in textbooks.Thus, we arrive at the conclusion that free particle dynamics with time-independent holonomicconstraints is geodesic flow. 2.2. Dirac’s bracket method
So, a natural question to ask is “How does one implement constraints in the Hamiltoniansetting, where phase space constraints depend on both the configuration space coordinate q andthe canonical momentum p ”? (See e.g. Sundermeyer (1982); Arnold et al. (1980) for a generaltreatment and Dermaret & Moncrief (1980) for a treatment in the context of the ideal fluid andrelativity and a selection of earlier references.) To this end we begin with the phase space actionprinciple S [ q , p ] = Z t t dt h p i ˙ q i − H ( q , p ) i , (2.12)where again repeated sum notation is used for i = , , . . . , N . Independent variation of S [ q , p ]with respect to q and p , with δ q ( t ) = δ q ( t ) = δ p , yields Hamilton’sequations, ˙ p i = − ∂ H ∂ q i and ˙ q i = ∂ H ∂ p i , (2.13)or equivalently ˙ z α = { z α , H } , (2.14)which is a rewrite of (2.13) in terms of the Poisson bracket on phase space functions f and g , { f , g } = ∂ f ∂ q i ∂ g ∂ p i − ∂ g ∂ q i ∂ f ∂ p i = ∂ f ∂ z α J αβ c ∂ g ∂ z β , (2.15)where in the second equality we have used z = ( q , p ), so α, β = , , . . . , N and the cosymplecticform (Poisson matrix) is J c = O N I N − I N O N ! , (2.16) P. J. Morrison, T. Andreussi, and F. Pegoraro with O N being an N × N block of zeros and I N being the N × N identity.Proceeding as in Section 2.1, albeit with phase space constraints D a ( q , p ), a = , , . . . , M < N , we vary S λ [ q , p ] = Z t t dt h p i ˙ q i − H ( q , p ) + λ a D a i , (2.17)and obtain ˙ p i = − ∂ H ∂ q i + λ a ∂ D a ∂ q i and ˙ q i = ∂ H ∂ p i − λ a ∂ D a ∂ p i . (2.18)Next, enforcing ˙ D a = a , will ensure that the constraints stay put. Whence, di ff erentiatingthe D a and using (2.18) yields ˙ D a = ∂ D a ∂ q i ˙ q i + ∂ D a ∂ p i ˙ p i = { D a , H } − λ b { D a , D b } ≡ . (2.19)We assume D ab : = { D a , D b } has an inverse, D − ab , which requires there be an even number ofconstraints, a , b = , , . . . , M , because odd antisymmetric matrices have zero determinant.Then upon solving (2.19) for λ b and inserting the result into (2.18) gives˙ z α = { z α , H } − D − ab { z α , D a }{ D b , H } . (2.20)From (2.20), we obtain a generalization of the Poisson bracket, the Dirac bracket, { f , g } ∗ = { f , g } − D − ab { f , D a }{ D b , g } . (2.21)which has the degeneracy property { f , D a } ∗ ≡ . (2.22)for all functions f and indices a = , , . . . , M .The generation of the equations of motion via a Dirac bracket, i.e.˙ z α = { z α , H } ∗ , (2.23)which is equivalent to (2.20), has the advantage that the Lagrange multipliers λ A have beeneliminated from the theory.Note, although the above construction of the Dirac bracket is based on the canonical bracketof (2.15), his construction results in a valid Poisson bracket if one starts from any valid Poissonbracket (cf. (2.76) of Section 2.4 and Section 3.3), which need not have a Poisson matrix ofthe form of (2.16) (see e.g. Morrison et al. et al. (1976); Sundermeyer (1982) for treatment of theseconcepts. 2.3. Holonomic constraints by Dirac’s bracket method
A connection between the approaches of Lagrange and Dirac can be made. From a setof Lagrangian constraints C A ( q ), where A = , , . . . , M , one can construct an additional M constraints by di ff erentiation, ˙ C A = ∂ C A ∂ q i ˙ q i = ∂ C A ∂ q i ∂ H ∂ p i , (2.24) agrangian and Dirac constraints for fluid and MHD D a ( q , p ) = (cid:16) C A ( q ) , ˙ C A ′ ( q , p ) (cid:17) , (2.25)where A = , , . . . , M , A ′ = M + , M + , . . . , M , and a = , , . . . , M . With the constraints of (2.25) the bracket D ab = { D a , D b } needed to construct (2.21) is easilyobtained, D = O M { C A , ˙ C B ′ }{ ˙ C A ′ , C B } { ˙ C A ′ , ˙ C B ′ } ! = : O M S − S A ! , (2.26)where O M is an M × M block of zeros and S is the following M × M symmetric matrix withelements S AB : = { C A , ˙ C B } = ∂ H ∂ p i ∂ p j ∂ C A ∂ q i ∂ C B ∂ q j , (2.27)and A is the following M × M antisymmetric matrix with elements A AB : = { ˙ C A ′ , ˙ C B ′ } (2.28) = ∂ H ∂ p i ∂ p k " ∂ H ∂ q i ∂ p j ∂ C A ∂ q j ∂ C B ∂ q k − ∂ C B ∂ q j ∂ C A ∂ q k ! + ∂ H ∂ p j ∂ C A ∂ q i ∂ q j ∂ C B ∂ q k − ∂ C B ∂ q i ∂ q j ∂ C A ∂ q k ! . Assuming the existence of D − , the 2 M × M inverse of (2.26), the Dirac bracket of (2.21)can be constructed. A necessary and su ffi cient condition for the existence of this inverse is thatdet S ,
0, and when this is the case the inverse is given by D − = S − AS − − S − S − O M . (2.29)Because of the block diagonal structure of (2.29), the Dirac bracket (2.21) becomes { f , g } ∗ = { f , g } + S − AB (cid:16) { f , C A }{ ˙ C B , g } − { g , C A }{ ˙ C B , f } (cid:17) + S − AC A CD S − DB { f , C A }{ C B , g } , (2.30)which has the form { f , g } ∗ = { f , g } − ( P ⊥ ) ij ∂ f ∂ q i ∂ g ∂ p j − ∂ g ∂ q i ∂ f ∂ p j ! + Q i j ∂ f ∂ p i ∂ g ∂ p j = ∂ f ∂ q i P ij ∂ g ∂ p j − ∂ g ∂ q i P ij ∂ f ∂ p j + Q i j ∂ f ∂ p i ∂ g ∂ p j , (2.31)where the matrices P = I − P ⊥ , with( P ⊥ ) ij = S − AB ∂ H ∂ p i ∂ p k ∂ C A ∂ q j ∂ C B ∂ q k , (2.32)and Q , a complicated expression that we will not record, are crafted using the constraints andHamiltonian so as to make { f , g } ∗ preserve the constraints.The equations of motion that follow from (2.30) are˙ q ℓ = { q ℓ , H } ∗ = ∂ H ∂ p ℓ + S − AB (cid:16) { q ℓ , C A }{ ˙ C B , H } − { H , C A }{ ˙ C B , q ℓ } (cid:17) + S − AC A CD S − DB { q ℓ , C A }{ C B , H } , (2.33)˙ p ℓ = { p ℓ , H } ∗ = − ∂ H ∂ q ℓ + S − AB (cid:16) { p ℓ , C A }{ ˙ C B , H } − { H , C A }{ ˙ C B , p ℓ } (cid:17) + S − AC A CD S − DB { p ℓ , C A }{ C B , H } . (2.34) P. J. Morrison, T. Andreussi, and F. Pegoraro
Given the Dirac bracket associated with the D of (2.27), dynamics that enforces the con-straints takes the form of (2.23). Any system generated by this bracket will enforce Lagrange’sholonomic constraints; however, only initial conditions compatible with D a ≡ , ∀ a = M + , M + , . . . , M , (2.35)or equivalently ˙ C A = ∂ C A ∂ q i ∂ H ∂ p i = { C A , H } ≡ , ∀ A = , , . . . , M , (2.36)will correspond to the system with holonomic constraints. Using (2.36) and { q ℓ , C A } ≡
0, (2.33)and (2.34) reduce to ˙ q ℓ = { q ℓ , H } ∗ = ∂ H ∂ p ℓ , (2.37)˙ p ℓ = { p ℓ , H } ∗ = − ∂ H ∂ q ℓ + S − AB { p ℓ , C A }{ ˙ C B , H } , (2.38)where { ˙ C B , H } = ∂ H ∂ q i ∂ p j ∂ C B ∂ q j + ∂ H ∂ p j ∂ C B ∂ q i q j ! ∂ H ∂ p i − ∂ C B ∂ q i ∂ H ∂ p i ∂ p j ∂ H ∂ q j . (2.39)Thus the Dirac bracket approach gives a relatively simple system for enforcing holonomicconstraints. It can be shown directly that if initially ˙ C A vanishes, then the system of (2.37) and(2.38) will keep it so for all time.2.3.1. Holonomic constraints and geodesic flow via Dirac
Let us now consider again the geodesic flow problem of Section 2.1.1: the N degree-of-freedom free particle system with holonomic constraints, but this time within the frameworkof Dirac bracket theory. For this problem the unconstrained configuration space is the Euclideanspace E N and we will denote by Q the constraint manifold within E N defined by the constancyof the constraints C A .The Lagrangian of (2.6) is easily Legendre transformed to the free particle Hamiltonian H = N X i = m i p i · p i , (2.40)where p i = m i ˙ q i . For this example the constraints of (2.25) take the form D a = C A ( q , q , . . . , q N ) , ∂ C A ′ ( q , q , . . . , q N ) ∂ q i · p i m i ! , (2.41)the M × M matrix S has elements S AB = N X i = m i ∂ C A ∂ q i · ∂ C B ∂ q i , (2.42)and the M × M matrix A is A AB = N X i , j = m i m j p i · " ∂ C A ∂ q i ∂ q j · ∂ C B ∂ q j − ∂ C B ∂ q i ∂ q j · ∂ C A ∂ q j . (2.43)The Dirac bracket analogous to (2.31) is { f , g } ∗ = N X i j = " ∂ f ∂ q i · ↔ P i j · ∂ g ∂ p j − ∂ g ∂ q i · ↔ P i j · ∂ f ∂ p j + ∂ f ∂ p i · ↔ Q i j · ∂ g ∂ p j , (2.44) agrangian and Dirac constraints for fluid and MHD ↔ P i j = ↔ I i j − ↔ P ⊥ i j with the tensors ↔ P ⊥ i j : = N X k = S − AB ∂ H ∂ p i ∂ p k · ∂ C B ∂ q k ∂ C A ∂ q j = S − AB m i ∂ C B ∂ q i ∂ C A ∂ q j , (2.45) ↔ Q i j : = N X k = S − AB " ∂ C A ∂ q j p k m k · ∂ C B ∂ q k ∂ q i − ∂ C A ∂ q i p k m k · ∂ C B ∂ q k ∂ q j + S − AC A CD S − DB ∂ C A ∂ q i ∂ C B ∂ q j (2.46) = : ↔ T i j − ↔ T ji + ↔ A i j , (2.47)where ↔ A i j is the term with S − AC A CD S − DB . Observe ↔ A i j = − ↔ A ji because A CD = − A DC and N X k = ↔ P ⊥ ik · ↔ P ⊥ k j = N X k = S − AB m i ∂ C B ∂ q i ∂ C A ∂ q k ! · S − A ′ B ′ m k ∂ C B ′ ∂ q k ∂ C A ′ ∂ q j ! = S − AB m i ∂ C B ∂ q i ! S − A ′ B ′ N X k = ∂ C A ∂ q k · m k ∂ C B ′ ∂ q k ∂ C A ′ ∂ q j = S − AB m i ∂ C B ∂ q i ! S − A ′ B ′ S AB ′ ∂ C A ′ ∂ q j = ↔ P ⊥ i j . (2.48)Also observe for the Hamiltonian of (2.40) N X j = ↔ P ⊥ i j · ∂ H ∂ p j = N X j = ↔ P ⊥ i j · p j m j ≡ , (2.49) N X j = ∂ H ∂ p j · ↔ T i j = N X j = p j m j · ↔ T i j ≡ , (2.50) N X j = ↔ A i j · ∂ H ∂ p j = − N X j = ↔ A ji · ∂ H ∂ p j ≡ , (2.51)when evaluated on the constraint ˙ C A , B =
0, while N X j = ∂ H ∂ p j · ↔ P ⊥ ji , N X i = ∂ H ∂ p i · ↔ T i j , , (2.52)when evaluated on the constraint ˙ C A , B =
