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Hidden supersymmetry of classical systems (hydrodynamics and conservation laws)
31 July 1995
PHYSICS
EISEVIER
LETTERS
A
Physics Letters A 203 (1995) 357-361
Hidden supersymmetry of classical systems ( hydrodynamics and conservation laws) D.V. Volkov a, A.V. Tur b, V.V. Yanovsky
’
a Kharkov Institute of Physics and Technology, Kharkov 310108, Ukraine ’ Obseruatoire Midi-PyrtMes, 14 Avenue Edouard Belin, 31400 Toulouse, France ’ Institute for Single Crystals, Academy of Sciences of Ukraine, Lenin ave., Kharkov 310001, Ukraine Received 19 January 1995; accepted for publication Communicated by A.R. Bishop
18 April 1995
Abstract
Hidden supersymmetry is found for a wide class of classical dynamical systems and in particular for hydrodynamics. The dynamics of the co-differentials plays the role of odd variables. It is shown that the hydrodynamical invariants form the natural super Lie algebra with respect to the odd Buttin bracket and all integral invariants are Fourier transforms of super invariants.
1. The concept of supersymmetry, which was initially proposed for the unification of the Bose and Fermi fields, is aspiring now to a leading role in theoretical physics. It was successfully applied especially to quantum field theory (see e.g. Refs. [1,2] and references therein), quantum mechanics [3] and quantum string theory (see e.g. Refs. [4] and references therein). However, the application of supersymmetric ideas to classical systems is not very developed and is restricted to some examples only (see e.g. Refs. [5,6]). In fact, as we will show in this paper, a wide class of dynamical systems, like classical hydrodynamics, have a hidden natural supersymmetric structure. We would like to emphasize that the subject of this paper is not a supersymmetric extension of some classical systems but a supersymmetrical reformulation of classical systems. Some of the conservation laws in hydrodynamics have been known for a long time [7-91. The algebra of the 0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00324-X
conservation laws for hydrodynamical systems was found recently [lo] and many new invariants have been constructed. In this paper we shall show that this algebra is a part of a natural super Lie algebra with respect to an odd Poisson bracket. We are introducing here the super Hamiltonian formalism for a wide class of classical dynamical systems. The field of co-differentials plays here the role of odd variables. It is important to emphasize that the odd variables dynamics are also induced by the dynamics of the basic dynamical system. We introduce the super Hamiltonian formalism with the help of wellknown Poisson brackets (Buttin bracket [ll]) (see also e.g. Refs. [12,13]). In particular, it will be shown that super invariants in these super Hamiltonian systems are strongly connected with the invariants of dissipationless hydrodynamic systems. As a consequence the super Lie algebra of the constructed super Hamiltonian system induces the super Lie
358
D.V. Volkou et al. /Physics
algebra of the hydrodynamical invariants. This super Lie algebra leads to some universal relations among the hydrodynamical invariants. (Some of them have been found earlier in Ref. [lo].> Using these relations one can construct the new set of invariants as a consequence of some known ones. The connections between super Hamiltonian systems on the one hand and hydrodynamical and dynamical systems on the other hand can be very fruitful according to our reckoning. 2. It is known that there are four types of local hydrodynamical invariants in all hydrodynamical media (see Ref. [lo]). These invariants obey the following equations, a,r+(v.
V)Z=O,
(1)
atJ+(V4’)J-(Jd)V=O,
(2) k
a,s+(v-v)s+s,f&=o,
(3)
a,p+div(
(4)
pV)
=O,
where V(x, t) is the velocity field of the hydrodynamical medium, I is the Lagrangian invariant, J is a frozen-in field, S is the S-invariant according to the terminology of Ref. [lo] (or frozen-in surface), p is the invariant density. The geometrical point of view of these invariants was presented in detail in Ref. [lo] with the help of differential forms. Let R3*3 be super space, z = (xLI, P,) E R3V3 (a = 1, 2, 3) are points of the super space that are naturally graded: g(x”) = 0, g(P,) = 1. Let A(z) and B(z) be super functions on R3*3. We can now define the odd Poisson bracket (Buttin bracket) (see Refs. [ll13lk
Letters A 203 (1995) 357-361
={A,
{A, Bj = -( c
[I+~(A)I~(B)B.
