- Email: [email protected]
Canonical description of incompressible fluid: Dirac brackets approach
Physica A 272 (1999) 48–55
www.elsevier.com/locate/physa
Canonical description of incompressible
uid: Dirac brackets approach Sonnet Nguyen a; ∗ , Lukasz A. Turskib a Center
for Theoretical Physics, Polish Academy of Science. Al. Lotnikow 32 46, 02-668 Warszawa, Poland b Center for Theoretical Physics, Polish Academy of Science and College of Science, Al. Lotnikow 32 46, 02-668 Warszawa, Poland Received 12 May 1999
Abstract We present a novel canonical description of the incompressible uid dynamics. This description uses the dynamical constraints, in our case re ecting incompressibility assumption, and leads to replacement of usual hydrodynamical Poisson brackets for density and velocity elds with Dirac brackets. The resulting equations are then known nonlinear, and nonlocal in space, equations for c 1999 Elsevier Science B.V. All rights reserved. incompressible uid velocity. PACS: 47.40; 47.90 Keywords: Fluid dynamics; Poisson brackets; Hamiltonian formulation; Dirac constraints
1. Introduction Canonical description of classical, compressible, isothermal uid has been developed in the past [1–7] for various purposes, for example description of super uid 4 He [1– 4] or in kinetics of the rst-order phase transformations [5,7]. An attempt to extend this formulation for the adiabatic ows has been proposed [8] and used to analyze the dynamical properties of thermally driven ows. The isothermal ow canonical description can be generalized for the case of viscous uids [9] within the framework of the metriplectic dynamics [10]. The Madelung representation for the wave function results in hydrodynamic-like picture of quantum mechanics, where the “only” di erences from Euler equations are hidden in the quantum pressure term, which is proportional to ˝2 , ∗
Corresponding author. E-mail address: [email protected] (S. Nguyen)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 1 9 4 - 6
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
49
and in the quantization of the circulation, = n(˝=m). Apart of that the canonical description of the quantum uid is then identical to that of the classical one. The dissipative generalization of the Schrodinger equation [11–14] also allows for metriplectic interpretation, which di ers from the classical one [15]. The fundamental point in all of the above listed formulations of uid dynamics is that the uid density % is one out of the pair of canonically conjugated variables. For potential ows the other canonical variable is the velocity potential . In case of general ow the two additional Clebsh potentials ; [16] in the velocity eld representation C = −∇ − ∇ are canonically conjugated to each other. None of these descriptions can be applied to the case of incompressible uids. The canonical description of incompressible ow is of considerable importance, for example in the formulation of statistical mechanics of turbulent ow [9]. Many attempts to provide such a canonical formalism [17] failed to do so. The other important point is that real turbulent ows are hardly incompressible, and the issues like compressibility corrections to scaling laws for turbulent ows are still open [18]. In this paper we propose a novel formulation of a canonical description of the incompressible uid based on the concept of the Dirac brackets [19]. The mathematical introduction to this formalism valid for general dynamic system subjected to some set of constraints {a = 0; a = 1 : : : N } can be found in [20]. Dirac bracket approach to description of incompressible membranes, within Lagrangian coordinates formulation of continuous mechanics of membranes, was given in [21]. We are unaware of any other application of that formalism to continuum mechanics problems. In separate publication we shall present other application of the Dirac constraints formalism in classical mechanics [22]. The Dirac brackets, for incompressible uid, are presented, in what follows, within the Poisson bracket formulation of uid mechanics [5,7,9], which avoids cumbersome introduction of the Clebsh potentials. Thus, the state of uid is described by specifying its density and velocity elds.