0. Thus, the bracket of (2.44) yields the equations ofmotion ˙ q i = { q i , H } ∗ = ∂ H ∂ p i = p i m i , (2.53)˙ p i = − ∂ C A ∂ q i S − AB N X j , k = p j m j · ∂ C B ∂ q j q k · p k m k , (2.54)or ¨ q i = − m i ∂ C A ∂ q i S − AB N X j , k = ˙ q j · ∂ C B ∂ q j q k · ˙ q k = − N X j , k = ˙ q j · b Γ i , jk · ˙ q k , (2.55)where b Γ i , jk : = m i ∂ C A ∂ q i ⊗ S − AB ∂ C B ∂ q j q k , (2.56)0 P. J. Morrison, T. Andreussi, and F. Pegoraro is used to represent the normal force.Observe, (2.55) has two essential features: as noted, its righthand side is a normal force thatprojects to the constraint manifold defined by the constraints C A and within the constraint man-ifold it describes a geodesic flow, all done in terms of the original Euclidean space coordinateswhere the initial conditions place the flow on Q by setting the values C A for all A = , , . . . , M .We will show this explicitly.First, because the components of vectors normal to Q are given by ∂ C A /∂ q i for A = , , . . . , M , this prefactor on the righthand side of (2.55) projects as expected. Upon comparing(2.55) with (2.7) we conclude that the coe ffi cient of this prefactor must be the Lagrangemultipliers, i.e., λ A = − S − AB N X k , j = ˙ q j · ∂ C B ∂ q j q k · ˙ q k . (2.57)Thus, we see that Dirac’s procedure explicitly solves for the Lagrange multiplier.Second, to uncover the geodesic flow we can proceed as usual by projecting explicitly onto Q . To this end we consider the transformation between the Euclidean configuration space E N coordinates ( q , q , . . . , q i , . . . , q N ) , where i = , , . . . , N (2.58)and another set of coordinates( q , q , . . . , q a , . . . , q N ) , where a = , , . . . , N , (2.59)which we tailor as follows:( q , q , . . . , q α . . . , q n , C , C , . . . , C A , . . . , C M ) , (2.60)where α = , , . . . , n , A = , , . . . , M , and n = N − M . Here we have chosen q n + A = C A and n is the actual number of degrees of freedom on the constraint surface Q . We can freely transformback and forth between the two coordinates, i.e.( q , q , . . . , q i , . . . , q N ) ←→ ( q , q , . . . , q a , . . . , q N ) . (2.61)Note, the choice q n + A = C A could be replaced by q n + A = f A ( C , C , . . . , C M ) for arbitraryindependent f A , but we assume the original C A are optimal. Because q α are coordinates within Q , tangent vectors to Q have the components ∂ q i /∂ q α , and there is one for each α = , , . . . , n .The pairing of the normals with tangents is expressed by N X i = ∂ q i ∂ q α · ∂ C A ∂ q i = , α = , , . . . , n ; A = , , . . . , M . (2.62)Let us now consider an alternative procedure that the Dirac constraint method provides. Proceed-ing directly we calculate ˙ q a = N X i = ∂ q a ∂ q i · ˙ q i . (2.63)Observe that on E N the matrix ∂ q a /∂ q i is invertible and the full metric tensor and its inverse inthe new coordinates are given as follows: g ab = N X i = m i ∂ q a ∂ q i · ∂ q b ∂ q i and g ab = N X i = m i ∂ q i ∂ q a · ∂ q i ∂ q b . (2.64)The metric tensor on Q of (2.9) is obtained by restricting g ab to a , b n and g αβ is obtained by agrangian and Dirac constraints for fluid and MHD g αβ and not by restricting g ab . Proceeding by di ff erentiating again we obtain¨ q a = N X i = ∂ q a ∂ q i · ¨ q i + N X i , j = ˙ q i · ∂ q a ∂ q i ∂ q j · ˙ q j , a = , , . . . , N . (2.65)Now inserting (2.55) into (2.65) gives¨ q a = − N X i = m i ∂ q a ∂ q i · ∂ C A ∂ q i g AB N X j , k = ˙ q j · ∂ C B ∂ q j q k · ˙ q k + N X i , j = ˙ q j · ∂ q a ∂ q i ∂ q j · ˙ q i , (2.66)where we have recognized that g AB = S AB = N X i = m i ∂ C A ∂ q i · ∂ C B ∂ q i (2.67)and, as was necessary for the workability of the Dirac bracket constraint theory, g AB = S − AB mustexist. This quantity is obtained by inverting S AB and not by restricting g ab .Equation (2.66) is an expression for the full system on E N . However, for a > n , we know¨ q a = ¨ C A =
0, so the two terms of (2.66) should cancel. To see this, in the first term of (2.66) weset q a = C C and observe that this first term becomes − N X i = m i ∂ C C ∂ q i · ∂ C A ∂ q i g AB N X j , k = ˙ q j · ∂ C B ∂ q j q k · ˙ q k = − g CA g AB N X j , k = ˙ q j · ∂ C B ∂ q j q k · ˙ q k = − N X j , k = ˙ q j · ∂ C C ∂ q j q k · ˙ q k . (2.68)Now, for a n , say α , the righthand side gives a Christo ff el symbol expression for the geodesicflow; viz. ¨ q α = − N X i = m i ∂ q α ∂ q i · ∂ C A ∂ q i g AB N X j , k = ˙ q j · ∂ C B ∂ q j q k · ˙ q k + N X j , k = ˙ q j · ∂ q α ∂ q j ∂ q k · ˙ q k = − Γ αµν ˙ q µ ˙ q ν , (2.69)where Γ αµν = N X i = m i ∂ q α ∂ q i · ∂ C A ∂ q i S − AB N X j , k = ∂ q j ∂ q µ · ∂ C B ∂ q j q k · ∂ q k ∂ q ν + N X j , k = ∂ q j ∂ q µ · ∂ q α ∂ q j ∂ q k · ∂ q k ∂ q ν (2.70)is an expression for the Christo ff el symbol in terms of the original Euclidean coordinates, theconstraints, and the choice of coordinates on Q .Using (2.70) one can calculate an analogous expression for the Riemann curvature tensor on Q from the usual expression R αβγδ = ∂Γ αδβ ∂ q γ − ∂Γ αγβ ∂ q δ + Γ αγλ Γ λδβ − Γ αδλ Γ λγβ , (2.71)using ∂/∂ q γ = P i ( ∂ q i /∂ q γ ) · ∂/∂ q i . This gives the curvature written in terms of the originalEuclidean coordinates, the constraints, and the chosen coordinates on Q .2.4. d + field theory The techniques of Sections 2.1, 2.2, and 2.3 have natural extensions to field theory.Given independent field variables Ψ A ( µ, t ), indexed by A = , , . . . , ℓ , where the independent2 P. J. Morrison, T. Andreussi, and F. Pegoraro variable µ = ( µ , µ , . . . , µ d ). The field theoretic version of Hamilton’s principle of (2.1) isembodied in the action S [ Ψ ] = Z t t dt L [ Ψ, ˙ Ψ ] , with L [ Ψ, ˙ Ψ ] = Z d d µ L ( Ψ, ˙ Ψ , ∂Ψ ) , (2.72)where we leave the domain of µ and the boundary conditions unspecified, but freely drop surfaceterms obtained upon integration by parts. The Lagrangian density L is assumed to depend on thefield components Ψ and ∂Ψ , which is used to indicate all possible partial derivatives with respectof the components of µ . Hamilton’s principle with (2.72) gives the Euler-Lagrange equations, δ S [ Ψ ] δΨ A ( µ, t ) = ⇒ ddt ∂ L ∂ ˙ Ψ A + ∂∂µ ∂ L ∂∂Ψ A − ∂ L ∂Ψ A = , (2.73)where the overdot implies di ff erentiation at constant µ . Local holonomic constraints C A ( Ψ, ∂Ψ )are enforced by Lagrange’s method by amending the Lagrangian L λ [ Ψ, ˙ Ψ ] = Z d d µ (cid:0) L ( Ψ, ˙ Ψ , ∂Ψ ) + λ A C A ( Ψ, ∂Ψ ) (cid:1) , (2.74)with again A = , , . . . , M and proceeding as in the finite-dimensional case.In the Hamiltonian field theoretic setting, we could introduce the conjugate momentum densi-ties π A , A = , , . . . , ℓ , with the phase space action S λ [ Ψ, π ] = Z t t dt Z d d µ h π A ˙ Ψ A − H + λ A D A i , (2.75)with Hamiltonian density H and local constraints D a depending on the values of the fields andtheir conjugates. Instead of following this route we will jump directly to a generalization of thefield theoretic Dirac bracket formalism that would result.Consider a Poisson algebra composed of functionals of field variables χ A ( µ, t ) with a Poissonbracket of the form { F , G } = Z d d µ F χ · J ( χ ) · G χ , (2.76)where F χ is a shorthand for the functional derivative of a functional F with respect to the field χ (see e.g. Morrison 1998) and F χ · J · G χ = F χ A J AB G χ B , again with repeated indices summed.Observe the fields χ A ( µ, t ) need not separate into coordinates and momenta, but if they dothe Poisson operator J has a form akin to that of (2.16). By a Poisson algebra we mean a Liealgebra realization on functionals, meaning the Poisson bracket is bilinear, antisymmetric, andsatisfies the Jacobi identity and that there is an associative product of functionals that satisfiesthe Leibniz law. From the Poisson bracket the equations of motion are given by ˙ χ = { χ, H } , forsome Hamiltonian functional H [ χ ].Dirac’s constraint theory is generally implemented in terms of canonical Poisson brackets(see e.g. Dirac 1950; Sudarshan & Makunda 1974; Sundermeyer 1982), but it is not di ffi cult toshow that his procedure also works for noncanonical Poisson brackets (see e.g. an Appendix ofMorrison et al. D a ( µ ) = constant, ashorthand for D a [ χ ( µ )], with the index a = , , . . . , M , bearing in mind that they depend onthe fields χ and their derivatives. As in the finite-dimensional case, the Dirac bracket is obtainedfrom the matrix D obtained from the bracket of the constraints, D ab ( µ, µ ′ ) = { D a ( µ ) , D b ( µ ′ ) } , where we note that D ab ( µ, µ ′ ) = − D ba ( µ ′ , µ ). If D has an inverse, then the Dirac bracket is defined agrangian and Dirac constraints for fluid and MHD { F , G } ∗ = { F , G } − Z d d µ Z d d µ ′ { F , D a ( µ ) } D − ab ( µ, µ ′ ) { D b ( µ ′ ) , G } , (2.77)where the coe ffi cients D − ab ( µ, µ ′ ) satisfy Z d d µ ′ D − ab ( µ, µ ′ ) D bc ( µ ′ , µ ′′ ) = Z d µ ′ D cb ( µ, µ ′ ) D − ba ( µ ′ , µ ′′ ) = δ ca δ ( µ − µ ′′ ) , consistent with D − ab ( µ, µ ′ ) = − D − ba ( µ ′ , µ ).We note, the procedure is e ff ective only when the coe ffi cients D − ab ( µ, µ ′ ) can be found. If D isnot invertible, then one needs, in general, secondary constraints to determine the Dirac bracket.2.4.1. Field theoretic geodesic flow
In light of Section 2.1.1, any field theory with a Lagrangian density of the form L =
12 ˙ Ψ A ( µ, t ) η AB ˙ Ψ B ( µ, t ) , (2.78)with the metric η AB = δ AB being the Kronecker delta, subject to time-independent holonomicconstraints can be viewed as geodesic flow on the constraint surface. This is a natural infinite-dimensional generalization of the idea of Section 2.1.1.