B)C+(-1)
{x’$.}
= s;,
{ n’,xj}
{B, q)
d{A,
B))
= 1+
g(A)
These equations are commutative identity, tion condition. To introduce the shall define the super
+ g(B).
respectively the Leibniz rule, Jacobi identity and a graduasuper Hamiltonian Hamiltonian H,:
system
we
H, = Vi( x, t) Pi. We shall define also a super invariant I(z) which obeys the equation dl G
=
as a function
a,z,( 2,t)+ (H,, Zs(z, t)} = 0.
(6)
This definition is wider than the usual ones because we will have to include the hydrodynamical invariants (l)--(4). The physical sense of Eq. (6) is reduced to the transfer of the function f, along the Hamiltonian vector field which corresponds to the function H, . The invariant which obeys Eq. (6) will be named the local super Lagrangian invariant. It is well known that every super function can be written in the form [I41 ls(z,t)=a,(x, t) +alk(x, t)pk +a,,(x, t)EkmnP,,,Pn +
a3(
x,
f)P1PZP3.
Let us consider now the equation for the components of the super invariant. Using definition (5) and some properties of the odd Poisson brackets one can easily show
’ v)vk)Pk=o,
conjugated
= {Pipi} = 0.
= 0,
(A,B,C)
(a,alk+(v'v)alk-(al The variables xi and Pi are canonically with respect to bracket (5):
~1,
11+g(AHb +g(B)lf B, A),
-1)
Il+dA)lll+dB)l(A,
C-1)
{A,
avp (
a,a2k
+
tv’
‘)%k
-Sk
div V+S,s 1
x EkmnPm P, = 0,
Using definition (5) it is easy to prove the following properties (see also Refs. [13,15]),
(a,u,+(V~V)a,-a,
{A, B.C)
Hence we have proved that Eq. (6) is equivalent to the system of equations (l)-(4). Besides, the super
div V)P,P,P,=O.
D. V. Volkov et al. /Physics
35Y
Letters A 203 (19951357-361
Lagrangian invariant I, is connected with local hydrodynamics invariants I, J, 5, p as follows,
a canonical form proper [13,15] with the graduation
Z,(z, t) =Z(x,
{A,
t) +.ZQ,
+ S’(x7 p( x,
f)Pk
r) Ekmnp t>
n
1 +---
P,P?.P3,
P(X,
t)
(7)
Otherwise the component with zero degree of odd variables is the Lagrangian invariant, the component with first degree is a frozen-in field, the next, second and third degree give the S and p-l invariants, respectively. Hence we have proved that Lagrangian invariant, frozen-in field, S-field and invariant density are the components of the unique physical object-s3er Lagrangian invariant, which is conserved by the super Hamiltonian phase flux. 3. Let us consider now some applications of Eqs. (6) and (7). We can build up new dynamical invariants on the basis of a certain number of known ones. Suppose, that we have two super invariants: Z,r, Zs2. It is easily seen that 4lZs2 = Zs39
(8)
43 = I41 7 421
(9)
are also super invariants. It is important to emphasize that in contrast to the usual Hamiltonian formalism the odd Poisson bracket Z, = {Z+ Z,,}
(10)
is not zero. Eqs. (8)~(10) are the source of many useful relations among hydrodynamical invariants. In particular, evaluating the components of Eqs. (8) and (10) we get all relations among hydrodynamical invariants that were obtained earlier [lo, pp. 77-811. Let us turn now to the more general equations (9). It is well known that the super functions form the super Lie algebra in which the multiplication of elements corresponds to their odd brackets [13,35]. Actually we shall introduce a new natural brackets graduation, g(A) g({A,
=g(A)
+2((%
@) =2(A)
Then, the commutation
+2(B).