2. Compressible uid Consider in nite three-dimensional volume of the isothermal uid with density %(r; t) and velocity C(r; t). The Hamiltonian for such a system is given as Z H {%; C} =
[%C2 =2 + f(%)] d 3 r ;
(1)
where f(%) is the uid Helmholtz free energy per unit volume, related to the uid pressure by p=%
@f − f(%) : @%
(2)
50
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
The Poisson bracket relations between elds %(r; t) and v(r; t) are [9] {%(x; t); %(y; t)} = 0 ; {%(x; t); vi (y; t)} = −
@ (x − y) ; @xi
{vi (x; t); v j (y; t)} = (x − y)
1 ij k (∇ × v)k (x; t) : %(x; t)
(3)
The continuity equation is obtained by evaluating the Poisson bracket {%; H }, and the Euler equation by {C; H } @t %(r; t) = {%; H } = −∇ · %C ; @t C(r; t) = {C; H } = −C · ∇C − (1=%)∇p(%) :
(4)
The above formulation of uid mechanics can be derived from the least action principle provided we choose the proper lagrangian. As shown by Thellung [1– 4] this lagrangian density is the local pressure. The incompressible uid, although an obvious simpli cation, is adequate for all the
ows when the local Mach number is small. The incompressibility condition then is that the density %(r; t) − %0 = 0 which also implies that ∇ · C = 0. Within the canonical formulation framework both these conditions are regarded as Dirac constraints [19] a (r; v; t) = 0; a = 1; 2. Next section contains a brief overview of the Dirac brackets theory.
3. Dirac brackets The de nition of the Dirac brackets, we shall use in the following is a natural generalization for the original construction proposed by Dirac [19] and discussed in detail in [20]. When the physical system with phase space P is subject to a set of TN constraints {a = 0} then its motion proceeds on a submanifold P ⊃ S = a=1 {z ∈ P | a (z)=0}. If the Poisson bracket for two arbitrary (suciently smooth, etc.,) phase space functions F and G was {F; G}, then the Dirac bracket v F; G w is de ned as v F; G w ={F; G} = −
N X
{F; a }Mab {b ; G} ;
(5)
a;b
where Mab is the inverse of the constraints Poisson bracket matrix Wab = {a ; b }. The generalization of the Dirac bracket to the case of continuous variables, like in hydrodynamics, is straightforward. The sum over the indices a is replaced by sum and integration over the space variables and the inverse of the matrix Wab (r; r 0 ) = {a (r); b (r 0 )} is de ned as XZ dr 0 Wac (r; r 0 )Mcb (r 0 ; r 00 ) = ab (r − r 00 ) : (6) c
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
51
Dirac brackets, given by Eq. (5) replace the original Poisson brackets in the equation of motion for the constrained system. Thus for a phase space function F the time evolution on the submanifold S is governed by @F = v F; H w ; (7) @t S where H is system Hamiltonian. Next section will contain the application of the Dirac brackets to the description of the incompressible uid.
4. Dirac brackets for incompressible uid The constraints used in constructing the incompressible uid dynamics are 1 ≡ %(r) − %0 = 0;
2 ≡ ∇ · C(r) = 0 :
(8)
