On the Hamiltonian formulation of incompressible ideal fluids and magnetohydrodynamics via Dirac's theory of constraints
aa r X i v : . [ phy s i c s . p l a s m - ph ] O c t On the Hamiltonian formulation of incompressible idealfluids and magnetohydrodynamics via Dirac’s theory ofconstraints
C. Chandre a , P. J. Morrison b , E. Tassi a a Centre de Physique Th´eorique, CNRS – Aix-Marseille Universit´e, Campus de Luminy,case 907, F-13288 Marseille cedex 09, France b Department of Physics and Institute for Fusion Studies, The University of Texas atAustin, Austin, TX 78712-1060, USA
Abstract
The Hamiltonian structures of the incompressible ideal fluid, including entropyadvection, and magnetohydrodynamics are investigated by making use of Dirac’stheory of constrained Hamiltonian systems. A Dirac bracket for these systems isconstructed by assuming a primary constraint of constant density. The resultingbracket is seen to naturally project onto solenoidal velocity fields.
1. Introduction
From the early work of Lagrange [1] it became clear that ideal fluid systemspossess the canonical Hamiltonian form when one adopts a fluid element descrip-tion, the so-called Lagrangian variable description. Because the Lagrangiandescription is particle-like in nature, it is amenable to action functional andHamiltonian formulations. However, when Eulerian variables are incorporatedthe canonical Hamiltonian structure for all ideal kinetic and fluid theories isaltered because the transformation from Lagrangian to Eulerian variables isnot canonical. This results in a Hamiltonian theory in terms of noncanonicalPoisson brackets (see, e.g., [2, 3, 4, 5, 6, 7] for review).
Email addresses: [email protected] (C. Chandre), [email protected] (P. J. Morrison), [email protected] (E. Tassi)
Preprint submitted to Elsevier October 31, 2018 he present paper concerns the proper treatment of the incompressibilityconstraint of fluid mechanics in the context of the Eulerian Hamiltonian the-ory in terms of noncanonical Poisson brackets. We do this by applying Dirac’smethod for incorporating constraints in Hamiltonian theories, a central elementof which is a Dirac bracket. In the past, researchers have used Dirac brackets forvarious reasons in fluid mechanics [8, 9, 10, 11, 6, 12], but the first works to use itto explicitly enforce the incompressibility constraint for Euler’s equation in threedimensions appear to be Refs. [13, 14, 15]. Here we first extend the work of theseauthors by constructing the Dirac bracket for the ideal fluid with the inclusion ofentropy advection, which allows for the inclusion of any advected quantity likesalt concentration in the ocean. This generalization reveals that Dirac bracketsof the kind considered in Refs. [13, 14, 15], as well as our generalization, canbe written in a considerably simplified and perspicuous form in terms of theprojection operator that takes a general vector field to a solenoidal one. Withthis realization we then construct the Dirac bracket for incompressible magne-tohydrodynamics (MHD), thereby making clear its Hamiltonian structure. Wepresent these results together by starting from the full compressible ideal MHDequations, ˙ v = − v · ∇ v − ρ − ∇ (cid:18) ρ ∂U∂ρ (cid:19) + ρ − ( ∇ × B ) × B , (1)˙ ρ = −∇ · ( ρ v ) , (2)˙ B = ∇ × ( v × B ) , (3)˙ s = − v · ∇ s , (4)where v ( x , t ) is the velocity field, ρ ( x , t ) is the mass density, B ( x , t ) is themagnetic field, and s ( x , t ) is the entropy per unit mass. All of these dynamicalvariables are functions of x ∈ U ⊂ R as well as time. We suppose boundary con-ditions are such that no surface terms appear in subsequent calculations which,e.g., would be the case on a periodic box or all space. The observables of theMHD system are functionals of these fields, denoted generically by F [ ρ, v , B , s ].2n terms of these variables, this system has the following Hamiltonian (energy): H [ ρ, v , B , s ] = Z d x (cid:18) ρv + ρU ( ρ, s ) + B (cid:19) , (5)where v = | v | and B = | B | . With the MHD noncanonical Poisson bracketof Refs. [16, 2] { F, G } = − Z d x (cid:0) F ρ ∇ · G v + F v · ∇ G ρ − ρ − ( ∇ × v ) · ( F v × G v )+ ρ − ∇ s · ( F s G v − F v G s ) + (cid:0) ρ − F v · [ ∇ G B ] − ρ − G v · [ ∇ F B ] (cid:1) · B + B · (cid:0) [ ∇ (cid:0) ρ − F v (cid:1) ] · G B − [ ∇ (cid:0) ρ − G v (cid:1) ] · F B (cid:3)(cid:1) , (6)where F v denotes the functional derivative of F with respect to v , i.e. F v = δF/δ v , and the same holds for F s , F B and F ρ . Here the notation a · [ M ] · b = b · ( a · [ M ]) is a scalar explicitly given by a i M ij b j (with repeated indices summed)for any vectors a and b and any matrix (or dyad) [ M ]. The bracket (6) withHamiltonian (5) gives the MHD equations (1)–(4) in the form ˙ F = { F, H } .(Assuming ∇ · B = 0.)The paper is organized as follows: In Sec. 2 we review Dirac’s formalism forconstrained Hamiltonian systems. Then, in Sec. 3 this theory is used to ob-tain the noncanonical Poisson-Dirac bracket for the incompressible ideal MHDequations including entropy advection. Here we impose a primary constraintthat is a constant and uniform density and the rest follows from Dirac’s algo-rithm. In particular, it is seen that the corresponding secondary constraint isthat the velocity field be solenoidal. We verify that Poisson-Dirac bracket in-deed produces the correct equations of motion. This is followed in Sec. 4 by adetailed comparison to previous attempts at incorporating incompressibility inHamiltonian formulations of incompressible ideal fluids. Finally, in Sec. 5, wesummarize and conclude. The paper also has several appendices that addressvarious issues that arise in the text.
