Casimir invariants and the Jacobi identity in Dirac's theory of constrained Hamiltonian systems
aa r X i v : . [ m a t h - ph ] M a r Casimir invariants and the Jacobi identity in Dirac’stheory of constrained Hamiltonian systems
C. Chandre
Centre de Physique Th´eorique, CNRS – Aix-Marseille Universit´e, Campus deLuminy, case 907, 13009 Marseille, FranceE-mail: [email protected]
Abstract.
We consider constrained Hamiltonian systems in the framework of Dirac’stheory. We show that the Jacobi identity results from imposing that the constraints areCasimir invariants, regardless of the fact that the matrix of Poisson brackets betweenconstraints is invertible or not. We point out that the proof we provide ensures thevalidity of the Jacobi identity everywhere in phase space, and not just on the surfacedefined by the constraints. Two examples are considered: A finite dimensional systemwith an odd number of constraints, and the Vlasov-Poisson reduction from Vlasov-Maxwell equations. asimir invariants and the Jacobi identity in Dirac’s theory
1. Introduction
Imposing constraints on a Hamiltonian system is routinely done using Dirac’s theory [1,2, 3, 4]. The key point is to compute the matrix C of the Poisson brackets betweentwo constraints (i.e., whose elements are C nm = { Φ n , Φ m } where Φ n ( z ) = 0 are theconstraints) and to invert this matrix. Under this hypothesis, it has been shown thatthe following bracket, called Dirac bracket, { F, G } ∗ = { F, G } − { F, Φ n } D nm { Φ m , G } , (1)where D = C − , is a Poisson bracket. The technical difficulty is to prove the Jacobiidentity, and this has been done by Dirac in Ref. [1], and his proof relies heavily on theinvertibility of C (see Eq. (59) which results from Eq. (35) in Ref. [1], or Appendix Bin Ref. [5]). As a consequence of the fact that D is the inverse of C , the constraints areCasimir invariants, i.e., { Φ n , G } ∗ = 0 for any observable G .What if the matrix C is not invertible? This could happen for instance if there is anodd number of constraints (since C is antisymmetric) or if the constraints are reducible orredundant [6, 9, 7, 8]. The purpose of the present article is to address this question. Ourmain result is to show that the Jacobi identity is a property which results from imposingthat the constraints are Casimir invariants, regardless of the invertibility of C . Thisresult holds for canonical or non-canonical Hamiltonian systems (see Refs. [10, 11, 12]for an introduction to non-canonical Hamiltonian systems). We notice that the proofwe provide shows the validity of the Jacobi identity everywhere in phase space, and notjust on the surface defined by the constraints as found in Refs. [6, 7, 8].We consider a finite dimensional Hamiltonian system whose variables are denoted z = ( z , z , . . . , z N ). It is given by a Hamiltonian H ( z ), a scalar function of the variables,and a Poisson bracket written as { F, G } = ∂F∂ z · J ( z ) ∂G∂ z , (2)where the Poisson matrix J ( z ) is such that the bracket (2) is antisymmetric and satisfiesthe Jacobi identity [in addition to the bilinearity and the Leibnitz rule which are alreadyensured by the form of the bracket (2)].We impose a set of M < N − m ( z ) = 0 for m = 1 , . . . , M whichare scalar functions of the variables z . We consider brackets of the form (1) with anantisymmetric matrix D which is not necessarily the inverse of C (and consequentlythese brackets are not not Poisson brackets in general). The matrix associated with thebracket (1) is given by [13] J ∗ = J − J ˆ Q † D ˆ Q J , (3)where the matrix ˆ Q has elements ˆ Q ni = ∂ Φ n /∂z i . The matrix D is chosen such that theconstraints are Casimir invariants. This leads to the following condition on D : J ˆ Q † ( − DC ) = 0 , (4)where is the M × M identity matrix and C = ˆ Q J ˆ Q † . A first situation is when C isinvertible, and hence a possible solution to Eq. (4) is D = C − . This is the main case asimir invariants