The constraints Poisson bracket matrix Wab (r; r 0 ) can be evaluated using Eq. (3), and it reads: # ! " 0 h −ij i 0 i j 0 (r − r ) : (9) Wab (r; r ) = ∇r ∇r 0 1 ij k ij ; %(r) (∇ × C(r))k In the Dirac formalism, one needs the inverse of the matrix Wab (r; r 0 ) de ned in (6). The matrix elements Mab (r; r 0 ) obey the set of partial di erential equations, written explicitly in Appendix A. Solving these equations we nd matrix Mab (r; r 0 ) in the form M {G}; −G(r − r 0 ) ; (10) Mab (r; r 0 ) = 0 G(r − r 0 ); where G(r − r 0 ) = |r − r 0 |=4 is the Green function for the Laplace operator in the in nite volume, and Z 1 0 0 0 0 0 ∇x0 G(x − r ) × (∇ × C(x )) : (11) M {G} = − d x G(r − x )∇x0 · %(x0 ) 5. Dirac equations of motion for incompressible uid Using the de nition and the explicit form of the Dirac brackets given in previous section and in Appendix A, we rst calculate the Dirac bracket v %; H w. From de nitions (5) and (7) we obtain @%(r; t) = v %(r; t); H w ={%(r; t); H } @t XZ − d z d z 0 {%(r; t); a (z)}Mab (z; z 0 ){b (z 0 ); H } : ab
(12)
52
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
Explicit evaluation of the right-hand side of Eq. (12) is a bit tedious, but using results from Appendix A one nds that it vanishes. Thus, the continuity equation for incompressible uid, within the Dirac formalism reads v %(r; t); H w =0. The algebra needed to derive equation of motion for the velocity eld C is slightly more complex than those leading to the continuity equation. Following Dirac procedure we obtain @C(r; t) = v C(r; t); H w ={C(r; t); H } @t XZ − d z d z 0 {C(r; t); a (z)}Mab (z; z 0 ){b (z 0 ); H } :
(13)
ab
Evaluation of the right-hand side of (13), using expression (10) gives @C(r; t) = C(r; t) × (∇ × C(r; t)) @t Z −∇x d z G(r − z)∇z · {C(z) × (∇ × C(r; t))} :
(14)
Thus, we have obtained nonlinear, nonlocal equation for the velocity eld known from previous work [7,8]. The above exercise in the Dirac brackets calculation provides a novel formulation of the Euler incompressible uid. the viscous uid equations can now easily be derived by replacing the Dirac brackets by the metriplectic brackets discussed in [9]. We can also use the Dirac brackets as starting point in the perturbation theory in which compressibility corrections are calculated. To do so one formally associates small parameter to the matrix elements Mab and expresses the Poisson brackets by the Dirac one. To the rst order in the expression is identical to that in (5) with reversed role of the Poisson and Dirac brackets. In conclusion, we have shown in the above that the Poisson brackets formulation of the uid dynamics can be used to derive the canonical theory of the incompressible
uid following the Dirac prescription. The application of this theory will be discussed in following publication. Acknowledgements We would like to thank Cyril Malyshev for contributing discussions during the earlier stage of this work. Appendix A Matrix M: The matrix elements Mab satisfy the following system of partial di erential equations (Ai). Two of them
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
(A1) z M12 (x; z) = −(x − z);
53
x M21 (x; z) = (x − z),
(A2) z M22 (x; z) = 0; x M22 (x; z) = 0, have obvious solutions: (a1) M12 (x; z) = −M21 (x; z) = −G(x − z), (a2) M22 (x; z) = 0. The remaining two (A3) ∇x · [∇x M12 (x; z) +
1 %(x) ∇x M22 (x; z)
× (∇ × C(x))] = −(x − z),
1 ∇x M21 (x; z) × (∇ × C(x))] = 0, (A4) ∇x · [∇x M11 (x; z) + %(x) can also be solved, using (a2) Eq. (A3) reduces to (A1), and using (a1) solution of equation (A4), reads: R (a3) M11 (x; z) = − d x0 G(x − x0 )∇x0 · [ %(x10 ) ∇x0 G(x0 − z) × (∇ × C(x0 ))] ≡ M {G}.