=
-(-l)g(A)b(B)(B,
(11)
rule and Jacobi identity have
Lie algebra
A),
g(A)g(C){A, {B, C}) = 0,
c (-1) (A.B.C)
p m
B}
to a super
(12)
where the last equation is a super Jacobi identity. The brackets graduation (11) plays the role of canonical graduation. Also the super invariants form the subalgebra of an infinite dimensional super Lie algebra with respect to the odd Poisson bracket. Taking into account the relation existing between super invariants and local conservation laws, one can consider as proven the fact that local hydrodynamical invariants form a super Lie algebra with respect to odd brackets. The presentation of this super algebra by differential operators may be easily found using Eq. (9). To this end we have to express the coefficients of Is3 through Is,, Zs2. Then follows, from (9) Zs3( z+ t)
* V)l,(x,
= [(J,
t) -(J,
. V)l,( x, t,]
+ (J1 * V)J; - ( J2 . V)J: [ +L(s,xvI,)‘+&s, +
Pk
xvz2$ Pl
P2
1
‘&I, - LkZ2 i P? Pl
+ $ i
:ciiv( 2
-S&Jf’
p2J1) - (J, . V)S,,
I
s,, -- 1 -W Pl ( Pl
- ( Jz . VP,,
52” E kmnPmP, II
-S&
+
P, J2)
Idiv(
i Pt
s,
pzJ,) - Idiv(
p,, J,)
P:
s,
s,
s,
+ - rot” + * rotP,P,P,. P2 P2 PI PI 1
(13)
D. V. Volkou et al. /Physics
360
Note also that the super Lie algebra contains the sub Lie algebra of frozen-in fields. The multiplication in this subalgebra is defined by the commutator of the vector fields. It is important to emphasize now that the super invariants unify in one geometrical object scalar function, vector fields, covector fields and densities. Hence in the super algebra (13) we have a unification of all of the most important kinds of geometrical objects which are used for the description of physical systems. The graduation of each field is defined by graduation of the corresponding super invariant. As well, this super algebra can be used in hydrodynamics models to construct new invariants using the set of known ones. It is easy to verify that a connection exists between super invariants and the Euler equation. Let Z, be an additional super invariant of the super Hamiltonian Z-Z,, Z, = P,P2P3 + (curl V)“P,. In this case, two components of Eq. (6) are equivalent to the Euler equation. Let us now discuss the reasons that allowed us to connect the super Hamiltonian systems and hydrodynamical invariants. Consider for simplicity the general dynamical system that defines the phase flux in Euclidian space R3, dx’ = Vi( x, t). dt
(14)
In fact the approach proposed below can be generalised for arbitrary manifolds. This is why we shall distinguish covariant and contravariant indices. The system of equations (14) induces the equations for differentials,
Letters A 203 (1995) 357-361
The following theorem is proved without difficulty: Eq. (14) and the equation for co-differentials (16) form a super Hamiltonian system with super Hamiltonian ZZ, = PiVi(x, t), if g is a Killing metric (V is the Killing vector field). The proof is the following. Using direct calculations we get f’= {ZZ”, xi} = V’(x, t),
avk(x, t)
( 174
(1%)
Eq. (17a) corresponds identically to Eq. (14). When substituting (16) into (17b) and taking into account (15) we get
a,g+L,g=o.
(18)
Here L, is the Lie derivative. That means that g is a Lie dragged metric. Thus the proof is accomplished. Consequently, an arbitrary flux in R3 could be extended up to super Hamiltonian ones in the corresponding super space. At the same time the odd variables (co-differentials) have a clear physical meaning. In particular any velocity fields of hydrodynamical media could be included in this super Hamiltonian formalism. As a result the hydrodynamical invariants turn into the super invariants of the corresponding Hamiltonian system. In the same way any dynamical system could be extended up to a super Hamiltonian system in the corresponding super space. This construction will be presented in the next paper. Note also that if we consider Pi and d xk as independent variables that satisfy Eq. (171, (15) then
&Pi
dx’) =0,
i
;dx’=
$
dxk.
(15)
The system of equations (15) describes the dynamics of the differentials induced by the phase flux of the system (14). Let us introduce the metric tensor gik, which is necessary to define the field of co-differentials, Pi = gik dx’.