and the Jacobi identity in Dirac’s theory C is not invertible, and, to the best of ourknowledge, there is no proof of the Jacobi identity in these cases which holds everywherein phase space. There are two properties which are strongly dependent on the particularchoice of matrix D , which are ( i ) the Jacobi identity and ( ii ) the fact that the constraintsare Casimir invariants. The purpose of this article is to show that ( ii ) implies ( i ), or inother words that imposing that the constraints are Casimir invariants, i.e., Eq. (4), issufficient to ensure that the Dirac bracket (1) is a Poisson bracket. The proof is donein Sec. 2. In Sec. 3 we illustrate the computation of the Dirac bracket in cases where C is not invertible, with two examples: a finite-dimensional Hamiltonian system withan odd number of constraints, and the Vlasov-Poisson reduction from Vlasov-Maxwellequations.Before going into the proof of the Jacobi identity, we would like to address twoquestions:1) Does the Jacobi identity for the bracket (1) imply that the constraints are Casimirinvariants, i.e., does ( i ) implies ( ii )? The answer is no, and we provide a counter examplebelow. Consider a Poisson matrix J written in block form as J = C
00 ¯ J ! , where C and ¯ J satisfy individually the Jacobi identity so that the matrix J too. Thebracket (1) is characterized by a matrix J ∗ = C ( − DC ) 00 ¯ J ! , for ˆ Q = ( , C is invertible, we choose D = C − (1 − λ ) so that˜ C = C ( − DC ) = λ C which satisfies the Jacobi identity inherited from C . In order tohave the constraints as Casimir invariants, ˜ C must vanish. Therefore the bracket (1)satisfies the Jacobi identity in this case, but does not have the constraints ( z , . . . , z M )(where M is the number of columns of C ) as Casimir invariants.2) Given that Eq. (4) might have more than one solution, does it lead to differentexpressions for the Dirac bracket? The answer is no. We consider D a solution of Eq. (4).We notice that any matrix ˜ D = D + (cid:1) with J ˆ Q † (cid:1)C = 0 also satisfies Eq. (4). The Diracbracket is obtained from the Dirac projector [13] P = 1 − J ˆ Q † D ˆ Q , i.e., J ∗ = P J P † . Ifwe consider the other projector associated with ˜ D , i.e., ˜ P = P − J ˆ Q † (cid:1) ˆ Q , then we showthat ˜ P J ˜ P † = P J P † , where we use the identity P J ˆ Q † = 0. This identity ensures that the Dirac bracket isunique, even if there might be more than one solution to Eq. (4). asimir invariants and the Jacobi identity in Dirac’s theory
2. Proof of Jacobi identity
A proof of a weak version of the Jacobi identity, i.e., the validity of the Jacobi identityon the surface defined by the constraints, has been detailed, e.g., in Refs. [6, 7, 8]. It isbased on showing that { F, { G, H } ∗ } ∗ ≈ { F ′ , { G ′ , H ′ }} , where F ′ = F − { F, Φ n } D nm Φ m in order to deduce that { F, { G, H } ∗ } ∗ + { H, { F, G } ∗ } ∗ + { G, { H, F } ∗ } ∗ ≈ . Here we show that the Jacobi identity holds everywhere in phase space, i.e., the weakequality can be made a strong one, i.e., { F, { G, H } ∗ } ∗ + { H, { F, G } ∗ } ∗ + { G, { H, F } ∗ } ∗ = 0 . As a consequence, even in the case where the matrix C is not invertible, the Diracbracket (1) [if it can be constructed using Eq. (4)] is a Poisson bracket.First we perform a local change of coordinates such that the new variables are theconstraint functions. This can be done at least locally under the assumption of thechange of coordinates. In other terms, we assume that Φ k ( z ) = z k for k ∈ [1 , M ]. Weassume that J satisfies the Jacobi identity. In the variables ( Φ , w ) the Poisson matrixis expressed by blocks J = C − B † B ¯ J ! . The Poisson matrix associated with the bracket (1) is given by J ∗ = C ( − DC ) − ( − CD ) B † B ( − DC ) ¯ J + BDB † ! . We assume that D is chosen such that the constraints are Casimir invariants of J ∗ . Thiscondition becomes: C ( − DC ) = 0 , (5) B ( − DC ) = 0 . (6)The Jacobi