A.1. Details of the Dirac brackets evaluations for the ideal uid Consider Hamiltonian (1), the Dirac bracket v %(x); H w reads: XZ d z1 d z2 {%(x); i (z1 )}Mij (z1 ; z2 ){j (z2 ); H } v %(x); H w = {%(X ); H } − i; j
Z = ∇x · J (x) −
d zx M21 (x; z)[∇z · J (z)]
+x M22 (x; z)[∇z · (C(z) × (∇ × C(z))) − z ((C; %))] = 0 : (A.1) One sees immediately that the right-hand side of (A.1) vanishes due to (a1) – (a3). Here J = %C denotes the uid particle current and (C; %) = |C|2 =2 + @f(%(z))=@% is the moving uid chemical potential. The continuity equation if then @ %(x; t)= v %(x; t); H w =0 ; (A.2) @t as expected. Evaluating Dirac bracket v vi (x); H w we obtain XZ d z1 d z2 {vi (x); a (z1 )}Mab (z1 ; z2 ){b (z2 ); H } v vi (x); H w = {vi (x); H } − = Ai0
−
X
a;b
Aiab
:
a;b
After straightforward but lengthy calculations we obtain Ai0 = {vi (x); H } = [C(x) × (∇ × C(x))]i − ∇i [(v; %)] ; Z i i A11 = −∇x d z M11 (x; z)[∇z · J (z)] ;
(A.3)
54
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
Ai22 = 0; Ai12 = ∇ix Ai21 = −
Z d z G(x − z){∇z · [C(z) × (∇ × C(z))] − ∇z [(v; %)]} ;
Z dz
1 [∇x G(x − z) × (∇ × C(x))]i [∇z · J (z)] : %(x)
(A.4)
Using above, together with Eqs. (a1) and (a2) we obtain @ C(x; t) = C(x) × (∇ × C(x)) @t Z 1 ∇x G(x − z) × (∇ × C(x)) [∇z · J (z)] + d z ∇x M11 (x; z) + %(x) Z (A.5) − d z[∇x G(x − z)]{∇z · [C(z) × (∇ × C(z))]} : Acting on both sides of Eq. (A.5) with operator div = ∇x ·, and using Eq. (A1) one gets @ [∇x · C(x; t)] = v ∇x · C(x; t); H w @t
@C (x; t) : (A.6) @t Using Eq. (A.5), one sees that this expression vanishes. Thus 2 (x) = ∇ · C(x; t) is a constant of motions, as expected. Condition % = %0 implies that ∇x · J (x; t) is also a constant of motions, and ∇x · J (x; t) = 0. Now, using Eqs. (A.6) one easily sees that Eq. (A.5) reduces to = ∇x · v C(x; t); H w =∇x ·
@C(x; t) = v(x; t) × (∇ × C(x; t)) @t Z − ∇x
d z G(x − z)∇z · [C(z; t) × (∇ × C(z; t))]
;
(A.7)
which is exactly the Euler equation for an ideal, incompressible uid in its integral form. References [1] A. Thellung, Physica 19 (1953) 217. [2] L.A. Turski, Physica A 57 (1972); in: Z.M. Galasiewicz (Ed.), Liquid Helium and Many Body Problem, University of Wroclaw Press, 1970. [3] P.J. Morrison, J.M. Greene, Phys. Rev. Lett. 45 (1980) 790, ibid., 48 (1982) 300. [4] J.E. Marsden, A. Weinstein, Physica D 7 (1982) 305. [5] J.S. Langer, L.A. Turski, Phys. Rev. A 46 (1973) 53 230. [6] L.A. Turski, J.S. Langer, Phys. Rev. A 22 (1980) 2189. [7] B. Kim, G.F. Mazenko, J. Stat. Phys. 64 (1991) 631. [8] D. Bedeaux, P. Mazur, W. van Saarlos, Physica A 107 (1981) 109. [9] C.P. Enz, L.A. Turski, Physica A 96 (1979) 369. [10] L.A. Turski, in: G.A. Maugin (Ed.), Continuum Models and Discrete Systems, Vol. 1, Longman, London, 1990.