(16)
We shall consider the differentials d x k and values Pi as odd or (Grassmanian) variables. Let us introduce the odd bracket (5) on super space 2 = (xm, P,,,>.
i.e. P,dx’ is invariant. In this case we can define the tensor g using its values with t = 0. Eq. (19) is connected with PoincarC integral invariants that allow us to pass from local invariants to integral ones. For hydrodynamical invariants this means a transition to integral conservation laws (see Ref. [lo]). Now, let us do this in explicit form. This means that we have to change the Pi odd variables to d xi odd ones. Taking into account the conservation P,dx’ (see (19)) one can perform this transition by integrating over odd variables. The more elegant form of such a transition is accomplished with a Fourier
D.V. Volkov et al. /Physics
transform. The integration has to be done over the invariants dynamical measure. Note, that the expression p-’ P, P2P3 is a super invariant (see (7)); it is obvious that we can take pdP’dP2dP3 as an invariant measure, because their product is invariant also (here p is the invariant density). Define now the Fourier transform of the super invariant Z(z), I
I,( z,
t)p(
x,
t)
exp(P, dx’) dP’ dP2 dP3
= Ii,
(20) and consider now the integration over odd variables. Taking into account the Berezin rules (JdP’ = 0, JP,dP’= 1) ( see for example Refs. [12,14]) and developing the exponent in a power series we obtain Ii,=
/
pZ, dx’ Adx2Adx3
+jp.ZdxAdx+jS.dx+l.
(21)
As a result we obtain the ordinary integral conservation laws: “mass” conservation law, frozen-in field flux conservation law and conservation of the circulation S-invariant respectively (see also Ref. [lo]).
Letters A 203 (1995) 357-361
361
The generalisation of Eqs. (13) and (20) for dynamical systems with an arbitrary number of degrees of freedom will be presented in another paper.
References [ll [2] [3] [41
P. Fayet and S. Ferrara, Phys. Rep. 32 (1977) 249. M.F. Sohnius, Phys. Rep. 128 (1985) 39. E. Witten, Nucl. Phys. B 185 (1981) 513. M. Green, J. Schwarz and E. Witten, Superstring theory, Vols. 1, 2 (Cambridge Univ. Press, Cambridge, 1987). [5] G. Parisi and N. Sourlas, Nucl. Phys. B 206 (1982) 321. 161 B.A. Dubrovin and S.P. Novikov, Sov. Phys. JETP 79 (1980) 106. [7] L. Voltijer, Proc. Nat. Acad. Sci. USA. 44 (1958) 489. 181 J.J. Moreau, CR. Acad. Sci. Paris 252 (1961) 2810. [9] H.K. Moffatt, J. Fluid Mech. 35 (1969) 117. [lo] A.V. Tur and V.V. Yanovsky, J. Fluid Mech. 248 (1993) 67. [ll] C. Buttin, C.R. Acad. Sci. Paris 269 (1969) 87. [12] B. Dewitt, Supermanifolds (Cambridge Univ. Press, Cambridge, 1992). 1131 D.V. Volkov, JETP L.&t. 1 (1983) 508. [141 F.A. Berezin, Introduction to superanalysis (Reidel, Dordrecht, 1987). [151 D.V. Volkov and V.A. Soroka, Nucl. Phys. 46 (1987) 116 [in Russian].
PHYSICS
EISEVIER
LETTERS
A
Physics Letters A 203 (1995) 357-361
Hidden supersymmetry of classical systems ( hydrodynamics and conservation laws) D.V. Volkov a, A.V. Tur b, V.V. Yanovsky
’
a Kharkov Institute of Physics and Technology, Kharkov 310108, Ukraine ’ Obseruatoire Midi-PyrtMes, 14 Avenue Edouard Belin, 31400 Toulouse, France ’ Institute for Single Crystals, Academy of Sciences of Ukraine, Lenin ave., Kharkov 310001, Ukraine Received 19 January 1995; accepted for publication Communicated by A.R. Bishop
18 April 1995
Abstract
Hidden supersymmetry is found for a wide class of classical dynamical systems and in particular for hydrodynamics. The dynamics of the co-differentials plays the role of odd variables. It is shown that the hydrodynamical invariants form the natural super Lie algebra with respect to the odd Buttin bracket and all integral invariants are Fourier transforms of super invariants.