3. Unconstrained Hamiltonian and action for fluid
Fluid action in Lagrangian variable description
The Lagrangian variable description of a fluid is described in Lagrange’s famous work(Lagrange 1788), while historic and additional material can be found in Serrin (1959); Newcomb(1962); Van Kampen & Felderhof (1967); Morrison (1998). Because the Lagrangian descriptiontreats a fluid as a continuum of particles, it naturally is amenable to the Hamiltonian form. TheLagrangian variable is a coordinate that gives the position of a fluid element or parcel, as itis sometimes called, at time t . We denote this variable by q = q ( a , t ) = ( q , q , q ), which ismeasured relative to some cartesian coordinate system. Here a = ( a , a , a ) denotes the fluidelement label, which is often defined to be the position of the fluid element at the initial time, a = q ( a , a is a continuum analog of thediscrete index that labels a generalized coordinate in a finite degree-of-freedom system. If D is adomain that is fully occupied by the fluid, then at each fixed time t , q : D → D is assumed to be1-1 and onto. Not much is really known about the mathematical properties of this function, butwe will assume that it is as smooth as it needs to be for the operations performed. Also, we willassume we can freely integrate by parts dropping surface terms and drop reference to D in ourintegrals.When discussing the ideal fluid and MHD we will use repeated sum notation with upper andlower indices even though we are working in cartesian coordinates. And, unlike in Section 2,latin indices, i , j , k , ℓ etc. will be summed over 1,2, and 3, the cartesian components, rather thanto N . This is done to avoid further proliferation of indices and we trust confusion will not arisebecause of context.Important quantities are the deformation matrix, ∂ q i /∂ a j and its Jacobian determinant J : = det( ∂ q i /∂ a j ), which is given by J = ǫ k j ℓ ǫ imn ∂ q k ∂ a i ∂ q j ∂ a m ∂ q ℓ ∂ a n , (3.1)where ǫ i jk = ǫ i jk is the purely antisymmetric (Levi-Civita) tensor density. We assume a fluidelement is uniquely determined by its label for all time. Thus, J , q = q ( a , t )4 P. J. Morrison, T. Andreussi, and F. Pegoraro to obtain the label associated with the fluid element at position x at time t , a = q − ( x , t ). Forcoordinate transformations q = q ( a , t ) we have ∂ q k ∂ a j A ik J = δ ij , (3.2)where A ik is the cofactor of ∂ q k /∂ a i , which can be written as follows: A ik = ǫ k j ℓ ǫ imn ∂ q j ∂ a m ∂ q ℓ ∂ a n . (3.3)Using q ( a , t ) or its inverse q − ( x , t ), various quantities can be written either as a function of x or a . For convenience we list additional formulas below for latter use: J = A k ℓ ∂ q ℓ ∂ a k , (3.4) A ji = ∂ J ∂ ( ∂ q i /∂ a j ) , (3.5) ∂ ( A ki f ) ∂ a k = A ki ∂ f ∂ a k , (3.6) δ J = A ki ∂δ q i ∂ a k or ˙ J = A ki ∂ ˙ q i ∂ a k , (3.7) δ A k ℓ J ∂ q ℓ ∂ a u = − A ki J ∂∂ a u δ q i or δ A k ℓ J = − A ki A u ℓ J ∂∂ a u δ q i , (3.8) A u ℓ ∂∂ a u A ki J ∂ f ∂ a k = A ki ∂∂ a k " A u ℓ J ∂ f ∂ a u , ∀ f , (3.9)which follow from the standard rule for di ff erentiation of determinants and the expression forthe cofactor matrix. For example, the commutator expression of (3.9) follows easily from (3.8),which in turn follows upon di ff erentiating (3.2). These formulas are all of classical origin, e.g.,the second equation of (3.7) is the Lagrangian variable version of a formula due to Euler (seee.g. Serrin 1959).Now we are in a position to recreate and generalize Lagrange’s Lagrangian for the ideal fluidaction principle. On physical grounds we expect our fluid to possess kinetic and internal energies,and if magnetized, a magnetic energy. The total kinetic energy functional of the fluid is naturallygiven by T [ ˙ q ] : = Z d a ρ ( a ) | ˙ q | , (3.10)where ρ is the mass density attached to the fluid element labeled by a and ˙ q denotes timedi ff erentiation of q at fixed label a . Note, in (3.10) | ˙ q | = ˙ q i ˙ q i , where in general ˙ q i = g i j ˙ q i , butwe will only consider the cartesian metric where g i j = δ i j = η i j .Fluids are assumed to be in local thermodynamic equilibrium and thus can be described by afunction U ( ρ, s ), an internal energy per unit mass that depends on the specific volume ρ − andspecific entropy s . If a magnetic field B ( x , t ) were present, then we could add dependence on | B | as in Morrison (1982) to account for pressure anisotropy. (See also Morrison et al. et al. U we compute thetemperature and pressure according to the usual di ff erentiations, T = ∂ U /∂ s and p = ρ ∂ U /∂ρ .For MHD, the magnetic energy H B = R d x | B | / agrangian and Dirac constraints for fluid and MHD V [ q ] : = Z d a ρ U ( ρ / J , s ) . (3.11)Here we have used the fact that a fluid element carries a specfic entropy s = s ( a ) and a massdetermined by ρ = ρ ( a ) / J . In Section 3.3 we will describe in detail the map from Lagrangianto Eulerian variables.Thus, the special case of the action principle of (2.72) for the ideal fluid has Lagrange’sLagrangian L [ q , ˙ q ] = T − V . Variation of this action gives the Lagrangian equation of motion forthe fluid ρ ¨ q i = − A ji ∂∂ a j ρ J ∂ U ∂ρ , (3.12)with an additional term that describes the J × B force in Lagrangian variables for MHD. See, e.g.,Newcomb (1962); Morrison (1998, 2009) for details of this calculation and the MHD extension.3.2. Hamiltonian formalism in Lagrangian description
Upon defining the momentum density as usual by π i = δ L δ ˙ q i = ρ ˙ q i , (3.13)we can obtain the Hamiltonian by Legendre transformation, yielding H [ π , q ] = T + V = Z d a | π | ρ + ρ U ( ρ / J , s ) ! , (3.14)where | π | = π i π i = π i η i j π j . This Hamiltonian with the canonical Poisson bracket, { F , G } = Z d a δ F δ q i δ G δπ i − δ G δ q i δ F δπ i ! , (3.15)yields ˙ q i = { q i , H } = π i /ρ , (3.16)˙ π i = { π i , H } = − A ji ∂∂ a j ρ J ∂ U ∂ρ . (3.17)Equations (3.16) and (3.17) are equivalent to (3.12). For MHD a term H B is added to (3.14) (seeNewcomb 1962; Morrison 2009). We will give this explicitly in the constraint context in Section4.2.1 after discussing the Lagrange to Euler map.3.3. Hamiltonian formalism in Eulerian description via the Lagrange to Euler map
In order to understand how constraints in terms of the Lagrangian variable description relate tothose in terms of the Eulerian description, in particular ∇ · v =
0, it is necessary to understand themapping from Lagrangian to Eulerian variables. Thus, we record here the relationship betweenthe two unconstrained descriptions, i.e., how the noncanonical Hamiltonian structure of thecompressible Euler’s equations relates to the Hamiltonian structure described in Section 3.2.For the ideal fluid, the set of Eulerian variables can be taken to be { v , ρ, s } , where v ( x , t ) is thevelocity field at the Eulerian observation point, x = ( x , y , z ) = ( x , x , x ) at time t and, as as notedin Section 3.1, ρ ( x , t ) is the mass density and s ( x , t ) is the specific entropy. In order to describemagnetofluids the magnetic field B ( x , t ) would be appended to this set. It is most important todistinguish between the Lagrangian fluid element position and label variables, q and a , and theEulerian observation point x , the latter two being independent variables. Confusion exists in the6 P. J. Morrison, T. Andreussi, and F. Pegoraro literature because some authors use the same symbol for the Lagrangian coordinate q and theEulerian observation point x .The Lagrangian and Eulerian descriptions must clearly be related and, indeed, knowing q ( a , t )we can obtain v ( x , t ). If one were to insert a velocity probe into a fluid at ( x , t ) then one wouldmeasure the velocity of the fluid element that happened to be at that position at that time. Thusit is clear that ˙ q ( a , t ) = v ( x , t ), where recall the overdot means the time derivative at constant a .But, which fluid element will be at x at time t ? Evidently x = q ( a , t ), which upon inversion yieldsthe label of that element that will be measured, a = q − ( x , t ). Thus, the Eulerian velocity field isgiven by v ( x , t ) = ˙ q ( a , t ) | a = q − ( x , t ) = ˙ q ◦ q − ( x , t ) . (3.18)Properties can be attached to fluid elements, just as a given mass is identified with a given particlein mechanics. For a continuum system it is natural to attach a mass density, ρ ( a ), to the elementlabeled by a . Whence the element of mass in a given volume is given by ρ d a and this amount ofmass is preserved by the flow, i.e. ρ ( x , t ) d x = ρ ( a ) d a . Because the locus of points of materialsurfaces move with the fluid are determined by q , an initial volume element d a maps into avolume element d x at time t according to d x = J d a , (3.19)Thus, using (3.19) we obtain ρ = ρ J as used in Section 3.1.Other quantities could be attached to a fluid element; for the ideal fluid, entropy per unit mass, s ( x , t ), is such a quantity. The assumption that each fluid element is isentropic then amounts to s = s . Similarly, for MHD a magentic field, B ( a ), can be attached, and then the frozen fluxassumption yields B · d x = B · d a . An initial area element d a maps into an area element d x at time t according to ( d x ) i = A ji ( d a ) j . (3.20)Using (3.20) we obtain J B i = B j ∂ q i /∂ a j .Sometimes it is convenient to use another set of Eulerian density variables: { M , ρ, σ, B } , where σ = ρ s is the entropy per unit volume, and M = ρ v is the momentum density. These Eulerianvariables can be represented by using the Dirac delta function to ‘pluck out’ the fluid elementthat happens to be at the Eulerian observation point x at time t . For example, the mass density ρ ( x , t ) is obtained by ρ ( x , t ) = Z d a ρ ( a ) δ ( x − q ( a , t )) = ρ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = q − ( x , t ) . (3.21)The density one observes at x at time t will be the one attached to the fluid element that happensto be there then, and this fluid element has a label given by solving x = q ( a , t ). The secondequality of (3.21) is obtained by using the three-dimensional version of the delta function identity δ ( f ( x )) = P i δ ( x − x i ) / | f ′ ( x i ) | , where f ( x i ) =