identity for J translates into four sets of equations C il ∇ l C jk − B li ∂ l C jk + (cid:9) ( ijk ) = 0 for i, j, k ∈ [1 , M ] , (7) C jl ∇ l B ki − C il ∇ l B kj + B li ∂ l B kj − B lj ∂ l B ki + B kl ∇ l C ij + ¯ J kl ∂ l C ij = 0 for i, j ∈ [1 , M ] and k ∈ [1 , N − M ] , (8) B jl ∇ l B ki − B kl ∇ l B ji + ¯ J jl ∂ l B ki − ¯ J kl ∂ l B ji + C il ∇ l ¯ J jk − B li ∂ l ¯ J jk = 0 for i ∈ [1 , M ] and j, k ∈ [1 , N − M ] , (9) B il ∇ l ¯ J jk + ¯ J il ∂ l ¯ J jk + (cid:9) ( ijk ) = 0 for i, j, k ∈ [1 , N − M ] , (10) asimir invariants and the Jacobi identity in Dirac’s theory (cid:9) ( ijk ) designates the terms obtained by circular permutations of the indices( i, j, k ), ∇ l = ∂/∂ Φ l and ∂ l = ∂/∂w l . So when the index l is involved with ∇ , theimplicit sum runs from l = 1 to M . When it is involved with ∂ , the implicit sum runsfrom l = 1 to N − M .Given Eqs. (5)-(6), the Jacobi identity for J ∗ reduces to(¯ J + BDB † ) il ∂ l (¯ J + BDB † ) jk + (cid:9) ( ijk ) = 0 . (11)The aim is to use Eqs. (7)-(10) together with Eq. (6) in order to prove Eq. (11).Equation (11) can be decomposed into three sets of terms : S ijk = ¯ J il ∂ l ¯ J jk + ( BDB † ) il ∂ l ¯ J jk + (cid:9) ( ijk ) , (12) T ijk = ¯ J il ∂ l ( BDB † ) jk + (cid:9) ( ijk ) , (13) U ijk = ( BDB † ) il ∂ l ( BDB † ) jk + (cid:9) ( ijk ) . (14)Here we notice that all indices i , j , k belong to [1 , N − M ]. Using Eq. (10), the S termscan be rewritten as S ijk = − B il ∇ l ¯ J jk + B im D mn B ln ∂ l ¯ J jk + (cid:9) ( ijk ) . By rewriting B ln ∂ l ¯ J jk using Eq. (9), a cancellation is obtained from Eq. (6), and the S terms are rewritten as S ijk = B im D im ( B jl ∇ l B kn − B kl ∇ l B jn + ¯ J jl ∂ l B kn − ¯ J kl ∂ l B jn )+ (cid:9) ( ijk ) . By noticing a cancellation in the terms
BDJ ∂B in S and T (using a circular permutationof the indices ( i, j, k ) and the antisymmetry of D ), we obtain S ijk + T ijk = B im D mn ( B jl ∇ l B kn − B kl ∇ l B jn )+ ¯ J il B jm B kn ∂ l D mn + (cid:9) ( ijk ) . (15)Concerning the U terms, we decompose them into two parts : U (1) ijk = B im D mn B ln D pq B jp ∂ l B kq + B im D mn B ln D pq B kq ∂ l B jp + (cid:9) ( ijk ) ,U (2) ijk = B im D mn B ln B jp B kq ∂ l D pq + (cid:9) ( ijk ) . The second term of U (1) is rewritten as B jp D pq D mn B im B lq ∂ l B kn using a circularpermutation of the indices ( i, j, k ). Therefore, we have U (1) ijk = B im D mn D pq B jp ( B ln ∂ l B kq − B lq ∂ l B kn )+ (cid:9) ( ijk ) , where we have used the antisymmetry of D . From Eq. (8), U (1) is rewritten as U (1) ijk = − B im D mn ( BDC ) jl ∇ l B kn + B jm D mn ( BDC ) il ∇ l B kn − B im B jp B kl D mn D pq ∇ l C nq − B im B jp D mn D pq ¯ J kl ∂ l C nq + (cid:9) ( ijk ) . (16)From Eq. (6) and using a circular permutation on the indices ( i, j, k ), the first line of theprevious equation cancels with the BDB ∇ B terms in Eq. (15). Concerning the fourthterm in Eq. (16), we show that − ¯ J kl B jp D pq B im D mn ∂ l C nq = − ¯ J kl B im B jn ∂ l D mn . asimir invariants and the Jacobi identity in Dirac’s theory B iq − B im D mn C nq = 0 with respect to z l together with Eq. (6) and the antisymmetry of D . Using a circular permutation of( i, j, k ) these terms cancel with the J BB∂D terms of Eq. (15).After these steps, the Jacobi identity is rewritten as S ijk + T ijk + U ijk = B im D mn B ln B jp B kq ∂ l D pq − B im B jp B kl D mn D pq ∇ l C nq + (cid:9) ( ijk ) . (17)Next the strategy is to get rid of the ∂D terms. In order to do this, we insert C terms through a B coefficient in the first term of the previous equation, i.e., inserting B jp = B jα D αβ C βp . From the identity C βp ∂ l D pq B kq = ∂ l B kβ − ∂ l C βp D pq B kq − C βp D pq ∂ l B kq , which is obtained by differentiating Eq. (6) with respect to z l (and using theantisymmetry of D and C ), the first term in Eq. (17) is rewritten as B im D mn B ln B jp B kq ∂ l D pq = − B im D mn B ln B jα D αβ D pq B kq ∂ l C βp + B im