1. The concept of supersymmetry, which was initially proposed for the unification of the Bose and Fermi fields, is aspiring now to a leading role in theoretical physics. It was successfully applied especially to quantum field theory (see e.g. Refs. [1,2] and references therein), quantum mechanics [3] and quantum string theory (see e.g. Refs. [4] and references therein). However, the application of supersymmetric ideas to classical systems is not very developed and is restricted to some examples only (see e.g. Refs. [5,6]). In fact, as we will show in this paper, a wide class of dynamical systems, like classical hydrodynamics, have a hidden natural supersymmetric structure. We would like to emphasize that the subject of this paper is not a supersymmetric extension of some classical systems but a supersymmetrical reformulation of classical systems. Some of the conservation laws in hydrodynamics have been known for a long time [7-91. The algebra of the 0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00324-X
conservation laws for hydrodynamical systems was found recently [lo] and many new invariants have been constructed. In this paper we shall show that this algebra is a part of a natural super Lie algebra with respect to an odd Poisson bracket. We are introducing here the super Hamiltonian formalism for a wide class of classical dynamical systems. The field of co-differentials plays here the role of odd variables. It is important to emphasize that the odd variables dynamics are also induced by the dynamics of the basic dynamical system. We introduce the super Hamiltonian formalism with the help of wellknown Poisson brackets (Buttin bracket [ll]) (see also e.g. Refs. [12,13]). In particular, it will be shown that super invariants in these super Hamiltonian systems are strongly connected with the invariants of dissipationless hydrodynamic systems. As a consequence the super Lie algebra of the constructed super Hamiltonian system induces the super Lie
358
D.V. Volkou et al. /Physics
algebra of the hydrodynamical invariants. This super Lie algebra leads to some universal relations among the hydrodynamical invariants. (Some of them have been found earlier in Ref. [lo].> Using these relations one can construct the new set of invariants as a consequence of some known ones. The connections between super Hamiltonian systems on the one hand and hydrodynamical and dynamical systems on the other hand can be very fruitful according to our reckoning. 2. It is known that there are four types of local hydrodynamical invariants in all hydrodynamical media (see Ref. [lo]). These invariants obey the following equations, a,r+(v.
V)Z=O,
(1)
atJ+(V4’)J-(Jd)V=O,
(2) k
a,s+(v-v)s+s,f&=o,
(3)
a,p+div(
(4)
pV)
=O,
where V(x, t) is the velocity field of the hydrodynamical medium, I is the Lagrangian invariant, J is a frozen-in field, S is the S-invariant according to the terminology of Ref. [lo] (or frozen-in surface), p is the invariant density. The geometrical point of view of these invariants was presented in detail in Ref. [lo] with the help of differential forms. Let R3*3 be super space, z = (xLI, P,) E R3V3 (a = 1, 2, 3) are points of the super space that are naturally graded: g(x”) = 0, g(P,) = 1. Let A(z) and B(z) be super functions on R3*3. We can now define the odd Poisson bracket (Buttin bracket) (see Refs. [ll13lk
Letters A 203 (1995) 357-361
={A,
{A, Bj = -( c
[I+~(A)I~(B)B.
B)C+(-1)
{x’$.}
= s;,
{ n’,xj}
{B, q)
d{A,
B))
= 1+
g(A)
These equations are commutative identity, tion condition. To introduce the shall define the super
+ g(B).
respectively the Leibniz rule, Jacobi identity and a graduasuper Hamiltonian Hamiltonian H,:
system
we
H, = Vi( x, t) Pi. We shall define also a super invariant I(z) which obeys the equation dl G
=
as a function
a,z,( 2,t)+ (H,, Zs(z, t)} = 0.
(6)
This definition is wider than the usual ones because we will have to include the hydrodynamical invariants (l)--(4). The physical sense of Eq. (6) is reduced to the transfer of the function f, along the Hamiltonian vector field which corresponds to the function H, . The invariant which obeys Eq. (6) will be named the local super Lagrangian invariant. It is well known that every super function can be written in the form [I41 ls(z,t)=a,(x, t) +alk(x, t)pk +a,,(x, t)EkmnP,,,Pn +
a3(
x,
f)P1PZP3.
Let us consider now the equation for the components of the super invariant. Using definition (5) and some properties of the odd Poisson brackets one can easily show
’ v)vk)Pk=o,
conjugated
= {Pipi} = 0.