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55 [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
55
N. Gisin, M. Rigo, J. Phys. A 28 (1995) 7375. N. Gisin, J. Phys. A 14 (1981) 2259. N. Gisin, Physica A 111 (1982) 364. M. Razavy, A. Pimpale, Phys. Rev. 168 (1988) 307. L.A. Turski, in: Z. Petru, J. Przystawa, K. Rapcewicz (Eds.), Springer Lectures in Physics, Vol. 4777, Springer, New York, 1996. H. Lamb, Hydrodynamics, 6th Edition, Cambridge University Press, Cambridge, 1932. V.I. Arnold, Uspekhi Mat. Nauk 24 (1969) 225. W.C. Lvov, A.V. Mikhailov, Sov. Phys. JETP 74 (1978) 1445. P.A.M. Dirac, Canad. J. Math. 2 (1950) 129. J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetries, Springer, Heidelberg, 1994. F. David, Europhys. Lett. 6 (1988) 603. S. Nguyen, L.A. Turski, Some applications of Dirac formalism in Classical Mechanics, in preparation.
www.elsevier.com/locate/physa
Canonical description of incompressible
uid: Dirac brackets approach Sonnet Nguyen a; ∗ , Lukasz A. Turskib a Center
for Theoretical Physics, Polish Academy of Science. Al. Lotnikow 32 46, 02-668 Warszawa, Poland b Center for Theoretical Physics, Polish Academy of Science and College of Science, Al. Lotnikow 32 46, 02-668 Warszawa, Poland Received 12 May 1999
Abstract We present a novel canonical description of the incompressible uid dynamics. This description uses the dynamical constraints, in our case re ecting incompressibility assumption, and leads to replacement of usual hydrodynamical Poisson brackets for density and velocity elds with Dirac brackets. The resulting equations are then known nonlinear, and nonlocal in space, equations for c 1999 Elsevier Science B.V. All rights reserved. incompressible uid velocity. PACS: 47.40; 47.90 Keywords: Fluid dynamics; Poisson brackets; Hamiltonian formulation; Dirac constraints
1. Introduction Canonical description of classical, compressible, isothermal uid has been developed in the past [1–7] for various purposes, for example description of super uid 4 He [1– 4] or in kinetics of the rst-order phase transformations [5,7]. An attempt to extend this formulation for the adiabatic ows has been proposed [8] and used to analyze the dynamical properties of thermally driven ows. The isothermal ow canonical description can be generalized for the case of viscous uids [9] within the framework of the metriplectic dynamics [10]. The Madelung representation for the wave function results in hydrodynamic-like picture of quantum mechanics, where the “only” di erences from Euler equations are hidden in the quantum pressure term, which is proportional to ˝2 , ∗
Corresponding author. E-mail address: [email protected] (S. Nguyen)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 1 9 4 - 6
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
49
and in the quantization of the circulation, = n(˝=m). Apart of that the canonical description of the quantum uid is then identical to that of the classical one. The dissipative generalization of the Schrodinger equation [11–14] also allows for metriplectic interpretation, which di ers from the classical one [15]. The fundamental point in all of the above listed formulations of uid dynamics is that the uid density % is one out of the pair of canonically conjugated variables. For potential ows the other canonical variable is the velocity potential . In case of general ow the two additional Clebsh potentials ; [16] in the velocity eld representation C = −∇ − ∇ are canonically conjugated to each other. None of these descriptions can be applied to the case of incompressible uids. The canonical description of incompressible ow is of considerable importance, for example in the formulation of statistical mechanics of turbulent ow [9]. Many attempts to provide such a canonical formalism [17] failed to do so. The other important point is that real turbulent ows are hardly incompressible, and the issues like compressibility corrections to scaling laws for turbulent ows are still open [18]. In this paper we propose a novel formulation of a canonical description of the incompressible uid based on the concept of the Dirac brackets [19]. The mathematical introduction to this formalism valid for general dynamic system subjected to some set of constraints {a = 0; a = 1 : : : N } can be found in [20]. Dirac bracket approach to description of incompressible membranes, within Lagrangian coordinates formulation of continuous mechanics of membranes, was given in [21]. We are unaware of any other application of that formalism to continuum mechanics problems. In separate publication we shall present other application of the Dirac constraints formalism in classical mechanics [22]. The Dirac brackets, for incompressible uid, are presented, in what follows, within the Poisson bracket formulation of uid mechanics [5,7,9], which avoids cumbersome introduction of the Clebsh potentials. Thus, the state of uid is described by specifying its density and velocity elds.
