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Lie symmetry and Hojman conserved quantity of Nambu system

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2008 Chinese Phys. B 17 4361

(http://iopscience.iop.org/1674-1056/17/12/004)

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Vol 17 No 12, December 2008 c© 2008 Chin. Phys. Soc.

1674-1056/2008/17(12)/4361-04 Chinese Physics B and IOP Publishing Ltd

Lie symmetry and Hojman conservedquantity of Nambu system

Lin Peng(蔺 鹏), Fang Jian-Hui(方建会)†, and Pang Ting(庞 婷)

College of Physics and Technology, China University of Petroleum, Dongying 257061, China

(Received 6 March 2008; revised manuscript received 26 April 2008)

This paper studies the Lie symmetry and Hojman conserved quantity of the Nambu system. The determining

equations of Lie symmetry for the system are given. The conditions for existence and the form of the Hojman conserved

quantity led by the Lie symmetry for the system are obtained. Finally, an example is given to illustrate the application

of the results.

Keywords: Nambu system, Lie symmetry, Hojman conserved quantityPACC: 0320

1. Introduction

The conservation laws for a dynamical systemplay an important role in the fields of mathemat-ics, mechanics and physics. A recent developmentdirection is to study the conserved quantities by us-ing symmetrical methods of the systems in analyti-cal mechanics, which is very important to learn thephysical states and properties of the systems. Thereare three main symmetrical methods used to seek theconserved quantity of the dynamical system: Noethersymmetry,[1] Lie symmetry,[2] and Mei symmetry.[3−5]

Recently, many important achievements have been ob-tained in this popular research field.[3−11]

In 1973, Nambu proposed a generalization ofordinary Hamilton mechanics which is now calledNambu mechanics.[12] Mukunda et al studied the con-nections between the Nambu system and Hamiltonsystem.[13−15] Ogawa et al presented a Lagrangianformulation for Nambu mechanics and advanced anapplication Euler equation.[16] Now, Nambu mechan-ics system is widely used in particle physics, cosmog-raphy, string theory, hydrodynamics, mathematicalphysics, and computational mathematics, etc.

The symmetries and conserved quantities of theNambu system with high dimensions has been rarelystudied so far. In this paper, we study the Lie sym-metry and Hojman conserved quantity of the Nambusystem. We construct the determining equations ofLie symmetry for the Nambu system. The conditions

for existence and the form of the Hojman conservedquantity led by the Lie symmetry for the system areobtained.

2. The Nambu mechanical sys-

tem

Suppose that the configuration of a mechanicalsystem in n-dimensional phase space is determined byn generalized coordinates qi (i = 1, 2, . . . , n). Forany functions Ai(q), we define the Nambu-Poissonbracket:

A1(q), A2(q), . . . , An(q)

≡

∂A1

∂q1

∂A1

∂q2. . .

∂A1

∂qn

∂A2

∂q1

∂A2

∂q2. . .

∂A2

∂qn

......

. . ....

∂An

∂q1

∂An

∂q2. . .

∂An

∂qn

. (1)

Introducing n − 1 Nambu–Hamilton functionsHk(q) (k = 1, 2, . . . , n − 1), we can write the equa-tions of motion of the Nambu mechanical system as

qi = qi,H1(q), . . . , Hn−1(q),(i = 1, 2, . . . , n). (2)

For any function F (q1, . . . , qn) in which the timeis invariable, we have

†Corresponding author. E-mail: Fangjh@hdpu.edu.cnhttp://www.iop.org/journals/cpb　http://cpb.iphy.ac.cn

4362 Lin Peng et al Vol. 17

dF

dt=

∂(F, H1, . . . , Hn−1)∂(q1, q2, . . . , qn)

= F, H1, . . . , Hn−1.The action of Nambu mechanical system is∫

Cn−1

Ω (n−1) =∫

Cn−1

q1dq2 ∧ . . . ∧ dqn

−H1dH2 ∧ . . . ∧ dHn−1 ∧ dt. (3)

Apparently, when we take n = 2, this system de-generates to a Hamilton system.

3. Lie symmetry of Nambu sys-

tem

The Lie symmetry is an invariance of differentialequations under the infinitesimal transformations.[7]

Introducing the infinitesimal transformations

t∗ = t + ∆t, q∗i (t∗) = qi(t) + ∆qi,

(i = 1, 2, . . . , n), (4)

we get the expanded formulae

t∗ = t + εξ0(t, q),

q∗i (t∗) = qi(t) + εξi(t, q), (5)

where ε is an infinitesimal parameter, and ξ0, ξi arethe generators of infinitesimal transformations.

Expanding Eq.(2), we get

qi = gi(t, q), (i = 1, 2, . . . , n). (6)

According to the invariance of differential equa-tions under the infinitesimal transformations, the in-variance of the Nambu differential equations (6) is

X(1)(qi − gi(t, q)) = 0, (i = 1, 2, . . . , n), (7)

where

X(1) = ξ0∂

∂t+ ξi

∂

∂qi+

(dξi

dt− gi

dξ0

dt

)∂

∂qi, (8)

ddt

=∂

∂t+ gi

∂

∂qi.

Equation (7) can be expanded as

dξi

dt− gi

dξ0

dt= ξ0

∂gi

∂t+ ξj

∂gi

∂qj,

(i, j = 1, 2, . . . , n). (9)

We call Eq.(9) the set of determining equations of Liesymmetry of the Nambu system.