= 0,
(A,B,C)
(a,alk+(v'v)alk-(al The variables xi and Pi are canonically with respect to bracket (5):
~1,
11+g(AHb +g(B)lf B, A),
-1)
Il+dA)lll+dB)l(A,
C-1)
{A,
avp (
a,a2k
+
tv’
‘)%k
-Sk
div V+S,s 1
x EkmnPm P, = 0,
Using definition (5) it is easy to prove the following properties (see also Refs. [13,15]),
(a,u,+(V~V)a,-a,
{A, B.C)
Hence we have proved that Eq. (6) is equivalent to the system of equations (l)-(4). Besides, the super
div V)P,P,P,=O.
D. V. Volkov et al. /Physics
35Y
Letters A 203 (19951357-361
Lagrangian invariant I, is connected with local hydrodynamics invariants I, J, 5, p as follows,
a canonical form proper [13,15] with the graduation
Z,(z, t) =Z(x,
{A,
t) +.ZQ,
+ S’(x7 p( x,
f)Pk
r) Ekmnp t>
n
1 +---
P,P?.P3,
P(X,
t)
(7)
Otherwise the component with zero degree of odd variables is the Lagrangian invariant, the component with first degree is a frozen-in field, the next, second and third degree give the S and p-l invariants, respectively. Hence we have proved that Lagrangian invariant, frozen-in field, S-field and invariant density are the components of the unique physical object-s3er Lagrangian invariant, which is conserved by the super Hamiltonian phase flux. 3. Let us consider now some applications of Eqs. (6) and (7). We can build up new dynamical invariants on the basis of a certain number of known ones. Suppose, that we have two super invariants: Z,r, Zs2. It is easily seen that 4lZs2 = Zs39
(8)
43 = I41 7 421
(9)
are also super invariants. It is important to emphasize that in contrast to the usual Hamiltonian formalism the odd Poisson bracket Z, = {Z+ Z,,}
(10)
is not zero. Eqs. (8)~(10) are the source of many useful relations among hydrodynamical invariants. In particular, evaluating the components of Eqs. (8) and (10) we get all relations among hydrodynamical invariants that were obtained earlier [lo, pp. 77-811. Let us turn now to the more general equations (9). It is well known that the super functions form the super Lie algebra in which the multiplication of elements corresponds to their odd brackets [13,35]. Actually we shall introduce a new natural brackets graduation, g(A) g({A,
=g(A)
+2((%
@) =2(A)
Then, the commutation
+2(B).
=
-(-l)g(A)b(B)(B,
(11)
rule and Jacobi identity have
Lie algebra
A),
g(A)g(C){A, {B, C}) = 0,
c (-1) (A.B.C)
p m
B}
to a super
(12)
where the last equation is a super Jacobi identity. The brackets graduation (11) plays the role of canonical graduation. Also the super invariants form the subalgebra of an infinite dimensional super Lie algebra with respect to the odd Poisson bracket. Taking into account the relation existing between super invariants and local conservation laws, one can consider as proven the fact that local hydrodynamical invariants form a super Lie algebra with respect to odd brackets. The presentation of this super algebra by differential operators may be easily found using Eq. (9). To this end we have to express the coefficients of Is3 through Is,, Zs2. Then follows, from (9) Zs3( z+ t)
* V)l,(x,
= [(J,
t) -(J,
. V)l,( x, t,]
+ (J1 * V)J; - ( J2 . V)J: [ +L(s,xvI,)‘+&s, +
Pk
xvz2$ Pl
P2
1
‘&I, - LkZ2 i P? Pl
+ $ i
:ciiv( 2
-S&Jf’
p2J1) - (J, . V)S,,
I
s,, -- 1 -W Pl ( Pl
- ( Jz . VP,,
52” E kmnPmP, II
-S&
+
P, J2)
Idiv(
i Pt
s,
pzJ,) - Idiv(
p,, J,)
P:
s,
s,
s,
+ - rot” + * rotP,P,P,. P2 P2 PI PI 1
(13)
D. V. Volkou et al. /Physics
360
Note also that the super Lie algebra contains the sub Lie algebra of frozen-in fields. The multiplication in this subalgebra is defined by the commutator of the vector fields. It is important to emphasize now that the super invariants unify in one geometrical object scalar function, vector fields, covector fields and densities. Hence in the super algebra (13) we have a unification of all of the most important kinds of geometrical objects which are used for the description of physical systems. The graduation of each field is defined by graduation of the corresponding super invariant. As well, this super algebra can be used in hydrodynamics models to construct new invariants using the set of known ones. It is easy to verify that a connection exists between super invariants and the Euler equation. Let Z, be an additional super invariant of the super Hamiltonian Z-Z,, Z, = P,P2P3 + (curl V)“P,. In this case, two components of Eq. (6) are equivalent to the Euler equation. Let us now discuss the reasons that allowed us to connect the super Hamiltonian systems and hydrodynamical invariants. Consider for simplicity the general dynamical system that defines the phase flux in Euclidian space R3, dx’ = Vi( x, t). dt