2. Compressible uid Consider in nite three-dimensional volume of the isothermal uid with density %(r; t) and velocity C(r; t). The Hamiltonian for such a system is given as Z H {%; C} =
[%C2 =2 + f(%)] d 3 r ;
(1)
where f(%) is the uid Helmholtz free energy per unit volume, related to the uid pressure by p=%
@f − f(%) : @%
(2)
50
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
The Poisson bracket relations between elds %(r; t) and v(r; t) are [9] {%(x; t); %(y; t)} = 0 ; {%(x; t); vi (y; t)} = −
@ (x − y) ; @xi
{vi (x; t); v j (y; t)} = (x − y)
1 ij k (∇ × v)k (x; t) : %(x; t)
(3)
The continuity equation is obtained by evaluating the Poisson bracket {%; H }, and the Euler equation by {C; H } @t %(r; t) = {%; H } = −∇ · %C ; @t C(r; t) = {C; H } = −C · ∇C − (1=%)∇p(%) :
(4)
The above formulation of uid mechanics can be derived from the least action principle provided we choose the proper lagrangian. As shown by Thellung [1– 4] this lagrangian density is the local pressure. The incompressible uid, although an obvious simpli cation, is adequate for all the
ows when the local Mach number is small. The incompressibility condition then is that the density %(r; t) − %0 = 0 which also implies that ∇ · C = 0. Within the canonical formulation framework both these conditions are regarded as Dirac constraints [19] a (r; v; t) = 0; a = 1; 2. Next section contains a brief overview of the Dirac brackets theory.
3. Dirac brackets The de nition of the Dirac brackets, we shall use in the following is a natural generalization for the original construction proposed by Dirac [19] and discussed in detail in [20]. When the physical system with phase space P is subject to a set of TN constraints {a = 0} then its motion proceeds on a submanifold P ⊃ S = a=1 {z ∈ P | a (z)=0}. If the Poisson bracket for two arbitrary (suciently smooth, etc.,) phase space functions F and G was {F; G}, then the Dirac bracket v F; G w is de ned as v F; G w ={F; G} = −
N X
{F; a }Mab {b ; G} ;
(5)
a;b
where Mab is the inverse of the constraints Poisson bracket matrix Wab = {a ; b }. The generalization of the Dirac bracket to the case of continuous variables, like in hydrodynamics, is straightforward. The sum over the indices a is replaced by sum and integration over the space variables and the inverse of the matrix Wab (r; r 0 ) = {a (r); b (r 0 )} is de ned as XZ dr 0 Wac (r; r 0 )Mcb (r 0 ; r 00 ) = ab (r − r 00 ) : (6) c
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
51
Dirac brackets, given by Eq. (5) replace the original Poisson brackets in the equation of motion for the constrained system. Thus for a phase space function F the time evolution on the submanifold S is governed by @F = v F; H w ; (7) @t S where H is system Hamiltonian. Next section will contain the application of the Dirac brackets to the description of the incompressible uid.