2. Dirac brackets
As stated above, Dirac’s theory is used for the derivation of the Hamiltonianstructure of Hamiltonian systems subjected to constraints. Dirac constructed3is theory in terms of canonical Poisson brackets and detailed expositions of histheory can be found in Refs. [17, 18, 19, 20, 5]. However, it is not difficult toshow that his procedure also works for noncanonical Poisson brackets (cf., e.g.,an Appendix of Ref. [10]). In this section, we recall a few basic facts about Diracbrackets in infinite dimensions in the context of noncanonical Poisson brackets.If we impose K local constraints Φ α ( x ) = 0 for α = 1 , . . . , K on a Hamil-tonian system with a Hamiltonian H and a Poisson bracket {· , ·} , the Diracbracket is obtained from the matrix C defined by the Poisson brackets betweenthe constraints, C αβ ( x , x ′ ) = { Φ α ( x ) , Φ β ( x ′ ) } , where we note that C αβ ( x , x ′ ) = − C βα ( x ′ , x ). If C has an inverse, then theDirac bracket is defined as follows: { F, G } ∗ = { F, G } − Z d x Z d x ′ { F, Φ α ( x ) } C − αβ ( x , x ′ ) { Φ β ( x ′ ) , G } , (7)where the coefficients C − αβ ( x , x ′ ) satisfy Z d x ′ C − αβ ( x , x ′ ) C βγ ( x ′ , x ′′ ) = Z d x ′ C αβ ( x , x ′ ) C − βγ ( x ′ , x ′′ ) = δ αγ δ ( x − x ′′ ) , which implies C − αβ ( x , x ′ ) = − C − βα ( x ′ , x ).This procedure is effective only when the coefficients C − αβ ( x , x ′ ) can befound. If C is not invertible, then one needs, in general, secondary constraintsto determine the Dirac bracket. The secondary constraint is given by theconsistency equation which states that ˙Φ ( x ) = 0 for the Hamiltonian H + R d x u ( x )Φ ( x ). This translates into Z d x { Φ ( x ) , H } µ ( x ) ≈ , (8)for all functions µ such that Z d x µ ( x ) C ( x , x ′ ) = 0 . Here the weak equality ≈ stands for an equality on the manifold defined byΦ ( x ) = 0. Equation (8) gives the expression which has to be satisfied by thesecondary constraint. 4 . Dirac bracket for ideal incompressible MHD To construct the Hamiltonian theory of ideal incompressible MHD, the first(primary) constraint is chosen to be a constant and uniform density ρ , i.e.Φ ( x ) = ρ ( x ) − ρ . However, the Dirac procedure can be performed for the case of a nonuniformbackground density (see Appendix A). Given that C ( x , x ′ ) = 0, at least onesecondary constraint is needed. This secondary constraint, denoted Φ ( x ), isgiven by { Φ ( x ) , H } = 0 which leads us naturally toΦ ( x ) = ∇ · v . From the Poisson bracket (6), we compute the elements C αβ ( x , x ′ ) as C ( x , x ′ ) = 0 ,C ( x , x ′ ) = ∆ δ ( x − x ′ ) ,C ( x , x ′ ) = − ∆ δ ( x − x ′ ) ,C ( x , x ′ ) = ∇ · (cid:0) ρ − ( ∇ × v ) × ∇ δ ( x − x ′ ) (cid:1) . From these expressions, we obtain the coefficients C − αβ ( x , x ′ ) as C − ( x , x ′ ) = ∆ − ∇ · (cid:0) ρ − ( ∇ × v ) × ∇ ∆ − δ ( x − x ′ ) (cid:1) ,C − ( x , x ′ ) = − ∆ − δ ( x − x ′ ) ,C − ( x , x ′ ) = ∆ − δ ( x − x ′ ) ,C − ( x , x ′ ) = 0 , where ∆ − acts on a function f as ∆ − f ( x ) = − (4 π ) − R d x ′ f ( x ′ ) / | x − x ′ | .Given the following expressions { Φ ( x ) , G } = −∇ · G v , { Φ ( x ) , G } = − ∆ G ρ − ∇ · (cid:0) ρ − ( ∇ × v ) × G v (cid:1) + ∇ · (cid:0) ρ − ∇ sG s (cid:1) −∇ · (cid:0) ρ − [ ∇ G B ] · B (cid:1) + ∇ · (cid:0) ρ − ∇ · [ B G B ] (cid:1) ,
5e deduce various contributions to the Dirac bracket (7):
Z Z d xd x ′ { F, Φ ( x ) } C − ( x , x ′ ) { Φ ( x ′ ) , G } = − Z d x ∇ · F v ∆ − ∇ · ( ρ − ( ∇ × v ) × ∇ ∆ − ∇ · G v ) , Z Z d xd x ′ { F, Φ ( x ) } C − ( x , x ′ ) { Φ ( x ′ ) , G } = Z d x ∇ · F v (cid:0) G ρ + ∆ − ∇ · ( ρ − ( ∇ × v ) × G v ) − ∆ − ∇ · ( ρ − ∇ sG s ) + ∆ − ∇ · ( ρ − [ ∇ G B ] · B ) − ∆ − ∇ · (cid:0) ρ − ∇ · [ B G B ] (cid:1)(cid:1) , Z Z d xd x ′ { F, Φ ( x ) } C − ( x , x ′ ) { Φ ( x ′ ) , G } = − Z d x (cid:0) F ρ + ∆ − ∇ · ( ρ − ( ∇ × v ) × F v ) − ∆ − ∇ · ( ρ − ∇ sF s ) + ∆ − ∇ · ( ρ − [ ∇ F B ] · B ) − ∆ − ∇ · (cid:0) ρ − ∇ · [ B F B ] (cid:1)(cid:1) ∇ · G v . From the contributions associated with C − and C − , we notice that the part − R d x ( F ρ ∇· G v + F v ·∇ G ρ ) of the