4. The Hojman conserved quan-

tity of Nambu system

The Lie symmetry of the Nambu system can leadto the Hojman conserved quantity under certain con-ditions.

Theorem In the Nambu system, if the genera-tors ξ0, ξi satisfy the conditions

(ξj − gjξ0)∂2gi

∂qj∂qi= 0, (i, j = 1, 2, . . . , n), (10)

the system possesses the following Hojman conservedquantity:

IH =∂

∂qi(ξi − ξ0gi) = const. (11)

Proof We have

dIH

dt=

ddt

(∂ξi

∂qi− gi

∂ξ0

∂qi− ξ0

∂gi

∂qi

)

=ddt

(∂ξi

∂qi+

∂ξ0

∂t

)− d

dt

dξ0

dt

− ddt

(ξ0

∂gi

∂qi

). (12)

Noticing the following equations:

ddt

∂ξ0

∂t=

∂

∂t

(dξ0

dt

)− ∂ξ0

∂qi

∂gi

∂t,

ddt

∂ξi

∂qi=

∂

∂qi

(dξi

dt

)− ∂ξs

∂qi

∂gi

∂qs, (13)

we get

ddt

(∂ξ0

∂t+

∂ξi

∂qi

)

=∂

∂t

(dξ0

dt

)− ∂ξ0

∂qi

∂gi

∂t+

∂

∂qi

(dξi

dt

)− ∂ξs

∂qi

∂gi

∂qs

=∂

∂t

(dξ0

dt

)+

∂

∂qi

(dξi

dt

)− ∂

∂qi

(ξ0

∂gi

∂t+ ξs

∂gi

∂qs

)

+ξ0∂2gi

∂qi∂t+ ξs

∂2gi

∂qi∂qs

=∂

∂t

(dξ0

dt

)+

∂

∂qi

(dξi

dt

)− ∂

∂qi

(dξi

dt− gi

dξ0

dt

)

+ξ0∂2gi

∂qi∂t+ ξs

∂2gi

∂qi∂qs

=ddt

(dξ0

dt

)+

∂gi

∂qi

(∂ξ0

∂t+ gs

∂ξ0

∂qs

)

No. 12 Lie symmetry and Hojman conserved quantity of Nambu system 4363

+ξ0∂2gi

∂qi∂t+ ξs

∂2gi

∂qi∂qs

=ddt

(dξ0

dt

)+

∂

∂t

(ξ0

∂gi

∂qi

)− ξ0

∂2gi

∂qi∂t

+gs∂

∂qs

(ξ0

∂gi

∂qi

)− gsξ0

∂2gi

∂qi∂qs

+ξ0∂2gi

∂qi∂t+ ξs

∂2gi

∂qi∂qs

=ddt

(dξ0

dt

)+

ddt

(ξ0

∂gi

∂qi

)

+(ξs − gsξ0)∂2gi

∂qs∂qi. (14)

Using Eq.(14) and the conditions (10), we can ob-tain

dIH

dt= 0. (15)

5. Argument

The Hojman conserved quantity of the Nambusystem has more generalized significance. It can de-generate to the Hojman conserved quantity of theHamilton system.

When the dimension of the Nambu system is two,equations represented by Eq.(2) are just the canoni-cal equations of the Hamilton system, and equationsrepresented by Eq.(6) become

gs =∂H

∂ps, hs = −∂H

∂qs. (16)

The variables qi of the Nambu system (i = 1, 2)include the canonical variables qs and ps of Hamil-ton system, and ξi include the generators ξs and ηs.So, in Hamilton system, the conditions representedby Eq.(10) for the existence of the Hojman conservedquantity for the Nambu system become

(ξj − gjξ0)∂2gi

∂qj∂qi

= (ξj − gjξ0)∂

∂qj

(∂gs

∂qs+

∂hs

∂ps

)

= (ξj − gjξ0)∂

∂qj

[∂

∂qs

(∂H

∂ps

)+

∂

∂ps

(−∂H

∂qs

)]

= 0. (17)

For the Hamilton system, we can see that theseconditions can always be achieved no matter what the

generators are chosen. So Lie symmetry can alwayslead to the Hojman conserved quantity for Hamiltonsystem. Then the Hojman conserved quantity (11) ofthe Nambu system becomes

IH =∂ξs

∂qs+

∂ηs

∂ps+

∂ξ0

∂t− dξ0

dt, (18)

which is just the Hojman conserved quantity led byLie symmetry of the Hamilton system.[17]

6. Example

Suppose that the two Nambu–Hamilton functionsof the Nambu system with three dimensions are:

H1 =12(x2

1 + x22),

H2 =12(x2

1 − x22 + x3). (19)

We get the equations of motion of the system:

x1 = x2,

x2 = −x1,

x3 = −4x1x2. (20)

The determining equations of Lie symmetry are:

dξ1

dt− g1

dξ0

dt= ξ2,

dξ2

dt− g2

dξ0

dt= −ξ1,

dξ3

dt− g3

dξ0

dt= −4x2ξ1 − 4x1ξ2. (21)

Using Eq.(20), we can obtain the following solu-tions:

ξ0 = ξ1 = ξ2 = 0, ξ3 = ln(2x21 + x3). (22)

Apparently they satisfy the conditions Eq.(10). Sub-stituting them into Eq.(11), we can get the Hojmanconserved quantity led by the Lie symmetry of theNambu system

IH =1

2x21 + x3

= const. (23)

4364 Lin Peng et al Vol. 17

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