4. Dirac brackets for incompressible uid The constraints used in constructing the incompressible uid dynamics are 1 ≡ %(r) − %0 = 0;
2 ≡ ∇ · C(r) = 0 :
(8)
The constraints Poisson bracket matrix Wab (r; r 0 ) can be evaluated using Eq. (3), and it reads: # ! " 0 h −ij i 0 i j 0 (r − r ) : (9) Wab (r; r ) = ∇r ∇r 0 1 ij k ij ; %(r) (∇ × C(r))k In the Dirac formalism, one needs the inverse of the matrix Wab (r; r 0 ) de ned in (6). The matrix elements Mab (r; r 0 ) obey the set of partial di erential equations, written explicitly in Appendix A. Solving these equations we nd matrix Mab (r; r 0 ) in the form M {G}; −G(r − r 0 ) ; (10) Mab (r; r 0 ) = 0 G(r − r 0 ); where G(r − r 0 ) = |r − r 0 |=4 is the Green function for the Laplace operator in the in nite volume, and Z 1 0 0 0 0 0 ∇x0 G(x − r ) × (∇ × C(x )) : (11) M {G} = − d x G(r − x )∇x0 · %(x0 ) 5. Dirac equations of motion for incompressible uid Using the de nition and the explicit form of the Dirac brackets given in previous section and in Appendix A, we rst calculate the Dirac bracket v %; H w. From de nitions (5) and (7) we obtain @%(r; t) = v %(r; t); H w ={%(r; t); H } @t XZ − d z d z 0 {%(r; t); a (z)}Mab (z; z 0 ){b (z 0 ); H } : ab
(12)
52
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
Explicit evaluation of the right-hand side of Eq. (12) is a bit tedious, but using results from Appendix A one nds that it vanishes. Thus, the continuity equation for incompressible uid, within the Dirac formalism reads v %(r; t); H w =0. The algebra needed to derive equation of motion for the velocity eld C is slightly more complex than those leading to the continuity equation. Following Dirac procedure we obtain @C(r; t) = v C(r; t); H w ={C(r; t); H } @t XZ − d z d z 0 {C(r; t); a (z)}Mab (z; z 0 ){b (z 0 ); H } :
(13)
ab
Evaluation of the right-hand side of (13), using expression (10) gives @C(r; t) = C(r; t) × (∇ × C(r; t)) @t Z −∇x d z G(r − z)∇z · {C(z) × (∇ × C(r; t))} :
(14)
Thus, we have obtained nonlinear, nonlocal equation for the velocity eld known from previous work [7,8]. The above exercise in the Dirac brackets calculation provides a novel formulation of the Euler incompressible uid. the viscous uid equations can now easily be derived by replacing the Dirac brackets by the metriplectic brackets discussed in [9]. We can also use the Dirac brackets as starting point in the perturbation theory in which compressibility corrections are calculated. To do so one formally associates small parameter to the matrix elements Mab and expresses the Poisson brackets by the Dirac one. To the rst order in the expression is identical to that in (5) with reversed role of the Poisson and Dirac brackets. In conclusion, we have shown in the above that the Poisson brackets formulation of the uid dynamics can be used to derive the canonical theory of the incompressible
uid following the Dirac prescription. The application of this theory will be discussed in following publication. Acknowledgements We would like to thank Cyril Malyshev for contributing discussions during the earlier stage of this work. Appendix A Matrix M: The matrix elements Mab satisfy the following system of partial di erential equations (Ai). Two of them
S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
(A1) z M12 (x; z) = −(x − z);
53
x M21 (x; z) = (x − z),
(A2) z M22 (x; z) = 0; x M22 (x; z) = 0, have obvious solutions: (a1) M12 (x; z) = −M21 (x; z) = −G(x − z), (a2) M22 (x; z) = 0. The remaining two (A3) ∇x · [∇x M12 (x; z) +
1 %(x) ∇x M22 (x; z)
× (∇ × C(x))] = −(x − z),
1 ∇x M21 (x; z) × (∇ × C(x))] = 0, (A4) ∇x · [∇x M11 (x; z) + %(x) can also be solved, using (a2) Eq. (A3) reduces to (A1), and using (a1) solution of equation (A4), reads: R (a3) M11 (x; z) = − d x0 G(x − x0 )∇x0 · [ %(x10 ) ∇x0 G(x0 − z) × (∇ × C(x0 ))] ≡ M {G}.