0. Similarly, the entropy per unit volume is givenby σ ( x , t ) = Z d a σ ( a ) δ ( x − q ( a , t )) = σ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = q − ( x , t ) , (3.22)which is consistent with σ ( a ) = ρ ( a ) s ( a ) and s ( x , t ) = s ( a ) | a = q − ( x , t ) , where the latter means s is constant along a Lagrangian orbit. Proceeding, the momentum density, M = ( M , M , M ), isrelated to the Lagrangian canonical momentum (defined in Section 3.2) by M ( x , t ) = Z d a π ( a , t ) δ ( x − q ( a , t )) = π ( a , t ) J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = q − ( x , t ) , (3.23) agrangian and Dirac constraints for fluid and MHD π ( a , t ) = ( π , π , π ) = ρ ˙ q . Lastly, B i ( x , t ) = Z d a ∂ q i ( a , t ) ∂ a j B j ( a ) δ ( x − q ( a , t )) = ∂ q i ( a , t ) ∂ a j B j ( a ) J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = q − ( x , t ) , (3.24)for the components of the magnetic field. It may be unfamiliar to view the magnetic field asdensity, but in MHD it obeys a conservation law. Geometrically, however, it naturally satisfies theequations of a vector density associated with a di ff erential 2-form as was observed in Morrison(1982) and Tur & Yanovsky (1993).To obtain the noncanonical Eulerian Poisson bracket we consider functionals F [ q , π ] that arerestricted so as to obtain their dependence on q and π only through the Eulerian variables. Uponsetting F [ q , π ] = ¯ F [ v , ρ, σ ], equating variations of both sides, δ F = Z d a " δ F δ q · δ q + δ F δπ · δπ = Z d x " δ ¯ F δρ δρ + δ ¯ F δσ δσ + δ ¯ F δ M · δ M = δ ¯ F , (3.25)varying the expressions (3.21), (3.22), and (3.23), substituting the result into (3.25), and equatingthe independent coe ffi cients of δ q and δ π , we obtain δ F δ q = Z d x " ρ ∇ δ ¯ F δρ + σ ∇ δ ¯ F δσ + π i ∇ δ ¯ F δ M i δ ( x − q ) , (3.26) δ F δ π = Z d x δ ¯ F δ M δ ( x − q ) . (3.27)(See Morrison (1998) and Morrison & Greene (1980) for details.) Upon substitution of (3.26)and (3.27), expressions of the functional chain rule that relate Lagrangian functional derivativesto the Eulerian functional derivates, into (3.15) yields the following bracket expressed entirely interms of the Eulerian fields { M , ρ, σ } : { F , G } = − Z d x " M i δ F δ M j ∂∂ x j δ G δ M i − δ G δ M j ∂∂ x j δ F δ M i ! + ρ δ F δ M · ∇ δ G δρ − δ G δ M · ∇ δ F δρ ! + σ δ F δ M · ∇ δ G δσ − δ G δ M · ∇ δ F δσ ! . (3.28)In (3.28) we have dropped the overbars on the Eulerian functional derivatives. The bracket forMHD is the above with the addition of the following term, which is obtained by adding a B contribution to (3.25): { F , G } B = − Z d x " B · (cid:18) δ F δ M · ∇ δ G δ B − δ G δ M · ∇ δ F δ B (cid:19) + B · (cid:18) ∇ (cid:18) δ F δ M (cid:19) · δ G δ B − ∇ (cid:18) δ G δ M (cid:19) · δ F δ B (cid:19) , (3.29)where dyadic notation is used; for example, B · [ ∇ ( D ) · C ] = P i , j B i C j ∂ D j /∂ x i , for vectors B , D ,and C . Alternatively, the bracket in terms of { v , ρ, s , B } is obtained using chain rule expressions,e.g., δ F δρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v , s = δ F δρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M , s + M ρ · δ F δ M + σρ δ F δσ , (3.30)8 P. J. Morrison, T. Andreussi, and F. Pegoraro yielding { F , G } = − Z d x " δ F δρ ∇ · δ G δ v − δ G δρ ∇ · δ F δ v ! + ∇ × v ρ · δ G δ v × δ F δ v ! + ∇ s ρ · δ F δ s δ G δ v − δ G δ s δ F δ v ! , (3.31)and { F , G } B = − Z d x " B · ρ δ F δ v · ∇ δ G δ B − ρ δ G δ v · ∇ δ F δ B ! + B · ∇ ρ δ F δ v ! · δ G δ B − ∇ ρ δ G δ v ! · δ F δ B ! . (3.32)The bracket of (3.31) plus that of (3.32) with the Hamiltonian H [ ρ, s , v , B ] = Z d x ρ | v | + ρ U ( ρ, s ) + | B | ! (3.33)gives the Eulerian version of MHD in Hamiltonian form, ∂ v /∂ t = { v , H } , etc., and similarly using(3.28) plus (3.29) with the Hamiltonian expressed in terms of ( M , ρ, σ, B ). Ideal fluid followsupon neglecting the B terms.3.4. Constants of motion: Eulerian vs. Lagrangian
In oder to compare the imposition of constraints in the Lagrangian and Eulerian descriptions, itis necessary to compare Lagrangian and Eulerian conservations laws. This is because constraints,when enforced, are conserved quantities. The comparison is not trivial because time independentquantities in the Eulerian description can be time dependent in the Lagrangian description.Consider a Lagrangian function f ( a , t ), typical of the Lagrangian variable description, andthe relation x = q ( a , t ), which relates an Eulerian observation point x to a corresponding fluidelement trajectory value. The function f can be written in either picture by composition, asfollows: f ( a , t ) = ˜ f ( x , t ) = ˜ f ( q ( a , t ) , t ) , (3.34)where we will use a tilde to indicated the Eulerian form of a Lagrangian function. Application ofthe chain rule gives A ik J ∂ f ∂ a i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = q − ( x , t ) = ∂ ˜ f ∂ x k and A k ℓ J ∂∂ a k π ℓ J !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = q − ( x , t ) = ∇ · v , (3.35)with the second equality of (3.35) being a special case of the first. Similarly,˙ f (cid:12)(cid:12)(cid:12) a = q − ( x , t ) = ∂ ˜ f ∂ t + ˙ q i ( a , t ) ∂ ˜ f ∂ x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = q − ( x , t ) = ∂ ˜ f ∂ t + v · ∇ ˜ f ( x , t ) , (3.36)where recall an overdot denotes the time derivative at constant a , ∂/∂ t denotes the time derivativeat constant x , and ∇ is the Eulerian gradient with components ∂/∂ x i as used in (3.35). Becausethe Jacobian determinant J is composed of derivatives of q , we have J ( a , t ) | a = q − ( x , t ) = ˜ J ( x , t ),whence one obtains a formula due to Euler (see e.g. Serrin 1959), ∂ ˜ J ∂ t + v · ∇ ˜ J = ˜ J ∇ · v , (3.37)which can be compared to its Lagrangian version of (3.7). agrangian and Dirac constraints for fluid and MHD D L + ∂Γ i D L ∂ a i = , (3.38)where the density D L ( a , t ) has the associated flux Γ D L . Then, the associated conserved quantityis I D L = Z d a D L , (3.39)which satisfies d I D L / dt = D E and flux Γ D E is ∂ D E ∂ t + ∂Γ i D E ∂ x i = I D E = Z d x D E . (3.41)The relationship between the two conservation laws (3.38) and (3.40) can be obtained by defining˜ D L = JD E , ˜ Γ i D L = A ik ¯ Γ k D E , and Γ D E = ¯ Γ D E + v D E , (3.42)and their equivalence follows from (3.7), (3.36), and (3.37). Given a Lagrangian conservationlaw, one can use (3.42) to obtain a corresponding Eulerian conservation law. The density D E is obtained from the first equation of (3.42), a piece of the Eulerian flux ¯ Γ D E from the second,which then can be substitued into the third equation of (3.42) to obtain the complete Eulerianflux Γ D E . An Eulerian conservation law is most useful when one can write D E and Γ D E entirelyin terms of the Eulerian variables of the fluid.The simplest case occurs when D L only depends on a , in which case the corresponding flux iszero and ∂ D L /∂ t = d I D L / dt = a and this has atrivial conservation law of this form. However, such trivial Lagrangian conservation laws yieldnontrivial Eulerian conservation laws. Observe, even thought ¯ Γ D E ≡ Γ D E = v D E ,
0. Consider the case of the entropy where D L = s ( a ), whence s ( x , t ) = s ( a ( x , t )) and by (3.36), ∂ s ∂ t + v · ∇ s = , (3.43)with the quantity s = s / J being according to (3.42) the Eulerian conserved density, as can beverified using (3.37). But, as it stands, this density cannot be written in terms of Eulerian fluidvariables. However, σ = ρ s is also a trivial Lagrangian conserved density and according to(3.42) we have the Eulerian density ρ s / J = ρ s = σ that satisfies ∂σ∂ t + ∇ · ( v σ ) = . (3.44)Thus, it follows that any advected scalar has an associated conserved quantity obtained bymultiplication by ρ .As another example, consider the quantity B i ∂ q j /∂ a i . This quantity is the limit displacementbetween two nearby fluid elements, i.e., q ( a , t ) − q ( a + δ a , t ) along the initial magnetic field as δ a →
0. Evidently, ˙ B i ∂ q j ∂ a i ! = B i ∂ ˙ q j ∂ a i = ∂∂ a i (cid:16) B i ˙ q j (cid:17) , (3.45)where the second equality follows if the initial magnetic field is divergence free. This is of course0 P. J. Morrison, T. Andreussi, and F. Pegoraro another trivial conservation law, for the time derivative of a density that is a divergence willalways be a divergence. However, let us see what this becomes in the Eulerian description. Ac-cording to (3.42) the corresponding Eulerian density is D E = D L / J ; so, the density associatedwith this trivial conservation law (3.45) is B j ( x , t ) = B i J ∂ q j ∂ a i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = q − ( x , t ) . (3.46)which as we saw in Section 3.3 is the expression one gets for the MHD magnetic field becauseof flux conservation. That the divergence-free magnetic field satisfies a conservation law is clearfrom ∂ B ∂ t = − v · ∇ B + B · ∇ v − B ∇ · v = ∇ · ↔ T , (3.47)where the tensor ↔ T of the last equality is ↔ T = B ⊗ v − v ⊗ B . (3.48)Thus we have another instance where a trivial Lagrangian conservation law leads to a nontrivialEulerian one.Although B i ∂ q j /∂ a i does not map into an expression entirely in terms of our set of Eulerianvariables, we can divide it by ρ , a quantity that only depends on the label a , and obtain B i ρ ∂ q j ∂ a i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = q − ( x , t ) = B j ρ . (3.49)Eulerianizing the counterpart of (3.45) for this expression gives ∂∂ t B ρ ! + v · ∇ B ρ ! = B ρ · ∇ v , (3.50)which is not an Eulerian conservation law. This is to be expected because, unlike what we did toget (3.47), we have Eulerianized without using (3.42). In light of its relationship to q ( a , t ) − q ( a + δ a , t ), the quantity B /ρ has been described as a measure of the distance of points on a magneticfield line (see e.g. Kampen & Felderho ff
4. Constraint theories for the incompressible ideal fluid
The incompressible fluid in Lagrangian variables
In order to enforce incompressibility, Lagrange added to his Lagrangian the constraint J = λ ( a , t ), L λ [ q , ˙ q ] = T [ ˙ q ] + λ J , (4.1)with T given (3.10). Here we have dropped V because incompressible fluids contain no internalenergy. Upon insertion of (4.1) into the action of Hamilton’s principle it is discovered that λ corresponds to the pressure. The essence of this procedure was known to Lagrange. (SeeSerrin (1959) for historical details and Sommerfeld (1964) for an elementary exposition.) Thisprocedure yields ρ ¨ q i = − A ij ∂λ∂ a j , (4.2)where use has been made of (3.7). The Eulerian form of (4.2) is clearly ρ ( ∂ v /∂ t + v · ∇ v ) = −∇ λ ,whence it is clear that λ is the pressure. Although Lagrange knew the Lagrange multiplier was agrangian and Dirac constraints for fluid and MHD Lagrangian volume preserving geodesic flow