Poisson bracket (6) vanishes. We also noticethat the terms in the Dirac bracket only involve¯ G v := G v − ∇ ∆ − ∇ · G v =: P · G v . (9)Two equivalent expressions for P acting on a vector a are P · a = a − ∇ ∆ − ∇ · a = −∇ × ( ∇ × ∆ − a ). The linear projection operator P acting on vectors issymmetrical, in the sense that Z d x a · P · b = Z d x b · P · a , for any vector fields a ( x ) and b ( x ). In addition, it satisfies the following prop-erties: P = P , P · ∇ = 0 , P · ∇× = ∇× , ∇ × P = ∇× , ∇ · P = 0 . As a consequence, we notice that the functional derivatives ¯ G v are divergence-free, i.e. ∇· ¯ G v = 0. In terms of ¯ G v given by Eq. (9) the Dirac bracket is writtenin the following compact form: { F, G } ∗ = Z d x (cid:0) ρ − ( ∇ × v ) · (cid:0) ¯ F v × ¯ G v (cid:1) − ρ − ∇ s · (cid:0) F s ¯ G v − ¯ F v G s (cid:1) − (cid:0) ρ − ¯ F v · [ ∇ G B ] − ρ − ¯ G v · [ ∇ F B ] (cid:1) · B − B · (cid:0) [ ∇ (cid:0) ρ − ¯ F v (cid:1) ] · G B − [ ∇ (cid:0) ρ − ¯ G v (cid:1) ] · F B (cid:3)(cid:1) . (10)Upon comparison with bracket (6), we see that this bracket is precisely thatof Refs. [16, 2] with the functional derivatives F v and G v replaced by the6ivergence-free functional derivatives ¯ F v and ¯ G v according to Eq. (9). In thisprocedure, the terms of the bracket (6) in F ρ or G ρ disappear because ∇· ¯ G v = 0.We also note that if we drop all terms but the first in Eq. (10), then with somemanipulations one can show this bracket is equivalent to the one obtained inRef. [13], albeit in a significantly simplified and perspicuous form, and that thisterm corresponds to the bracket of Ref. [21].Because the Poisson bracket (10) is exactly the bracket of Ref. [16] with thereplacement of the functional derivatives by projected functional derivatives,one wonders if one can always construct Dirac brackets by this procedure. InAppendix B it is shown that not all projections produce good brackets, onlythose that define Hamiltonian vector fields (see also Appendix C).Given that ∇ · ¯ F v = 0 for all observables F , we obtain the following familyof Casimir invariants of the Poisson bracket (10): C [ s ] = Z d x f ( s ) , where f ( s ) is any function of the entropy, i.e. it commutes with all the ob-servables, { C [ s ] , G } ∗ = 0 for all G . This family originates from the family ofCasimir invariants of the original Poisson bracket (6) given by R d x ρf ( s ), andthe fact that the Dirac constraints are also Casimir invariants. This follows sinceDirac brackets built on brackets with Casimir invariants retain those invariants(cf. Ref. [10]). As a consequence, the term R d x ρU ( ρ, s ) in the Hamiltonianis now a Casimir invariant, so that it can be dropped from the Hamiltonianbecause it will not give any contribution to the equations of motion (contraryto the compressible fluid or compressible MHD cases). Upon setting ρ = 1, theHamiltonian becomes H = 12 Z d x (cid:0) v + B (cid:1) . (11)Therefore, the Hamiltonian theory for ideal Eulerian incompressible MHD isgiven by the bracket (10) with the Hamiltonian (11). The equations of motionfollow: For the entropy s , this yields˙ s = { s, H } ∗ = − ¯ v · ∇ s, v = P · v , and evidently ∇ · ¯ v = 0. Note s can be any advected quantity such as theconcentration of salt.Similarly, the dynamical equation for B is obtained˙ B = { B , H } ∗ = − ¯ v · ∇ B + B · ∇ ¯ v = ∇ × (¯ v × B ) . The equation for v is slightly more complicated, viz.˙ v = { v , H } ∗ = −P · [( ∇ × v ) × ¯ v ] + P · [( ∇ × B ) × B ] . (12)In particular, the property that ∇ · P = 0 implies that ∇ · ˙ v = 0, which isconsistent with the constraint Φ . We notice that the first term in Eq. (12) wasobtained in Ref. [13]. Since ¯ v ≈ v (weak equality with the constraint Φ ), theequations for v and B becomes˙ v = − v · ∇ v − ∇ P c + ( ∇ × B ) × B , ˙ B = ∇ × ( v × B ) , where the pressure-like term P c is given by P c := − v − ∆ − ∇ · (cid:0) ( ∇ × v ) × v (cid:1) + ∆ − ∇ · (cid:0) ( ∇ × B ) × B (cid:1) . Given the equation for the pressure, P c is not necessarily positive. Lastly, wepoint out that there is no equation for the mass density ρ , since it has beeneliminated altogether from the theory.The equations obtained above correspond to the traditional equations forincompressible MHD. It should be noted that ∇ · v = 0 is no longer a constrainton the flow since it is a conserved quantity. Actually, it is more than a con-served quantity since it is a Casimir invariant. If one choses an initial conditionsatisfying ∇· v = 0, then this quantity will remain constant under the dynamics.