A.1. Details of the Dirac brackets evaluations for the ideal uid Consider Hamiltonian (1), the Dirac bracket v %(x); H w reads: XZ d z1 d z2 {%(x); i (z1 )}Mij (z1 ; z2 ){j (z2 ); H } v %(x); H w = {%(X ); H } − i; j
Z = ∇x · J (x) −
d zx M21 (x; z)[∇z · J (z)]
+x M22 (x; z)[∇z · (C(z) × (∇ × C(z))) − z ((C; %))] = 0 : (A.1) One sees immediately that the right-hand side of (A.1) vanishes due to (a1) – (a3). Here J = %C denotes the uid particle current and (C; %) = |C|2 =2 + @f(%(z))=@% is the moving uid chemical potential. The continuity equation if then @ %(x; t)= v %(x; t); H w =0 ; (A.2) @t as expected. Evaluating Dirac bracket v vi (x); H w we obtain XZ d z1 d z2 {vi (x); a (z1 )}Mab (z1 ; z2 ){b (z2 ); H } v vi (x); H w = {vi (x); H } − = Ai0
−
X
a;b
Aiab
:
a;b
After straightforward but lengthy calculations we obtain Ai0 = {vi (x); H } = [C(x) × (∇ × C(x))]i − ∇i [(v; %)] ; Z i i A11 = −∇x d z M11 (x; z)[∇z · J (z)] ;
(A.3)
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S. Nguyen, L.A. Turski / Physica A 272 (1999) 48–55
Ai22 = 0; Ai12 = ∇ix Ai21 = −
Z d z G(x − z){∇z · [C(z) × (∇ × C(z))] − ∇z [(v; %)]} ;
Z dz
1 [∇x G(x − z) × (∇ × C(x))]i [∇z · J (z)] : %(x)
(A.4)
Using above, together with Eqs. (a1) and (a2) we obtain @ C(x; t) = C(x) × (∇ × C(x)) @t Z 1 ∇x G(x − z) × (∇ × C(x)) [∇z · J (z)] + d z ∇x M11 (x; z) + %(x) Z (A.5) − d z[∇x G(x − z)]{∇z · [C(z) × (∇ × C(z))]} : Acting on both sides of Eq. (A.5) with operator div = ∇x ·, and using Eq. (A1) one gets @ [∇x · C(x; t)] = v ∇x · C(x; t); H w @t
@C (x; t) : (A.6) @t Using Eq. (A.5), one sees that this expression vanishes. Thus 2 (x) = ∇ · C(x; t) is a constant of motions, as expected. Condition % = %0 implies that ∇x · J (x; t) is also a constant of motions, and ∇x · J (x; t) = 0. Now, using Eqs. (A.6) one easily sees that Eq. (A.5) reduces to = ∇x · v C(x; t); H w =∇x ·
@C(x; t) = v(x; t) × (∇ × C(x; t)) @t Z − ∇x
d z G(x − z)∇z · [C(z; t) × (∇ × C(z; t))]
;
(A.7)
which is exactly the Euler equation for an ideal, incompressible uid in its integral form. References [1] A. Thellung, Physica 19 (1953) 217. [2] L.A. Turski, Physica A 57 (1972); in: Z.M. Galasiewicz (Ed.), Liquid Helium and Many Body Problem, University of Wroclaw Press, 1970. [3] P.J. Morrison, J.M. Greene, Phys. Rev. Lett. 45 (1980) 790, ibid., 48 (1982) 300. [4] J.E. Marsden, A. Weinstein, Physica D 7 (1982) 305. [5] J.S. Langer, L.A. Turski, Phys. Rev. A 46 (1973) 53 230. [6] L.A. Turski, J.S. Langer, Phys. Rev. A 22 (1980) 2189. [7] B. Kim, G.F. Mazenko, J. Stat. Phys. 64 (1991) 631. [8] D. Bedeaux, P. Mazur, W. van Saarlos, Physica A 107 (1981) 109. [9] C.P. Enz, L.A. Turski, Physica A 96 (1979) 369. [10] L.A. Turski, in: G.A. Maugin (Ed.), Continuum Models and Discrete Systems, Vol. 1, Longman, London, 1990.
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