If the constraint is dropped from (4.1), we obtain free particle motion for an infinite-dimensional system, the ideal fluid case of (2.78) of Section 2.4, which is analogous to thefinite-dimensional case of Section 2.1.1. Because the constraint J = q , it is a configuration space constraint; thus, it is an holonomic constraint. Asis well-known and reviewed in Section 2.1.1, free particle motion with holonomic constaints isgeodesic flow. Thus, following Lagrange, it is immediate that the ideal incompressible fluid isan infinite-dimensional version of geodesic flow.Lagrange’s calculation was placed in a geometric / group theoretic setting in Arnold (1966) (seealso Appendix 2 of Arnold (1978) and Arnold & Khesin (1998)). Given that the transformation a ↔ q , at any time, is assumed to be a smooth invertible coordinate change, it is a Liegroup, one referred to as the di ff eomorphism group. With the additional assumption that thesetransformations are volume preserving, Lagrange’s constraint J =
1, the transformations forma subgroup, the group of volume preserving di ff eomorphisms. Thus, Lagrange’s work can beviewed as geodesic flow on the group of volume preserving di ff eomorphisms.Although Arnold’s assumptions of smoothness etc. are mathematically dramatic, his descrip-tion of Lagrange’s calculations in these terms has spawned a considerable body of research.Associated with a geodesic flow is a metric, and whence one can calculate a curvature. In hisoriginal work, Arnold added the novel calculation of the curvature in the mathematically moreforgiving case of two-dimensional flow with periodic boundary conditions.4.2. Lagrangian-Dirac constraint theory
More recently there have been several works (Tassi et al. et al. et al. D a ( a ′ ) = Z d a D a ( a ) δ ( a − a ′ ) , (4.3)where a = , D a ( a ) is a shorthand for a function of q ( a , t ) and π ( a , t ) and their derivativeswith respect to a . Then the matrix D is a 2 × D ab ( a , a ′ ) = { D a ( a ) , D b ( a ′ ) } , using the canonical bracket of (3.15). To construct the Dirac bracket { F , G } ∗ = { F , G } − Z d a Z d a ′ { F , D a ( a ) } D − ab ( a , a ′ ) { D b ( a ′ ) , G } , (4.4)2 P. J. Morrison, T. Andreussi, and F. Pegoraro we require the inverse, which satisfies Z d a D ac ( a ′ , a ) D − cb ( a , a ′′ ) = δ ab δ ( a ′ − a ′′ ) . (4.5)Rather than continuing with the general case, which is unwieldy, we proceed to the specialcase for the incompressible fluid, an infinite-dimensional version of the holonomic constraintsdiscussed in Section 2.3.1.4.2.1. Lagrangian-Dirac incompressibility holonomic constraint
Evidently we will want our holonomic incompressibility constraint to be J . However, it isconvenient to express this by choosing D = ln J ρ ! . (4.6)This amounts to the same constraint as J = D = − ln( ρ ). The scaling of J in (4.6) by ρ ( a ) is immaterial because it is a time-independent quantity. To obtain the secondconstraint we follow suit and set D = ˙ D = A k ℓ J ∂∂ a k π ℓ ρ ! = η ℓ j A k ℓ J ∂∂ a k π j ρ ! , (4.7)where recall we assume η ℓ j = δ ℓ j and π j is given by (3.13). That the constraint D is the timederivative of D requires the definition of π j of (3.13) that uses the Hamiltonian R d a | π | / (2 ρ ).Observe, that constraints D and D are local constraints in that they are enforced pointwise(see e.g. Flierl & Morrison 2011), i.e., they are enforced on each fluid element labeled by a .Equation (4.6) corresponds in the Eulerian picture to − ln( ρ ), while the second constraint of(4.7), the Lagrangian time derivative of the first constraint, corresponds in the Eulerian pictureto ∇ · v , which can be easily verified using the second equation of (3.35). Note, the particularvalues of these constraints of interest are, of course, J = ∇ · v =
0, but the dynamics theDirac bracket generates will preserve any values of these constraints. For example, we could set J = f ( a ) where the arbitrary function f is less than unity for some a and greater for others,corresponding to regions of fluid elements that experience contraction and expansion. Also note,because we have used π with the up index in (4.7); thus as seen in the second equality it dependson the metric. This was done to make it have the Eulerian form ∇ · v .For the constraints (4.6) and (4.7), D only depends on two quantities because D does notdepend on π , i.e. { D , D } = { D , D } = −{ D , D } . Thus the inverse has the form D − = D − D − D − ! , (4.8)giving rise to the conditions D − · D = I = D − · D and D − · D + D − · D = , (4.9)where I is the identity. Thus, the inverse is easily tractable if the inverse of D exists; whence, D − = − D − · D · D − . (4.10)In the above the symbol ‘ · ’ is used to denote the product with the sum in infinite dimensions,i.e., integration over d a as in (4.5). Equation (4.10) can be rewritten in an abbreviated form withimplied integrals on repeated arguments as D − ( a ′ , a ′′ ) = D − ( a ′ , ˆ a ) · D (ˆ a , ˇ a ) · D − (ˇ a , a ′′ ) . (4.11) agrangian and Dirac constraints for fluid and MHD D and its inverse, we need the functional derivatives of D and D . Theseare obtained directly by writing these local constraints as in (4.5), yielding δ D ( a ′ ) δ q i ( a ) = − A ki ∂∂ a k δ ( a − a ′ ) J , (4.12) δ D ( a ′ ) δπ i ( a ) = , (4.13)where use has been made of (3.7), and δ D ( a ′ ) δ q i ( a ) = ∂∂ a u A ki A u ℓ J ∂∂ a k π ℓ ρ ! δ ( a − a ′ ) , (4.14) δ D ( a ′ ) δπ i ( a ) = − η i j ρ ∂∂ a m A mj J δ (cid:0) a − a ′ (cid:1)! , (4.15)where use has been made of (3.8) and recalling we have (3.6) at our disposal.Let us now insert (4.12), (4.13), (4.14), and (4.15) into the canonical Poisson bracket (3.15),to obtain D ( a , a ′ ) = { D ( a ) , D ( a ′ ) } = − A ℓ i J ∂∂ a ℓ η i j ρ A kj ∂∂ a k δ ( a − a ′ ) J !! , (4.16)which corresponds to the symmetric matrix S of (2.27) and (2.42) and D ( a , a ′ ) = { D ( a ) , D ( a ′ ) } = A ki A u ℓ J ∂∂ a k π ℓ ρ ! ∂∂ a u " η i j ρ ∂∂ a m A mj J δ (cid:0) a − a ′ (cid:1)! − A mi J ∂∂ a m η i j ρ ∂∂ a u A kj A u ℓ J ∂∂ a k π ℓ ρ ! δ ( a − a ′ ) , (4.17)which corresponds to the antisymmetric matrix A of (2.28). Observe the symmetries correspond-ing to the matrices S and A , respectively, are here Z d a ′ D ( a , a ′ ) φ ( a ′ ) = Z d a ′ D ( a ′ , a ) φ ( a ′ ) , Z d a ′ D ( a , a ′ ) φ ( a ′ ) = − Z d a ′ D ( a ′ , a ) φ ( a ′ ) , for all functions φ . The first follows from integration by parts, while the second is obvious fromits definition.Using (4.5), the first condition of (4.9) is Z d a ′′ D ( a ′ , a ′′ ) D − ( a ′′ , ˆ a ) = δ ( a ′ − ˆ a ) , (4.18)which upon substitution of (4.16) and integration gives − A ℓ i J ∂∂ a ℓ η i j ρ A kj ∂∂ a k D − ( a , a ′′ ) J = δ ( a − a ′′ ) . (4.19)We introduce the formally self-adjoint operator (cf. (3.6)) ∆ ρ f : = A ℓ i J ∂∂ a ℓ " η i j ρ A kj ∂∂ a k f J ! , (4.20)4 P. J. Morrison, T. Andreussi, and F. Pegoraro i.e., an operator that satisfies Z d a f ( a ) ∆ ρ g ( a ) = Z d a g ( a ) ∆ ρ f ( a ) , (4.21)a property inherited by its inverse ∆ − ρ . Thus we can rewrite equation (4.19) as D − ( a , a ′′ ) = − G (cid:0) a , a ′′ (cid:1) = − ∆ − ρ δ ( a − a ′′ ) , (4.22)where G represents the Green function associated with (4.19).In order to obtain D − , we find it convenient to transform (4.22) to Eulerian variables. Using x = q ( a , t ) we find D − ( a , a ′ ) J = − G ( x , x ′ ) = − G ( q ( a ) , q ( a ′ )) , (4.23)where G satisfies ∇ · ρ ∇ G ! = − ∆ ρ D − ( a , a ′ ) J = J δ ( x − x ′ ) . (4.24)Here use has been made of identities (3.6) and (3.35). As noted in Section 1.1, under physicallyreasonable conditions, the operator ∆ ρ f = ∆ ρ ( J f ) = ∇ · ρ ∇ f ! (4.25)has an inverse. Thus we write D − ( a , a ′ ) = −J ∆ − ρ (cid:16) J δ (cid:0) q ( a , t ) − q ( a ′ , t ) (cid:1)(cid:17) . (4.26)Now, using D − = − D − , the element D − follows directly from (4.10).For convenience we write the Dirac bracket of (4.4) as follows: { F , G } ∗ = { F , G } − [ F , G ] D , (4.27)where [ F , G ] D : = X a , b = [ F , G ] Dab = Z d a Z d a ′ { F , D a ( a ) } D − ab ( a , a ′ ) n D b (cid:0) a ′ (cid:1) , G o . (4.28)Because D − = F , G ] D = − [ G , F ] D , we only need to calculate [ F , G ] D and [ F , G ] D .As above, we substitute (4.12), (4.13), (4.14), and (4.15) into the bracket (3.15) and obtain n F , D ( a ) o = − A ki J ∂∂ a k δ F δπ i ! , (4.29) n F , D ( a ) o = A k ℓ J ∂∂ a k η i ℓ ρ δ F δ q i ! + A ki J A u ℓ J ∂∂ a k π ℓ ρ ! ∂∂ a u δ F δπ i ! . (4.30)Then, exploiting the antisymmetry of the Poisson bracket, it is straightforward to calculateanalogous expressions for the terms n D , , G o .We first analyze the operator[ F , G ] D = Z d a Z d a ′ Z d ˆ a Z d ˇ a n F , D ( a ) o D − ( a , ˆ a ) D (ˆ a , ˇ a ) D − (ˇ a , a ′ ) n D (cid:0) a ′ (cid:1) , G o , (4.31)where we used the second condition of (4.9) to replace D − . Upon inserting (4.22) and (4.29), agrangian and Dirac constraints for fluid and MHD F , G ] D = − Z d a Z d a ′ Z d ˆ a Z d ˇ a A hj J ∂∂ a h δ F δπ j ! ∆ − ρ δ ( a − ˆ a ) a = a D (ˆ a , ˇ a ) " A sr J ∂∂ a s δ G δπ r ! ∆ − ρ δ (ˇ a − a ) a = a ′ , (4.32)where the subscripts on the right delimiters indicate that a is to be replaced after the derivativeoperations, including those that occur in J and A ji .Integrating this expression by parts with respect to a and a ′ yields[ F , G ] D = − Z d ˆ a Z d ˇ a ∆ − ρ A hj J ∂∂ a h δ F δπ j ! a = ˆ a D (ˆ a , ˇ a ) " ∆ − ρ A sr J ∂∂ a s δ G δπ r !! a = ˇ a , (4.33)and then substituting (4.17) gives[ F , G ] D = − Z d ˆ a Z d ˇ a ∆ − ρ A hj J ∂∂ a h δ F δπ j ! a = ˆ a A ki A u ℓ J ∂∂ a k π ℓ ρ ! ∂∂ a u " η in ρ ∂∂ a m A mn J δ ( a − ˇ a ) ! − A mi J ∂∂ a m " η in ρ ∂∂ a u A kn J A u ℓ J ∂∂ a k π ℓ ρ ! δ ( a − ˇ a ) ! a = ˆ a " ∆ − ρ A sr J ∂∂ a s δ G δπ r !! a = ˇ a . (4.34)Then, by means of integrations by parts we can remove the derivatives from the term δ ( a − ˇ a )and perform the integral. After relabeling the integration variable as a to simplify the notation,(4.34) becomes[ F , G ] D = Z d a ( ρ η ui A ku J ∂∂ a k π ℓ ρ ! P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ ! + η ni A u ℓ ∂∂ a u " A kn J ∂∂ a k π ℓ ρ ! P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ∆ − ρ J A hj J ∂∂ a h δ F δπ j ! − P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ∆ − ρ J A hj J ∂∂ a h δ G δπ j ! ) , (4.35)where we introduced the projection operator (cid:16) P ρ ⊥ (cid:17) ij z j = η i ℓ ρ A u ℓ ∂∂ a u ∆ − ρ J A hj J ∂∂ a h z j = : P ρ ⊥ z (cid:12)(cid:12)(cid:12) i , (4.36)where in the last equality we defined a shorthand for convenience; thus, P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ : = ρ A u ℓ ∂∂ a u ∆ − ρ J A hj J ∂∂ a h δ F δπ j . (4.37)It is straightforward to prove that P ρ ⊥ represents a projection, i.e. P ρ ⊥ (cid:16) P ρ ⊥ z (cid:17) = P ρ ⊥ z foreach z , which in terms of indices would have an i th component given by ( P ρ ⊥ ) ij ( P ρ ⊥ ) jk z k = ( P ρ ⊥ ) ik z k . Also, P ρ ⊥ is formally self-adjoint with respect to the following weighted inner prod-uct: Z d a ρ w i ( P ρ ⊥ ) ij z j = Z d a ρ z i ( P ρ ⊥ ) ij w j . (4.38)The projection operator complementary to P ρ ⊥ is given by P ρ = I − P ρ ⊥ , (4.39)6 P. J. Morrison, T. Andreussi, and F. Pegoraro where I is the identity.Now let us return to our evaluation of [ F , G ] D and analyze the contribution[ F , G ] D = Z d a Z d a ′ n F , D ( a ) o D − ( a , a ′ ) n D (cid:0) a ′ (cid:1) , G o . (4.40)Using (4.26), (4.29), and (4.30), this equation can be rewritten as[ F , G ] D = − Z d a Z d a ′ A ki J ∂∂ a k η in ρ δ F δ q n ! + A ki J A u ℓ J ∂∂ a k π ℓ ρ ! ∂∂ a u δ F δπ i ! × ∆ − ρ δ ( a − a ′ ) A hj J ∂∂ a h δ G δπ j ! a = a ′ (4.41)and, integrating by parts to simplify the δ ( a − a ′ ) term, results in[ F , G ] D = Z d a ( δ F δ q i P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i + ρ A ki J ∂∂ a k π ℓ ρ ! δ F δπ i P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ + A u ℓ ∂∂ a u A ki J ∂∂ a k π ℓ ρ ! δ F δπ i ∆ − ρ J A hj J ∂∂ a h δ G δπ j ! ) . (4.42)We can now combine the operators [ F , G ] D , [ F , G ] D , and [ F , G ] D = − [ G , F ] D , given by(4.35) and (4.42), to calculate the Dirac bracket (4.27). First, we rewrite (4.28) as[ F , G ] D = Z d a ( δ F δ q i P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − δ G δ q i P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i + ρ A ki J ∂∂ a k π ℓ ρ ! P ρ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ − P ρ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ ! + A u ℓ ∂∂ a u A ki J ∂∂ a k π ℓ ρ ! P ρ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ∆ − ρ J A hj J ∂∂ a h δ G δπ j ! − P ρ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ∆ − ρ J A hj J ∂∂ a h δ F δπ j ! ) . (4.43)Using the identity of (3.9) with z ℓ set to π ℓ /ρ , A u ℓ ∂∂ a u A ki J ∂∂ a k π ℓ ρ ! = A ki ∂∂ a k " A u ℓ J ∂∂ a u π ℓ ρ ! , (4.44)and integrating by parts, (4.43) becomes[ F , G ] D = Z d a ( δ F δ q i P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − δ G δ q i P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i + ρ A ki J ∂∂ a k π ℓ ρ ! P ρ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ − P ρ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ ! − ρ A u ℓ J ∂∂ a u π ℓ ρ ! P ρ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − P ρ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ! ) , (4.45)where we used A ki ∂∂ a k P ρ z (cid:12)(cid:12)(cid:12) i = A ki ∂∂ a k (cid:16) P ρ (cid:17) ij z j = , for all z , (4.46) agrangian and Dirac constraints for fluid and MHD A ki J ∂∂ a k (cid:16) P ρ ⊥ (cid:17) ij z j = A ki J ∂ z i ∂ a k , (4.47)and (4.39). Also, upon inserting P ρ ⊥ = I − P ρ in the last line of (4.45), symmetry implies wecan drop the P ρ ⊥ . Finally, upon substituting (4.45) into (4.27), we obtain { F , G } ∗ = Z d a ( δ F δ q i P ρ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − δ G δ q i P ρ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − ρ A ki J ∂∂ a k π ℓ ρ ! P ρ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ − P ρ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ ! + ρ A k ℓ J ∂∂ a k π ℓ ρ ! P ρ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i δ G δπ i − P ρ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i δ F δπ i ! ) . (4.48)Once more inserting P ρ ⊥ = I − P ρ , rearranging, and reindexing gives { F , G } ∗ = − Z d a ρ ( ρ δ G δ q i P ρ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − ρ δ F δ q i P ρ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i + A mn P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n + T mn δ F δπ m P ρ ⊥ δ G δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − δ G δπ m P ρ ⊥ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n ! ) , (4.49)where A nm : = η ℓ m D ℓ n − η ℓ n D ℓ m and T mn : = η ℓ n D ℓ m + η mn D , (4.50)with D ℓ m = A km J ∂∂ a k π ℓ ρ ! . (4.51)Note the trace D ℓℓ = D , which we will eventually set to zero. Equation (4.49) gives the Diracbracket for the incompressibility holonomic constraint. This bracket with the Hamiltonian H = Z d a | π | ρ = Z d a η mn π m π n ρ , (4.52)produces dynamics that fixes J and thus enforces incompressibility provided the constraint D = H the following: H B = Z d a η mn B j B k J ∂ q m ∂ a j ∂ q n ∂ a k . (4.53)We note, any Hamiltonian that is consistent with (4.7) can be used to define a constrained flow.Proceeding to the equations of motion, we first calculate ˙ q i ,˙ q i = { q i , H } ∗ = (cid:16) P ρ (cid:17) ij δ H δπ j = δ H δπ i − η i ℓ ρ A u ℓ ∂∂ a u ∆ − ρ J A hj J ∂∂ a h δ H δπ j = π i ρ − η i ℓ ρ A u ℓ ∂∂ a u ∆ − ρ J A hj J ∂∂ a h π j ρ . (4.54)The equation for ˙ π i is more involved. Using the adjoint property of (4.38), which is valid for both8 P. J. Morrison, T. Andreussi, and F. Pegoraro P ρ ⊥ and P ρ , we obtain˙ π i = { π i , H } ∗ = − ρ (cid:16) P ρ (cid:17) ji ρ δ H δ q j − ρ (cid:16) P ρ ⊥ (cid:17) mi A mn (cid:16) P ρ ⊥ (cid:17) nk δ H δπ k ! + ρ (cid:16) P ρ ⊥ (cid:17) ni T mn δ H δπ m ! − ρ T in (cid:16) P ρ ⊥ (cid:17) nk δ H δπ k = − ρ (cid:16) P ρ ⊥ (cid:17) mi A mn (cid:16) P ρ ⊥ (cid:17) nk π k ρ ! + ρ (cid:16) P ρ ⊥ (cid:17) ni T mn π m ρ ! − ρ T in (cid:16) P ρ ⊥ (cid:17) nk π k ρ , (4.55)which upon substitution of the definitions of P ρ , A mn , and T mn of (4.36) and (4.50) yields acomplicated nonlinear equation.Equations (4.54) and (4.55) are infinite-dimensional versions of the finite-dimensional systemsof (2.33) and (2.34) considered in Section 2.3.1. There, equations (2.33) and (2.34) were reducedto (2.37) and (2.38) upon enforcing the holomomic constraint by requiring that initially D = D of (4.7), which is compatible with the Hamiltonian(4.52). Instead of addressing this evaluation now, we find the meaning of various terms is muchmore transparent when written in terms of Eulerian variables, which we do in Section 4.3. Wethen return to these Lagrangian equations in Section 4.4 and make comparisons. Nevertheless,the solution of equations (4.54) and (4.55), q ( a , t ), with the initial conditions D = − ln ρ and D =
0, is a volume preserving transformation at any time t .4.3. Eulerian-Dirac constraint theory
Because we chose the form of constraints D , of (4.6) and (4.7) to be Eulerianizable, it followsthat we can transform easily the results of Section 4.2.1 into Eulerian form. This we do in Section4.3.1. Alternatively, we can proceed as in Nguyen & Turski (1999, 2001); Tassi et al. (2009);Chandre et al. (2013); Morrison et al. (2009), starting from the Eulerian noncanonical theoryof Section 3.3 and directly construct a Dirac bracket with Eulerian constraints. This is a validprocedure because Dirac’s construction works for noncanonical Poisson brackets, as shown, e.g.,in Morrison et al. (2009), but it does not readily allow for advected density. This direct methodwith uniform density is reviewed in Section 4.3.2, where it is contrasted with the results ofSection 4.3.1.4.3.1. Lagrangian-Dirac constraint theory in the Eulerian picture
In a manner similar to that used to obtain (3.26) and (3.27), we find the functional derivativestransform as δ F δπ i = ρ δ ¯ F δ i , ρ δ F δ q i = ∂∂ x i δ ¯ F δρ − ρ δ ¯ F δ s ∂ s ∂ x i − ρ δ ¯ F δ ℓ ∂ ℓ ∂ x i , (4.56)where the expressions on the left of each equality are clearly Lagrangian variable quantities,while on the right they are Eulerian quantities represented in terms of Lagrangian variables.Substituting these expressions into (2.77) and dropping the bar on F and G gives the following agrangian and Dirac constraints for fluid and MHD { F , G } ∗ = − Z d x ( P ρ δ F δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ∂∂ x i δ G δρ − P ρ δ G δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ∂∂ x i δ F δρ ! + ρ ∂ s ∂ x i δ F δ s P ρ δ G δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − δ G δ s P ρ δ F δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ! + ρ ∂ ℓ ∂ x i P ρ δ F δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ P ρ δ G δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − P ρ δ G δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ P ρ δ F δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ! + ρ ∂ ℓ ∂ x ℓ P ρ δ F δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i δ G δ i − P ρ δ G δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i δ F δ i ! ) , (4.57)where we used the relations (3.19) and (3.35) and we introduced the Eulerian projection operator P ρ δ F δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i = ( P ρ ) ij δ F δ j = ρ P ρ δ F δ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i and P ρ δ F δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i = η i j P ρ δ F δ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j , (4.58)with (cid:16) P ρ (cid:17) ij z j = δ ij − η ik ∂∂ x k " ∆ − ρ ∂∂ x j z j ρ ! , (4.59)which is easily seen to satisfy ( P ρ ) ij ( P ρ ) jk = ( P ρ ) ik . Observe, like its Lagrangian counterpart, P ρ isformally self-adjoint; however, this time we found it convenient to define the projection in sucha way that the self-adjointness is with respect to a di ff erent weighted inner product, viz. Z d x ρ w i ( P ρ ) ij z j = Z d x ρ z i ( P ρ ) ij w j . (4.60)In terms of usual cartesian vector notation P ρ δ G δ v = δ G δ v − ∇ ∆ − ρ ∇ · ρ δ G δ v ! . (4.61)Upon writing P ρ = I − P ρ ⊥ and decomposing an arbitrary vector field as z = −∇ Φ + ρ ∇ × A , this projection operator yields the component P ρ z = ρ ∇ × A . Therefore, if ∇ ρ × A =