4. Comparisons between brackets for incompressible fluids
We focus now on ordinary fluids and in particular we consider different for-mulations for describing the motion of an ideal incompressible fluid. In his8amous treatise Lagrange [1] provides descriptions of both incompressible andcompressible ideal fluids. Lagrange uses what is now generally called the La-grangian variable description, whereby the dynamics of fluid elements, points,are treated in a spatial domain, and he constructs the Lagrangian for thisinfinite-dimensional system. If we let q denote the position of a fluid elementlabeled by a , that lies in a domain U ⊂ R occupied by the fluid, then q : U → U at each time, or q ( a , t ). To describe incompressible fluids, Lagrange adds theconstraint det | ∂ q /∂ a | = 1. It naturally leads to what is now referred to asthe volume preserving diffeomorphism description of the incompressible fluid.A formal description of this was introduced in Refs. [22, 23] for the Euler equa-tions of an incompressible fluid. It was based on the fact that diffeomorphismsform an infinite parameter Lie group with a Lie algebra given by the commuta-tor of vector fields. If D denotes vector fields of R , then the commutator (Liebracket) [ V , W ] L = ( W · ∇ ) V − ( V · ∇ ) W , (13)is again a vector field for any V , W ∈ D , and it is an elementary exercisein vector calculus to show the Jacobi identity, [ U , [ V , W ] L ] L + (cid:9) = 0 for all U , V , W ∈ D , where (cid:9) denotes the two other terms obtained by cyclic permu-tation of ( U , V , W ). If one restricts D to contain only divergence-free vectorfields, ¯ D := { V ∈ D|∇ · V = 0 } , then ¯ D ⊂ D is a Lie subalgebra, as seen byanother elementary vector calculation that assures closure: ∇ · [ V , W ] L = 0 if V , W ∈ ¯ D . From the Lie bracket (13) one can construct the Lie-Poisson bracket(cf., e.g., Refs.[4, 5]) { F, G } L = Z d x v · [ F v , G v ] L , (14)which indeed satisfies the Jacobi identity for all functionals of v , it being of theLie-Poisson form. However, combined with the Hamiltonian H = R d x v /
2, itdoes not yield the correct equations of motion for incompressible fluid mechanicssince ∇ · v is not conserved by the flow.9nother bracket for incompressible fluids was proposed in Ref. [24]: { F, G } = Z d x ω · [( ∇ × F ω ) × ( ∇ × G ω )] . With this bracket and the Hamiltonian H = R d x v / ω = ∇ × v , theequation ∇ × H ω = v = H v leads to ∂ ω ∂t = ∇ × ( v × ω ) , which is the correct equation of motion for the vorticity in both compressible andincompressible barotropic fluids. However, two issues should be noted: (i) thisbracket does not satisfy the Jacobi identity for functionals defined on arbitraryvector fields ω . This is easily seen by the following counter example: F = 12 Z d x ω · ˆ x y , F = 12 Z d x ω · ˆ y z , , F = Z d x ω · ˆ z x , which yields, { F , { F , F } } + (cid:9) = − Z d x ω · ∇ ( yz ) = 0 , (15)and (ii) it is not stated how the constraint ∇ · v = 0 is to be applied, andindeed the procedure of this paper also gives the correct equation of motion forcompressible barotropic fluids.With regards to (i), if one considers vector fields that satisfy ω = ∇ × v ,then Eq. (15) gives zero (see Appendix D). Thus, one might attempt to restrictthe space of functionals on which this bracket is defined in order to get a Liealgebra realization on such functionals. Yet, since ω = ∇ × v , it seems naturaljust to use v as a variable to enforce the constraint ∇ · ω = 0. This leads to abracket similar to Eq. (14), namely { F, G } • = R d x v · [ F v , G v ] • , where[ V , W ] • := ∇ × ( V × W ) = [ V , W ] L + V ( ∇ · W ) − W ( ∇ · V ) . This bracket is not of Lie-Poisson type since [ · , · ] • does not satisfy the Jacobiidentity, as can be seen from the counterexample ( V , V , V ) = ( xy ˆ x , y ˆ y , ˆ z ),giving [[ V , V ] • , V ] • + (cid:9) = y ˆ z . Thus { F, G } • has to be discarded even thoughit has the interesting property {∇ · v , F } • = 0 for all observables F .10his returns us to issue (ii) above about enforcing ∇ · v = 0. The failureof Eq. (14) to give the correct equations of motion can be traced to the use of ∇ × F ω = F v , which cannot be true for all functionals because 0 = ∇ · ∇ × F ω = ∇ · F v = 0. Note, even if F [ v ] is defined on divergence-free vector fields, it doesnot follow that ∇ · F v = 0. This suggests introducing ∇ × F ω = F v + Υ , where Υ is chosen to enforce the constraint, i.e. ∇ × F ω = F v − ∇ ∆ − ∇ · F v = P · F v . (16)Now, inserting Eq. (16) into Eq. (14), we obtain the Dirac bracket of Sec. 3.So the correct Poisson bracket for incompressible fluids can be constructed as aLie-Poisson bracket, from a projection of the Lie bracket [ · , · ] L as follows:[ V , W ] P := [ P · V , P · W ] L , where we notice an important property for verifying the Jacobi identity is P · [ P · V , P · W ] L = [ P · V , P · W ] L (cf. Appendix C). Previously the need for theprojection for the incompressible fluid was observed in Refs. [5, 21]. However,in light of our work, when projection is handled appropriately, this amounts tothe Dirac bracket construction of Ref. [13], which we here generalized.In closing this section we make a few more remarks. In the two-dimensionalformulations of Refs. [25, 2, 26] there is no issue with projection: unlike {· , ·} and {· , ·} • , the bracket given there satisfies the Jacobi identity for all functionalsof the scalar vorticity. Also, in the compressible formulation the density isadded as a dynamical variable (cf. the first term of Eq. (9) of Ref. [16]) and thevariations with respect to density in the Jacobi identity compensate the failureof Jacobi for the second term alone (see footnotes 10 and 12 of Ref. [16]). Lastlywe point out that care must be taken when inserting projections on functionalderivatives into Poisson brackets, for the resulting Poisson bracket may notsatisfy the Jacobi identity (cf. Appendix B and Appendix C).11 . Conclusions Here we have generalized the Dirac bracket approach of Ref. [13] by includingentropy advection. This produces a Hamiltonian description of an importantmissing piece of the dynamics of incompressible fluids, viz. that of density ad-vection. Recall, ∇ · v = 0 does not imply constant ρ , but that ρ be advected. Ifone chooses ∇ · v as the primary constraint then one does not obtain a bracketfor an advected density. Thus, it would appear that density advection cannot beproduced by the Dirac bracket construction. However, having done the calcula-tion with entropy advection we observe that ρ drops out of the picture and weobtain a bracket that describes advection of a quantity s by a solenoidal velocityfield. Thus, if one just reinterprets s as ρ we obtain the missing dynamics ofdensity advection.Performing the Dirac construction for MHD with density advection showedus that this approach is equivalent to direct projection of the MHD bracketof Refs. [16, 2] to solenoidal vector fields. It is now evident how to constructbrackets for a variety of incompressible models. If it is Lie-Poisson, then onecan proceed as in Appendix C and if is not, then one can step through theDirac bracket construction. In fact, the Dirac construction is more general andcan be used to enforce any compatible constraints, as seen, e.g., in AppendixA. Evidently, Dirac brackets provide a powerful tool that extends well beyondthe results of this paper. Acknowledgments
We acknowledge financial support from the Agence Nationale de la Recherche(ANR GYPSI). This work was also supported by the European Community un-der the contract of Association between EURATOM, CEA, and the FrenchResearch Federation for fusion study. The views and opinions expressed hereindo not necessarily reflect those of the European Commission. Also, PJM wassupported by U.S. Dept. of Energy Contract
Appendix A. Generalization to ideal MHD with nonuniform back-ground density
Suppose the density is constant but nonuniform, which might, e.g., be animposed stratification caused by gravity. It is interesting to see where suchan assumption leads when one follows the Dirac construction. To this end weassume Φ ( x ) = ρ − ρ ( x ) = 0, where ρ is the time-independent backgrounddensity. Proceeding as in Sec. 3, because { Φ ( x ) , Φ ( x ′ ) } = 0 we obtain thesecondary constraint that has the formΦ ( x ) = ∇ · ( ρ ( x ) v ) = 0 . (A.1)Although Eq. (A.1) is valid for compressible equilibria, to justify such a con-straint on physical grounds would require a mechanism for maintaining theconstraint or a time scale argument of some sort. We will not pursue this here.From the Poisson bracket (6), we compute the elements C αβ ( x , x ′ ) as C ( x , x ′ ) = 0 ,C ( x , x ′ ) = A δ ( x − x ′ ) ,C ( x , x ′ ) = −A δ ( x − x ′ ) ,C ( x , x ′ ) = ∇ · (cid:0) ρ ρ − ( ∇ × v ) × ∇ δ ( x − x ′ ) (cid:1) , where A is the symmetric operator A f = ∇ · ( ρ ∇ f ). Provided A is invertible,we obtain the coefficients C − αβ ( x , x ′ ) as C − ( x , x ′ ) = A − ∇ · (cid:0) ρ ρ − ( ∇ × v ) × ∇A − δ ( x − x ′ ) (cid:1) ,C − ( x , x ′ ) = −A − δ ( x − x ′ ) ,C − ( x , x ′ ) = A − δ ( x − x ′ ) ,C − ( x , x ′ ) = 0 . { Φ ( x ) , G } = −∇ · G v , { Φ ( x ) , G } = −A G ρ − ∇ · (cid:0) ρ ρ − ( ∇ × v ) × G v (cid:1) + ∇ · (cid:0) ρ ρ − ∇ sG s (cid:1) −∇ · (cid:0) ρ ρ − [ ∇ G B ] · B (cid:1) + ∇ · (cid:0) ρ ρ − ∇ · [ B G B ] (cid:1) , we deduce various contributions to the Dirac bracket (7): Z Z d xd x ′ { F, Φ ( x ) } C − ( x , x ′ ) { Φ ( x ′ ) , G } = − Z d x ∇ · F v A − ∇ · ( ρ ρ − ( ∇ × v ) × ∇A − ∇ · G v ) , Z Z d xd x ′ { F, Φ ( x ) } C − ( x , x ′ ) { Φ ( x ′ ) , G } = Z d x ∇ · F v (cid:0) G ρ + A − ∇ · ( ρ ρ − ( ∇ × v ) × G v ) −A − ∇ · ( ρ ρ − ∇ sG s ) + A − ∇ · ( ρ ρ − [ ∇ G B ] · B ) − A − ∇ · (cid:0) ρ ρ − ∇ · [ B G B ] (cid:1)(cid:1) , Z Z d xd x ′ { F, Φ ( x ) } C − ( x , x ′ ) { Φ ( x ′ ) , G } = − Z d x (cid:0) F ρ + A − ∇ · ( ρ ρ − ( ∇ × v ) × F v ) −A − ∇ · ( ρ ρ − ∇ sF s ) + A − ∇ · ( ρ ρ − [ ∇ F B ] · B ) − A − ∇ · (cid:0) ρ ρ − ∇ · [ B F B ] (cid:1)(cid:1) ∇ · G v . The Dirac bracket now reads { F, G } ∗ = Z d x (cid:16) ρ − ( ∇ × v ) · (cid:16) ˆ F v × ˆ G v (cid:17) − ρ − ∇ s · (cid:16) F s ˆ G v − ˆ F v G s (cid:17) − (cid:16) ρ − ˆ F v · [ ∇ G B ] − ρ − ˆ G v · [ ∇ F B ] (cid:17) · B − B · (cid:16) [ ∇ (cid:16) ρ − ˆ F v (cid:17) ] · G B − [ ∇ (cid:16) ρ − ˆ G v (cid:17) ] · F B i(cid:17) , (A.2)where ˆ F v = P A · F v = F v − ρ ∇ (cid:0) A − ∇ · F v (cid:1) . Observe that ∇ · ˆ F v = 0 for theseequations, as was the case for the incompressible MHD. We also notice that theDirac bracket has the same form as that for incompressible MHD, with the onlydifference being divergence-free functional derivatives ˆ F v replacing ¯ F v .In the same way as in Sec. 3, one term in the Hamiltonian corresponds toa Casimir invariant. More precisely, from the property that ∇ · ˆ F v = 0 for anyobservable F , it is shown that C [ s ] = Z d x ρf ( ρ, s ) , is a family of Casimir invariants, where f is any function of s and ρ . Thereforethe Hamiltonian is H = 12 Z d x (cid:0) ρ v + B (cid:1) , and the internal energy U plays no role in the dynamics, just as was the casefor ideal incompressible MHD. 14he two dynamical equations for s and B are similar than the ones forincompressible MHD, and are given by˙ s = − v · ∇ s and ˙ B = ∇ × ( v × B ) , since ˆ v = v − ρ ρ − ∇ ( A − ∇ · ( ρ v )) ≈ v with the secondary constraint Φ . Thedynamical equation for v becomes˙ v = − v · ∇ v + ρ − ( ∇ × B ) × B − ∇ W c , where the Bernoulli-like term W c is given by W c = − v − A − ∇ · ( ρ ( ∇ × v ) × v ) + A − ∇ · (( ∇ × B ) × B ) . Again we notice that ∇ · ( ρ v ) is conserved by the flow since it is a Casimirinvariant.It should be noted that the second constraint Φ above had a constantbackground density. Another choice would be to use the constraint Φ with ρ replacing ρ , i.e. use the following set of constraints:Φ ( x ) = ρ − ρ ( x ) and Φ ( x ) = ∇ · ( ρ v ) . Proceeding as above, the definition of the operator A naturally becomes A f = ∇ · ( ρ ∇ f ), and the expression for the Dirac bracket obtained is identical toEq. (A.2) with ˜ F v := F v − ρ ∇ ( A − ∇ · F v ), which still satisfies ∇ · ˜ F v = 0. Appendix B. Hamiltonian-Dirac Vector Fields