D mn B ln B jα D αβ ( ∂ l B kβ − C βp D pq ∂ l B kq ) . (18)From Eq. (7) we replace ∂C by ∇ C . The first term becomes B im D mn B jα D αβ B kq D qp B ln ∂ l C βp + (cid:9) ( ijk ) = B im D mn B jα D αβ B kq D qp C nl ∇ l C βp + (cid:9) ( ijk ) . From the equation B im D mn C nl = B il [see Eq. (6)] we see that the previous term cancelswith the second term in Eq. (17), still using a circular permutation of the indices ( i, j, k ).In a similar way, the second term in Eq. (18) vanishes by inserting B jα D αβ C βp = B jp .Consequently, we have proved the Jacobi identity for the bracket (1) with Eqs. (5)-(6),i.e., S ijk + T ijk + U ijk = 0 .
3. Examples
First we describe a rather trivial example in order to illustrate the method. We considerthe following Poisson matrix J = − z z z − z − z z −
10 0 0 1 0 , which corresponds to the Poisson bracket { F, G } = − z · ∂F∂ z × ∂G∂ z − ∂F∂w ∂G∂w + ∂F∂w ∂G∂w . asimir invariants and the Jacobi identity in Dirac’s theory k ( z , w ) = z k for k = 1 , ,
3. The associated operator ˆ Q is given by ˆ Q = . The matrix C is a 3 × C = − z z z − z − z z . The following matrix D satisfies Eq. (4) : D = − . The solution of Eq. (4) is not unique, but all solutions lead to the same Dirac bracketwith Poisson matrix given by Eq. (11) : J ∗ = −
10 0 0 1 0 , which corresponds to the Dirac bracket { F, G } ∗ = − ∂F∂w ∂G∂w + ∂F∂w ∂G∂w . The second example concerns the Vlasov-Poisson reduction from the Vlasov-Maxwellequations. The field variables χ ( z ) = ( f ( x , v ) , E ( x ) , B ( x )) where z = ( x , v ), and theequations of motion are˙ f = − v · ∇ f − ( E + v × B ) · ∂ v f, ˙ E = ∇ × B − J , ˙ B = −∇ × E , where J = R d v v f . The Poisson bracket is given by { F, G } = Z d zF χ · J G χ , asimir invariants and the Jacobi identity in Dirac’s theory F χ is the functional derivative of the functional F with respect to the fieldvariables χ . The Poisson matrix is given by (see Refs. [14, 15, 13]) J = − [ f, · ] − ∂ v f − f ∂ v δ ( v ) ∇× − δ ( v ) ∇× , where the small bracket [ · , · ] is given by[ f, g ] = ∇ f · ∂ v g − ∂ v f · ∇ g + B · ( ∂ v f × ∂ v g ) , where ∇ (resp. ∂ v ) is the partial derivative operator with respect to x (resp. v ).In order to obtain the Vlasov-Poisson equations from the Vlasov-Maxwell equationswe impose the following constraints: Q [ f, E , B ]( x ) = ( ∇ × E , B − B ( x )) , (19)where B is a non-uniform background magnetic field. The operators ˆ Q is given byˆ Q = ∇×
00 0 1 ! . The operator C is given by C = ∇× ) − ( ∇× ) ! . (20)The operator C is not invertible; however, the Dirac procedure still applies with anappropriate choice for D given by D = − − ∆ − ! , (21)so that Eq. (4) is satisfied. We notice that ˆ Q ( − DC ) = 0 but J ˆ Q ( − DC ) = 0. Thisis due to the fact that ∇ · B is a Casimir invariant of J .As a result, the Poisson operator of the Vlasov-Poisson equations is given by J ∗ = − [ f, · ] −∇ ∆ − ∇ · ∂ v f −∇ ∆ − ∇ · ( f ∂ v ) 0 00 0 0 . It leads to the expression of the Poisson bracket [13], { F, G } ∗ = Z d z f [ F f − ∆ − ∇ · F E , G f − ∆ − ∇ · G E ] . In the case of the constraints given by Eq. (19), one of the constraints, ∇ · B , is alreadya Casimir invariant of the Vlasov-Maxwell bracket, and hence a first-class constraint.This kind of redundancy is a source for the non-invertibility of the matrix C . In the asimir invariants and the Jacobi identity in Dirac’s theory B , i.e., we consider the following constraints : Q [ f, E , B ]( x ) = ( ∇ × E , P ( B − B ( x ))) , where P is the projector on the solenoidal part given by P = 1 − ∇ ∆ − ∇· . Using similarcalculations as above, we find ˆ Q = ∇×
00 0 P ! , and C remains unchanged and is given by Eq. (20). The matrix D given by Eq. (21)satisfies ˆ Q † (1 − DC ) = 0. We have eliminated some redundancy at the origin of thenon-invertibility of C . As we shall see below, this kind of redundancy is not an issue inthe proposed approach to compute the Dirac bracket.