0, then thisoperator projects into the space of incompressible vector fields. For convenience we introducethe associated projector P ρ v : = v − ρ ∇ ∆ − ρ ∇ · v = ρ P ρ ( ρ v ) , (4.62)which has the desirable property ∇ · ( P ρ v ) = ∀ v compared to ∇ · ρ P ρ w ! = ∀ w . (4.63)Upon writing P ρ = I − P ρ ⊥ and decomposing an arbitrary vector field v as v = − ρ ∇ Φ + ∇ × A , this projection operator yields the component P ρ v = ∇ × A , while P ρ ⊥ v = ∇ Φ/ρ . Note, P ρ is theEulerianization of P ρ and it is not di ffi cult to write (4.64) in terms of this quantity.0 P. J. Morrison, T. Andreussi, and F. Pegoraro
Upon adopting this usual vector notation, the bracket (4.57) can also be written as { F , G } ∗ = − Z d x " ∇ δ G δρ · P ρ δ F δ v − ∇ δ F δρ · P ρ δ G δ v + ∇ s ρ · (cid:18) δ F δ s P ρ δ G δ v − δ G δ s P ρ δ F δ v (cid:19) + ∇ × v ρ · (cid:18) P ρ δ G δ v × P ρ δ F δ v (cid:19) + ∇ · v ρ (cid:18) δ F δ v · P ρ δ G δ v − δ G δ v · P ρ δ F δ v (cid:19) . (4.64)For MHD there is a magnetic field contribution to (4.56) and following the steps that lead to(4.64) we obtain { F , G } ∗ B = − Z d x " B · ρ P ρ δ F δ v · ∇ δ G δ B − ρ P ρ δ G δ v · ∇ δ F δ B ! + B · ∇ ρ P ρ δ F δ v ! · δ G δ B − ∇ ρ P ρ δ G δ v ! · δ F δ B ! . (4.65)With the exception of the last term of (4.64) proportional to ∇ · v and the presence of the Eulerianprojection operator P ρ , (4.64) added to (4.65) is identical to the noncanonical Poisson bracketfor the ideal fluid and MHD as given in Morrison & Greene (1980). By construction, we knowthat (4.64) satisfies the Jacobi identity – this follows because it was obtained by Eulerianizingthe canonical Dirac bracket in terms of Lagrangian variables. Guessing the bracket and provingJacobi for (4.64) directly would be a di ffi cult chore, giving credence to the path we have followedin obtaining it.To summarize, the bracket of (4.64) together with the Hamiltonian H = Z d x ρ | v | , (4.66)the Eulerian counterpart of (4.52), generates dynamics that can preserve the constraint ∇ · v =
0. If we add H B = R d x | B | / ∂ρ∂ t = { ρ, H } ∗ = −∇ · P ρ δ H δ v = −∇ ρ · P ρ v , (4.67) ∂ s ∂ t = { s , H } ∗ = − ∇ s ρ · P ρ δ H δ v = −∇ s · P ρ v , (4.68) ∂ v ∂ t = { v , H } ∗ = − ρ P ρ ρ ∇ δ H δρ ! + ρ P ρ (cid:18) ∇ s δ H δ s (cid:19) − ρ P ρ (cid:18) ( ∇ × v ) × P ρ δ H δ v (cid:19) − ∇ · v ρ P ρ δ H δ v + ρ P ρ (cid:18) ∇ · v δ H δ v (cid:19) = − P ρ ∇ | v | − P ρ (cid:16) ( ∇ × v ) × P ρ v (cid:17) − ( ∇ · v ) P ρ v + P ρ ( v ∇ · v ) . (4.69)If we include H B we obtain additional terms to (4.69) generated by (4.65) for the projected J × B force. Observe, equation (4.69) is not yet evaluated on the constraint D =
0, which in Eulerianvariables is ∇ · v =
0. As noted at the end of Section 4.2, we turn to this task in Section 4.4. agrangian and Dirac constraints for fluid and MHD
Eulerian-Dirac constraint theory direct with uniform density
For completness we recall the simpler case where the Eulerian density ρ is uniformly con-stant, which without loss of generality can be scaled to unity. This case was considered inNguyen & Turski (1999, 2001); Chandre et al. (2012, 2013) (although a trick of using entropyas density was employed in Chandre et al. (2013) to treat density advection). In these works theDirac constraints were chosen to be the pointwise Eulerian quantities D = ρ and D = ∇ · v , (4.70)and the Dirac procedure was e ff ected on the purely Eulerian level. This led to the projector P : = P ρ = = − ∇ ∆ − ∇ · , (4.71)where ∆ = ∆ ρ = , and the following Dirac bracket: { F , G } ∗ = − Z d x " ∇ s ρ · (cid:18) δ F δ s P δ G δ v − δ G δ s P δ F δ v (cid:19) − ∇ × v ρ · (cid:18) P δ F δ v × P δ G δ v (cid:19) . (4.72)Incompressible MHD with constant density is generated by adding the following to (4.72) { F , G } ∗ B = − Z d x " B ρ · (cid:18) P δ F δ v · ∇ δ G δ B − P δ G δ v · ∇ δ F δ B (cid:19) + B · ∇ ρ P δ F δ v ! · δ G δ B − ∇ ρ P δ G δ v ! · δ F δ B ! , (4.73)and adding | B | / ff ers from that of (4.64) in two ways: the projector P ρ is replaced bythe simpler projector P and it is missing the term proportional to ∇ · v . Given that ∇ · v cannotbe set to zero until after the equations of motion are obtained, this term gives rise to a significantdi ff erences between the constant and nonconstant density Poisson brackets and incompressibledynamics.4.4. Comparison of the Eulerian-Dirac and Lagrangian-Dirac constrained theories
Let us now discuss equations (4.67), (4.68) and (4.69). Given that ∇ · P ρ v = P ρ v ,as expected. However, the meaning of (4.69) remains to be clarified. To this end we take thedivergence of (4.69) and again use (4.63) to obtain ∂ ( ∇ · v ) ∂ t = −∇ · (cid:16) ∇ · v P ρ v (cid:17) = − (cid:16) P ρ v (cid:17) ·∇ ( ∇ · v ) . (4.74)Thus ∇ · v itself is advected by an incompressible velocity field. As with any advection equation,if initially ∇ · v = ∇ · v = ∂ v ∂ t = − P ρ ( v · ∇ v ) ; (4.75)this is the anticipated equation of motion, the momentum equation of (1.1) with the insertion ofthe pressure given by (1.5).Given the discussion of Lagrangian vs. Eulerian constants of motion of Section (3.4), that ∇ · v is advected rather than pointwise conserved is to be expected. Our development began with theconstraints D , of (4.6) and (4.7) both of which are pointwise conserved by the Dirac procedure,2 P. J. Morrison, T. Andreussi, and F. Pegoraro i.e. ˙ D L ≡
0. This means their corresponding fluxes are identically zero, i.e., in (3.38) we have Γ D L ≡ ¯ Γ D E of (3.42) vanishes and the Eulerian flux for both D and D have the form v D E . Because D and D Eulerianize to − ln( ρ ) and ∇ · v , respectively,we expected equations of the from of (4.74) for both. We will see in Section 4.5 that the equationfor D in fact follows also because the constraints are Casimir invariants.Let us return to (4.55) and compare with the results of Section 2.3.1. Because the incompress-ibility condition is an holonomic constraint and Section 2.3.1 concerns holonomic constraints forthe uncoupled N -body problem, both results are geodesic flows. In fact, one can think of the fluidcase as a continuum version of that of Section 2.3.1 with an infinity of holonomic constraints–thus we expect similarities between these results. However, because the incompressibility con-straints are pointwise constraints, the comparison is not as straightforward as it would be forglobal constraints of the fluid.To make the comparison we first observe that the term A mn of (4.49) must correspond tothe term ↔ A i j of (2.47), since their origin follows an analogous path in the derivation, both areantisymmetric, and both project from both the left and the right. The analog of (2.49) accordingto (4.36) is (cid:16) P ρ ⊥ (cid:17) ij π j ρ = η i ℓ ρ A u ℓ ∂∂ a u ∆ − ρ J A hj J ∂∂ a h π j ρ = η i ℓ ρ A u ℓ ∂∂ a u ∆ − ρ J (cid:16) D (cid:17) ≡ , (4.76)when evaluated on D =
0. Unlike (2.49) a sum, which would here be an integral over d a , doesnot occur because the constraint D is a pointwise constraint as opposed to a global constraint.Also, because the constraints are pointwise, the ↔ T i j is analogous to the terms with T mn that alsohave a factor of the projector P ρ ⊥ , giving the results analogous to (2.50). Just as in Section 2.3.1,we obtain π i = ρ ˙ q i from (4.54) when evaluated on the constraint D = T mn contributes to the momentum equation of motion (4.55). We obtain˙ π i = ρ η in (cid:16) P ρ ⊥ (cid:17) nr (cid:16) ˙ q m η rs T ms (cid:17) = A ui ∂∂ a u ∆ − ρ J A h ℓ J ∂∂ a h ˙ q m A km J ∂ ˙ q ℓ ∂ a k ! = A ui ∂∂ a u ∆ − ρ J A h ℓ J ∂∂ a h A km J ∂ ( ˙ q m ˙ q ℓ ) ∂ a k ! , (4.77)where the second equality follows upon substitution of T ms → η ℓ s A km J ∂ ˙ q ℓ ∂ a k , for D = , which follows from (4.50), while the third follows again from D = q i = η i ℓ A u ℓ ρ ∂∂ a u ∆ − ρ J A hj J ∂∂ a h A fk J ∂ ( ˙ q j ˙ q k ) ∂ a f = (cid:16) P ρ ⊥ (cid:17) ij A fk J ∂ ( ˙ q j ˙ q k ) ∂ a f = : − b Γ ijk ( ˙ q j , ˙ q k ) , (4.78)where in (4.78) we have defined b Γ ijk ( ˙ q j , ˙ q k ), the normal force operator for geodesic flow, analo-gous to that of (2.55).As was the case for the b Γ i , jk of (2.56), b Γ ijk possesses symmetry: given arbitrary vector fields V agrangian and Dirac constraints for fluid and MHD W b Γ ijk ( V j , W k ) : = − η i ℓ A u ℓ ρ ∂∂ a u ∆ − ρ J A hj J ∂∂ a h A fk J ∂ ( V j W k ) ∂ a f = b Γ ijk ( V k , W j ) . (4.79)where the second equality follows from the commutation relation of (3.9).Equation (4.78) defines geodesic flow on the group of volume preserving di ff eomorphisms,as was the case in Section 2.3.1, it does so in terms of the original coordinates, i.e., withoutspecifically transforming to normal coordinates on the constraint surfaces which here are infinite-dimensional.Now we are in position to close the circle by writing (4.78) in Eulerian form. We will dothis for the ideal fluid, but MHD follows similarly. As usual the term ¨ q i becomes the advectivederivative ∂ v /∂ t + v · ∇ v , the projector P ρ ⊥ becomes P ρ ⊥ (using ∆ − ρ = J ∆ − ρ ) when Eulerianized,and the b Γ ijk term becomes P ρ ⊥ ( ∇ · ( v ⊗ v )). Thus (4.78) is precisely the Lagrangian form of(4.75), written as follows: ∂ v ∂ t = − P ρ (cid:0) ∇ · ( v ⊗ v ) (cid:1) = − P ρ ( v · ∇ v ) . (4.80)Similarly, the Lagrangian version of (4.74) follows easily from (4.78). To see this we operatewith the counterpart of taking the Eulerian divergence on the first line of (4.77) and make use of(4.47), A hn J ∂∂ a h ˙ π n ρ = A hn J ∂∂ a h (cid:16) P ρ ⊥ (cid:17) nr (cid:16) ˙ q m η rs T ms (cid:17) = A hn J ∂∂ a h (cid:16) ˙ q m η ns T ms (cid:17) = δ n ℓ A hn J ∂∂ a h ˙ q m A km J ∂ ˙ q ℓ ∂ a k ! , (4.81)which in Eulerian variables becomes ∇ · ∂ v ∂ t + v · ∇ v ! = ∇ · ( v · ∇ v ) or ∇ · ∂ v ∂ t = ∂∂ t ∇ · v = . (4.82)In Lagrangian variables we have the trivial conservation laws˙ ρ = s = , (4.83)where the corresponding fluxes are identically zero. However, as is evident from (4.67) and(4.68) we obtain nontrivial conservation laws for ρ and s with nonzero fluxes. Thus we see again,consistent with Section 3.4, how Lagrangian and Eulerian conservation laws are not equivalent.For the special case where ρ = J = Incompressible algebra of invariants