Let Z denote a phase space manifold that is a symplectic or Poisson manifold,and is thus equipped with a bracket operation { · , · } : C ∞ ( Z ) × C ∞ ( Z ) → R .We suppose the bracket satisfies the usual Lie enveloping algebra properties andcan thus be written in coordinates as { f, g } = ∂f∂z a J ab ∂g∂z b for functions f, g ∈ C ∞ ( Z ), i.e. f, g : Z → R . Note the bracket above is ageneric Poisson bracket and may have any form or degeneracy. Only the Liealgebra properties are required. 15e impose an even number of constraints Φ α ∈ C ∞ ( Z ), α = 1 , . . . , m , andwish to project Hamiltonian vector fields on Z , elements of X ( Z ), to Hamilto-nian vector fields that are tangent to a submanifold M := ∩ α Φ α , X ( M ).As is well-known elements of X ( Z ) are linear operators, in particular, theelement generated by f ∈ C ∞ ( Z ) has the form L f = −{ f, · } = J ab ∂f∂z b ∂∂z a , and the commutator of two such elements satisfies [ L f , L g ] = − L { f,g } . Thusthere is an isomorphism between the Lie algebra of such linear operators andPoisson brackets. We wish to maintain this structure for Hamiltonian vectorfields projected onto X ( M ).To project a Cartesian vector into a surface defined by φ =constant, oneuses the normal ∇ φ to construct the following projection operator: P := I − ∇ φ ∇ φ |∇ φ | (B.1)where I is the identity. Evidently P · ∇ φ ≡
0. Essentially this same ideaoccurs in infinite dimensions in the context of Hilbert spaces and is efficaciousfor application in quantum mechanics. However, the problem at hand differsfrom these cases in that we are interested in Hamiltonian vector fields (finite orinfinite) and our manifold is symplectic with no intrinsic notion of metric. Thus,if we are to proceed without adding additional structure, we must construct aprojection operator using only the functions Φ α and cosymplectic form, J . WithEq. (B.1) as a guide we write P ab = δ ab − K αβ ∂ Φ α ∂z b ∂ Φ β ∂z c J ac where K αβ is chosen so that Hamiltonian vector fields generated by any of theΦ α are projected out, i.e. P · L Φ α ≡ α . Now it is desired to find such a K αβ in terms of the { Φ α } and J alone. Fortunately, a direct calculation revealsthat the desired quantity is given by K αβ = { Φ α , Φ β } − . Thus we have achievedour goal if this inverse exists. Assuming this is the case we obtain the following16amiltonian projection operator: P ab = δ ab − { Φ α , Φ β } − ∂ Φ α ∂z b ∂ Φ β ∂z c J ac . EvidentlyΛ af := P ab L bf = P ab J bd ∂f∂z d = J ad ∂f∂z d − { Φ α , Φ β } − ∂ Φ α ∂z b ∂ Φ β ∂z c J ac J bd ∂f∂z d , (B.2)and Λ Φ α ≡ α . Also, an elementary calculation reveals the P = P , asexpected for a projection operator.It remains to show that the set of projected vector fields of the form Λ f = P · L f are Hamiltonian on the constraint submanifold:[Λ f , Λ g ] = − Λ { f,g } ∗ , (B.3)for some well-defined Poisson bracket { f, g } ∗ . As the notation suggests thisturns out to be the Dirac bracket.It is evident from Eq. (B.2) that Λ af = −{ f, · } ∗ . Because Eq. (B.3) issatisfied for a generic Poisson bracket, then it must be true for the Dirac bracketas well. To see that it is true for a generic Poisson bracket we write[ L f , L g ] = L f L g − L g L f = J bc ∂ c f ∂ b ( J rs ∂ s g∂ r ) − ( f ↔ g )= [ J bc ∂ b J rs ∂ c f ∂ s g + J bc J rs ∂ c f ∂ b ∂ s g ] ∂ r − ( f ↔ g )= − [ J rb ∂ b J cs ∂ c f ∂ s g + J sc J rb ∂ b ( ∂ c f ∂ s g )= − L { f,g } , where ∂ b := ∂/∂z b and ∂ b operates only on the term immediately to its right un-less parenthesis are included. In obtaining the second equality, second