4. Concluding remarks
We consider an ensemble of M constraints Φ n of which the first K ones are Casimirinvariants of the original Poisson bracket (a particular family of first-class constraints).The matrix C is written as C = C ! , where ˜ C is assumed to be invertible. Given that Φ k for k = 1 , . . . , K are Casimirinvariants, J ˆ Q † = (0 , J ˆ Q † ) where ˆ Q is the matrix of the derivatives of Φ K +1 , . . . , Φ n with respect to the phase space variables. It is straightforward to show that any matrixof the form D = D D − D † ˜ C − ! , satisfies Eq. (4). Therefore in the definition of the constraints there is no need to worryabout possible combinations giving rise to Casimir invariants of the original Poissonbracket. Notice that if some of the constraints are first-class but not Casimir invariants,this might lead to some inconsistencies in Eq. (4) like the ones in Refs. [16, 6].Another case of redundancy is when one (or several) constraints can be obtainedfrom the other constraints. For instance, we assume that one constraint Φ is dependentof the other constraints, i.e., Φ = f (Φ , . . . , Φ M ). In this case, the usual Dirac’sprocedure cannot be carried out since the matrix C is not invertible. This matrix isgiven by C = − b † b ˜ C ! , where b = ( { Φ , Φ } , { Φ , Φ } , . . . , { Φ M , Φ } ) † and the coefficients of ˜ C are { Φ n , Φ m } for n, m ≥
2. In fact the vector ( − , d , . . . , d M ) where d m = ∂f /∂ Φ m , belongs to the asimir invariants and the Jacobi identity in Dirac’s theory C . We assume that ˜ C in invertible. Then the following matrix D satisfiesEq. (4) D = − d † d ˜ C − ! , where d = ( d , . . . , d M ). Here we have used the two properties, b † d = 0 and ˜ C d = b .As a consequence this kind of redundancy is properly handled using Eq. (4) instead ofthe too stringent requirement that the matrix C is invertible.In summary the requirement for the existence of a Dirac-like bracket (1) is obtainedby imposing Eq. (4), i.e., that the constraints are Casimir invariants. It translates intoKer C ⊂ Ker J ˆ Q † , (22)which is a necessary and sufficient condition for the existence of a solution to Eq. (4).Indeed we denote r the rank of C and we rewrite it as C = O C ! O † , where O is an orthogonal matrix and ˜ C is invertible. A possible antisymmetric solutionto Eq. (4) is given by D = O C − ! O † , given the condition (22).If the specific choice of constraints is such that Eq. (22) is satisfied, then the Diracbracket (1) can be computed from Eq. (4), and it is a Poisson bracket everywhere inphase space, and not just on the surface defined by the constraints. Otherwise someobstructions are present, and one should modify the set of constraints so as to reducethe kernel of C . Acknowledgments
This work was supported by the Agence Nationale de la Recherche (ANR GYPSI) and bythe European Community under the contract of Association between EURATOM, CEA,and the French Research Federation for fusion study. The views and opinions expressedherein do not necessarily reflect those of the European Commission. CC acknowledgesfruitful discussions with P.J. Morrison and with the ´Equipe de Dynamique Nonlin´eaireof the Centre de Physique Th´eorique of Marseille.
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