In closing this section, we examine the constants of motion for the constrained system. ThePoisson bracket together with the set of functionals that commute with the Hamiltonian, i.e.,that satisfy { H , I a } = a = , , . . . , d , constitute the d -dimensional algebra of invariants,a subalgebra of the infinite-dimensional Poisson bracket realization on all functionals. Thissubalgebra is a Lie algebra realization associated with a symmetry group of the dynamicalsystem, and the Poisson bracket with { I a , · } yields the infinitesimal generators of the symmetries,i.e., the di ff erential operator realization of the algebra. This was shown for compressible MHDin Morrison (1982), where the associated Lie algebra realization of the 10 parameter Galilean4 P. J. Morrison, T. Andreussi, and F. Pegoraro group on functionals was described. This algebra is homomorphic to usual representations ofthe Galilean group, with the Casimir invariants being in the center of the algebra composed ofelements that have vanishing Poisson bracket with all other elements.A natural question to ask is what happens to this algebra when incompressibility is enforcedby our Dirac constraint procedure. Obviously the Hamiltonian is in the subalgebra and { H , · } ∗ clearly generates time translation, and this will be true for any Hamiltonian, but here we useHamiltonian of (4.66).Inserting the momentum P = Z d x ρ v (4.84)into (4.64) with the Hamiltonian (4.66) gives { P , H } ∗ = ∇ · v =
0. To see this, we use (4.61) to obtain P ρ δ H δ v = ρ v − ∇ ∆ − ρ ∇ · v and P ρ δ P i δ j = ρ δ i j , (4.86)which when inserted into (4.64) gives { P i , H } ∗ = − Z d x (cid:20) ρ ∂ | v | ∂ x i + i ∇ · (cid:16) ρ v − ∇ ∆ − ρ ∇ · v (cid:17) + h ( ∇ × v ) × (cid:16) ρ v − ∇ ∆ − ρ ∇ · v (cid:17)i i + ( ∇ · v ) h(cid:16) ρ v − ∇ ∆ − ρ ∇ · v (cid:17) i − ρ i i (cid:21) = , (4.87)as expected. The result of (4.87) follows upon using standard vector identities, integration byparts, and the self-adjointness of ∆ − ρ .The associated generator of space translations that satisfies the constraints is given by theoperator { P , · } ∗ , which can be shown directly. And, it follows that { P i , P j } ∗ = , ∀ i , j = , , . (4.88)Because the momentum contains no s dependence the the second line of (4.64) vanishes andusing P ρ δ P i /δ j = ρ δ i j of (4.86) it is clear the last line involving ∇ · v of (4.64) also vanishes.The result of (4.88) is obtained because the first and third lines cancel.Next, consider the angular momentum L = Z d x ρ x × v . (4.89)We will show { L i , H } ∗ = . (4.90)Using P ρ δ L i /δ v = δ L i /δ v , which follows from (4.61) with ∂ ( ǫ ik ℓ x ℓ ) /∂ x ℓ =
0, the fact that { L i , H } = P ρ = I − P ρ ⊥ , we obtain { L i , H } ∗ = Z d x " − ∇ δ L i δρ · P ρ ⊥ ( ρ v ) + δ L i δ v × ∇ × v ρ + ∇ · v ρ δ L i δ v ! · P ρ ⊥ ( ρ v ) . (4.91) agrangian and Dirac constraints for fluid and MHD P ρ ⊥ ( ρ v ) = ∇ ∆ − ρ ∇ · v and integrating by parts, we obtain { L i , H } ∗ = Z d x ∆ − ρ ( ∇ · v ) " ∇ δ L i δρ − ∇ · δ L i δ v × ∇ × v ρ + δ L i δ v ∇ · v ρ ! . (4.92)Then upon inserting δ L i δρ = ǫ i jk x j k and δ L i δ j = ρ x k ǫ ik j , (4.93)and using standard vector analysis we obtain (4.90).Because P ρ δ L i /δ v = δ L i /δ v , the first and third lines of (4.64) produce { L i , L j } ∗ = ǫ i jk L k , (4.94)just as they do for the compressible fluid (and MHD), while the fourth line manifestly vanishes.Similarly, it follows that that { L , · } ∗ is the generator for rotations.To obtain the full algebra of invariants we need { L i , P j } ∗ . However because P ρ δ P i /δ v = δ P i /δ v and P ρ δ L i /δ v = δ L i /δ v , it follows as for the compressible fluid that { L i , P j } ∗ = ǫ i jk P k .Finally, consider the following measure of the position of the center of mass, the generator ofGalilean boosts, G = Z d x ρ ( x − v t ) . (4.95)Calculations akin to those above reveal { G i , G j } ∗ = , { G i , P j } ∗ = , { G i , H } ∗ = P i , { L i , G j } ∗ = ǫ i jk G k . (4.96)Thus the bracket (4.72) with the set of ten invariants { H , P , L , G } is at once a closed subalgebraof Poisson bracket realization on all functionals and produces an operator realization of theGalilean group (see e.g. Sudarshan & Makunda 1974) that is homomorphic to the operatoralgebra of { L i , · } ∗ , { P i , · } ∗ , etc. with operator commutation relations. This remains true forMHD with the only change being the addition of H B to the Hamiltonian.Thus, the Galilean symmetry properties of the ideal fluid and MHD are not a ff ected by thecompressibility constraint. However, based on past experience with advected quantities, we doexpect a new Casimir invariant of the formˆ C [ ρ, s ] = Z d x ˆ C ( ρ, s ) . (4.97)To see that { ˆ C , F } ∗ = F , where ˆ C ( ρ, s ) is an arbitrary function of itsarguments, we calculate { F , ˆ C } ∗ = − Z d x ρ " ρ ∇ ∂ ˆ C ∂ρ − ∂ ˆ C ∂ s ∇ s · P ρ δ F δ v , (4.98)and since ∇ × ( ρ ∇ ∂ ˆ C /∂ρ − ∂ ˆ C /∂ s ∇ s ) = ∇ p , giving for (4.98) { F , ˆ C } ∗ = − Z d x ρ ∇ p · P ρ δ F δ v . (4.99)Thus, integration by parts and use of (4.63) imply { F , ˆ C } ∗ = F . Note, withoutloss of generality we can write ˆ C ( ρ, s ) = ρ U ( ρ, s ), in which case p = ρ ∂ U /∂ρ . Thus, it isimmaterial whether or not one retains the internal energy term R d x ρ U ( ρ, s ) in the Hamiltonian.Now, (4.97) is not the most general Casimir. Because both ρ and ∇· v are Lagrangian pointwiseDirac constraints, we expect the following to be an Eulerian Casimirˆ C [ ρ, s , ∇ · v ] = Z d x C ( ρ, s , ∇ · v ) , (4.100)6 P. J. Morrison, T. Andreussi, and F. Pegoraro where C is an arbitrary function of its arguments. To see that { C , F } ∗ = F ,we first observe that δ C δ v = −∇ ∂ C ∂ ∇ · v (4.101)and, as is evident from (4.61), that ∇ · ( P ρ ∇ Φ ) = Φ ; hence, all the δ C /δ v terms vanishexcept the first term of the last line of (4.64). This term combines with the others to cancel, justas for the calculation of ˆ C .For constant density, entropy, and magnetic field, the bracket of (4.72) reduces to { F , G } ∗ = − Z d x ∇ × v ρ · (cid:18) P δ F δ v × P δ G δ v (cid:19) , (4.102)whence it is easily seen that the helicity C ·∇× = Z d x v · ∇ × v (4.103)is a Casimir invariant because P ( ∇ × v ) = ∇ × v . This Casimir is lost when entropy and densityare allowed to be advected, for it is no longer a Casimir invariant of (4.64).Now, let us consider invariants in the Lagrangian description. Without the incompressibilityconstraints, the Hamiltonian has a standard kinetic energy term and the internal energy dependson ∂ q /∂ a , an infinitesimal version of the two-body interaction, if follows that just like the N -bodyproblem the system has Galilean symmetry, and because the Poisson bracket in the Lagrangiandescription (3.15) is canonical there are no Casimir invariants. With the incompressibility con-straint, the generators of the algebra now respect the constraints, with Dirac constraints beingCasimirs and the algebra of constraints now having a nontrivial center. Because the Casimirsare pointwise invariants, we expect the situation to be like that for the Maxwell Vlasov equationMorrison (1982), where the following is a Casimir C ∇· B [ B ] = Z d x C ( ∇ · B , x ) , (4.104)with C being an arbitrary function of its arguments. Because both nabla ∇· B and J are pointwiseconstraints, analogous to (4.104) we expect the following Casimir:ˆ C [ J ] = Z d a ˆ C ( J , a ) . (4.105)Indeed, only the first term of (4.49) contributes when we calculate { ˆ C , G } ∗ and this term vanishesby (4.46) because δ ˆ C δ q i = − ∂∂ a ℓ A ℓ i ∂ ˆ C ∂ J ! , (4.106)which follows upon making use of (3.7). Similarly, it can be shown that the full Casimir isˆ C [ D , D ] = Z d a ˆ C ( D , D , a ) , (4.107)a Lagrangian Casimir consistent with (4.100).For MHD, the magnetic helicity, C A · B = Z d x A · B , (4.108)where B = ∇ × A is easily seen to be preserved and a Casimir up to the usual issues regardinggauge conditions and boundary terms (see Finn & Antonsen 1985). We know that the cross agrangian and Dirac constraints for fluid and MHD C · B = Z d x v · B , (4.109)is a Casimir of the compressible barotropic MHD equations, and it is easy to verify that it is alsoa Casimir of (4.72) added to (4.73), that is for uniform density. However, it is not a Casimir forthe case with advected density, i.e., for the bracket of (4.64) added to (4.65).
5. Conclusions
In this paper we have substantially investigated constraints, particularly incompressibility forthe ideal fluid and MHD, for the three dichotomies described in Section 1.1: the Lagrangian vs.Eulerian fluid descriptions, Lagrange multiplier vs. Dirac constraint methods, and Lagrangianvs. Hamiltonian formalisms. An in depth description of the interplay between the various fluidand MHD descriptions was given, with an emphasis on Dirac’s constraint method. Although wemainly considered geodesic flow for simplicity, the Dirac’s Poisson bracket method can be usedto find other forces of constraint in a variety of fluid and plasma contexts.Based on our results, many avenues for future research are presented. We mention a few. Sincethe Hamiltonian structure of extended and relativistic MHD are now at hand (Charidakos et al. et al. et al. et al. et al. et al. ff ects (e.g. Tassi et al. et al. et al. et al. (2016); Xiao et al. (2016); Kraus et al. (2017), but there is a large body ofadditional work by these and other authors. Given how the finite-dimensional material of Section2 so strongly parallels the infinite-dimensional material of Section 4, notably the structure ofgeodesic flow, a natural avenue for future research would be to develop numerical algorithmsthat preserve this structure.Lastly, we mention that there is considerable geometric structure behind our calculations thatcould be further developed. Our results can be restated in geometric / Lie group language (see e.g.Bloch 2002). Also, Arnold’s program for obtaining the Riemann curvature for geodesic flow onthe group of volume preserving di ff eomorphisms can be explored beginning from our results ofSection 4. We did not feel this special issue would be the appropriate place to explore these ideas. Acknowledgment
PJM was supported by U.S. Dept. of Energy under contract
P. J. Morrison, T. Andreussi, and F. Pegoraro
Numerical Plasma Physics Division of the IPP, Max Planck, Garching. FP would like to ac-knowledge the hospitality of the Institute for Fusion Studies of the University of Texas at Austin.
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