derivativeterms canceled in the usual way, and in obtaining the third equality, antisym-metry, the Jacobi identity, and relabeling were used.All of the above can be formally extended to infinite dimensions (see, e.g.,Ref. [2]) by replacing partial derivatives by functional derivatives, sums by in-tegrals, and matrix multiplication by operator action.17 ppendix C. Projections and Poisson brackets Consider the general Poisson bracket, { F, G } = Z dµ δFδχ J δGδχ , where J is a cosymplectic operator (generally dependent on χ ( µ )) that ensuresthis bracket satisfies the Jacobi identity. Now suppose P is some projectionoperator, and consider { F, G } = Z dµ P (cid:18) δFδχ (cid:19) J P (cid:18) δGδχ (cid:19) = Z dµ δFδχ P † J P δGδχ . (C.1)The bracket (C.1) does not in general satisfy the Jacobi identity. However, if P is independent of χ it may.If the bracket is Lie-Poisson, then projection onto subalgebras always pro-duces brackets that satisfy the Jacobi identity. Consider the Lie-Poisson bracket { F, G } = Z dµ δFδχ J δGδχ = h χ, [ F χ , G χ ] i . In this construction F χ ∈ g where g is a Lie algebra and hence F χ is a vector.Suppose P : g → k , where k is a vector subspace of g . Then, (i) the bracket { F, G } P = h χ, [ P F χ , P G χ ] i , (C.2)is defined, and (ii) it satisfies the Jacobi identity for arbitrary functionals of χ ,provided P [ P F χ , P G χ ] = [ P F χ , P G χ ], which is the case if k is a subalgebra of g . This follows from the general Jacobi identity theorem proven in Ref. [2] ormore immediately from the fact that Eq. (C.2) is a Lie-Poisson bracket for k . Appendix D. Direct proof of Jacobi identity
We consider the bracket { F, G } = Z d x ω · [( ∇ × F ω ) × ( ∇ × G ω )] (D.1)In order to prove the Jacobi identity for this kind of bracket, one only needs toconsider the explicit dependence of the bracket on ω when taking the functional18erivative δ { F, G } /δ ω . In what follows, we let f := ∇ × F ω . The functionalderivative of { F, G } with respect to ω contains three terms: one that comesfrom the explicit dependence of the bracket on ω and two other terms thatare the second order functional derivatives of F and G . It has been shown inRef. [2] that the only important term comes from the explicit dependence onthe variables, i.e. on ω . So from { F, G } ω = f × g , we get { F, { G, H }} = Z d x ω · ( f × ∇ × ( g × h )) . Since ∇ · f = 0, this becomes { F, { G, H }} = Z d x ω · [ f × ( h · ∇ ) g − f × ( g · ∇ ) h ] . (D.2)If ∇ · ω = 0, there exists a vector v such that ω = ∇ × v . By symmetry of theoperator ∇× , we obtain terms like ∇ × [ f × ( h · ∇ ) g − f × ( g · ∇ ) h ], which aretransformed by the identity ∇ × [ f × ( h · ∇ ) g ] = f ∇ · [( h · ∇ ) g ] + [( h · ∇ ) g · ∇ ] f − ( f · ∇ )[( h · ∇ ) g ] . Since ∇ · f = 0 (for all observable F ), the divergence terms in Eq. (D.2) vanish,i.e. ∇ · [( h · ∇ ) g ] = ∇ · [( g · ∇ ) h ]. In addition, from the following identity( f · ∇ )[( h · ∇ ) g ] = [( f · ∇ ) h · ∇ ] g + f i h j ∂ i ∂ j g , we have ∇× [ f × ( h ·∇ ) g − f × ( g ·∇ ) h ] = ( h , g , f ) − ( f , h , g ) − ( g , h , f )+( f , g , h ) − f i h j ∂ i ∂ j g + f i g j ∂ i ∂ j h , where ( f , g , h ) := [( f · ∇ ) g · ∇ ] h . By adding the cyclic permutations of F , G and H , we obtain ∇ × [ f × ( h · ∇ ) g − f × ( g · ∇ ) h ]+ (cid:9) = 0 . As a consequence, the bracket (D.1) satisfies the Jacobi identity if ∇ · ω = 0.From the bracket (D.1) with ∇ · ω = 0, we perform the following change ofvariables: ω = ∇ × v . This change of variables depends on a gauge since v + ∇ φ gives the same value for ω . For instance, if we choose ∇ · v = 0, we obtain ∇ × F ω = F v − ∇ ∆ − ∇ · F v . Thus, we end up with the Poisson bracket obtained with Dirac’s procedure forconstrained Hamiltonian systems. 19 eferences [1] J.L. Lagrange, M´ecanique Analytique, English Title: Analytical Mechan-ics, translated and edited by A. Boissonnade and V.N. Vagliente (KluwerAcademic, Imprint Dordrecht, Boston, Mass. 1997).[2] P.J. Morrison, in
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