(14)
In fact the approach proposed below can be generalised for arbitrary manifolds. This is why we shall distinguish covariant and contravariant indices. The system of equations (14) induces the equations for differentials,
Letters A 203 (1995) 357-361
The following theorem is proved without difficulty: Eq. (14) and the equation for co-differentials (16) form a super Hamiltonian system with super Hamiltonian ZZ, = PiVi(x, t), if g is a Killing metric (V is the Killing vector field). The proof is the following. Using direct calculations we get f’= {ZZ”, xi} = V’(x, t),
avk(x, t)
( 174
(1%)
Eq. (17a) corresponds identically to Eq. (14). When substituting (16) into (17b) and taking into account (15) we get
a,g+L,g=o.
(18)
Here L, is the Lie derivative. That means that g is a Lie dragged metric. Thus the proof is accomplished. Consequently, an arbitrary flux in R3 could be extended up to super Hamiltonian ones in the corresponding super space. At the same time the odd variables (co-differentials) have a clear physical meaning. In particular any velocity fields of hydrodynamical media could be included in this super Hamiltonian formalism. As a result the hydrodynamical invariants turn into the super invariants of the corresponding Hamiltonian system. In the same way any dynamical system could be extended up to a super Hamiltonian system in the corresponding super space. This construction will be presented in the next paper. Note also that if we consider Pi and d xk as independent variables that satisfy Eq. (171, (15) then
&Pi
dx’) =0,
i
;dx’=
$
dxk.
(15)
The system of equations (15) describes the dynamics of the differentials induced by the phase flux of the system (14). Let us introduce the metric tensor gik, which is necessary to define the field of co-differentials, Pi = gik dx’.
(16)
We shall consider the differentials d x k and values Pi as odd or (Grassmanian) variables. Let us introduce the odd bracket (5) on super space 2 = (xm, P,,,>.
i.e. P,dx’ is invariant. In this case we can define the tensor g using its values with t = 0. Eq. (19) is connected with PoincarC integral invariants that allow us to pass from local invariants to integral ones. For hydrodynamical invariants this means a transition to integral conservation laws (see Ref. [lo]). Now, let us do this in explicit form. This means that we have to change the Pi odd variables to d xi odd ones. Taking into account the conservation P,dx’ (see (19)) one can perform this transition by integrating over odd variables. The more elegant form of such a transition is accomplished with a Fourier
D.V. Volkov et al. /Physics
transform. The integration has to be done over the invariants dynamical measure. Note, that the expression p-’ P, P2P3 is a super invariant (see (7)); it is obvious that we can take pdP’dP2dP3 as an invariant measure, because their product is invariant also (here p is the invariant density). Define now the Fourier transform of the super invariant Z(z), I
I,( z,
t)p(
x,
t)
exp(P, dx’) dP’ dP2 dP3
= Ii,
(20) and consider now the integration over odd variables. Taking into account the Berezin rules (JdP’ = 0, JP,dP’= 1) ( see for example Refs. [12,14]) and developing the exponent in a power series we obtain Ii,=
/
pZ, dx’ Adx2Adx3
+jp.ZdxAdx+jS.dx+l.
(21)
As a result we obtain the ordinary integral conservation laws: “mass” conservation law, frozen-in field flux conservation law and conservation of the circulation S-invariant respectively (see also Ref. [lo]).
Letters A 203 (1995) 357-361
361
The generalisation of Eqs. (13) and (20) for dynamical systems with an arbitrary number of degrees of freedom will be presented in another paper.
References [ll [2] [3] [41
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