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ELSEVIER Physics Letters A 223 ( 1996) 355-358

16 December 1996

PHYSICS LETTERS A

Symmetries, integrating factors and Nambu mechanics

G. ha1 Faculty of Science and Letters, Istanbul Technical University, Maslak 80626, Istanbul. Turkey

Received 19 August 1996; accepted for publication 7 October 1996 Communicated by A.R. Bishop

Abstract

We show that if an ?l-dimensional autonomous dyna~cal system (DS) with a vector field (VF) which has const~t divergence possesses n - 1 independent first integrals, then it admits a symmetry VF which involves Nambu mechanics (NM). If the DS is conservative, then the Nambu VF happens to be a symmetry VF of the DS. We also show that the integrating factors can be constructed via NM. We illustrate our results on the Lotka-Volterra DS.

Recent works showed that Lie symmetries of DS are crucial in the study of normal forms [ 11, invariant solutions [ 21 and their bifurcations [ 31. Thus find- ing these symmetries has become important. Yet it is a formidable task to undertake. Our main objec- tive here is to look for symmetries which are intrin- sic property of an ~-dimensional DS with TV - 1 inde- pendent first integrals (i.e, integrable in the Liouville sense). Interestingly enough we have encountered NM in this quest. Parallel to this result, Sen and Tabor [ 41 found the Hamiltonian structure of a symmetry VF for completely integrable dissipative systems such as the Lorenz model for certain parameter values.

In Ref. [ 51 Nambu has generalized Hamiltonian dy- namics under the guidance of the Liouville theorem. One of the salient features of NM is that it is not re- stricted to even-dimensional phase spaces. Mukunda and Sudarshan [6] have shown that Nambus equa- tions can be derived from a singular Lagrangian and it can be embedded in a higher-dimensional Hamilto- nian phase space. In Ref. [7] the authors compared NM with the other mechanics. They have concluded that NM is indeed a new mechanics. Kentwell [ 81 dis-

cussed that the Nambu bracket form helps to construct the generalized Poisson bracket easily. In Ref. [ 91 the authors revealed that the DS arising in NM pass the Painlevd test.

We also show that if an integrating factor (density function, last multiplier) is known in addition to n - 1 independent first integrals, then the DS obtained from the original DS by multiplying it by the integrating factor admits the Nambu VF as a symmetry VF. It is also possible to obtain the integrating factors via NM. Here we consider an autonomous DS

&=fj(x) (i= I,2 )..., n),

which defines a VF of the form

(1)

We assume that unless stated otherwise the summa- tion convention applies to repeated indices. Through- out this Letter we also assume that the DS given by ( 1) has n - 1 independent first integrals, namely the II,. . . t &_ f. Associated with (2) we also consider the VF

0375-9601/96/$12.00 Copyright @ 1996 Published by Hsevier Science B.V. All rights reserved. PIISO375-9601(96)00771-2

356 G. &al/Physics Letters A 223 (1996) 355-358

N= a d

vi(x) aX_ = Eijk...l~l,j~2,k. . . I-l,lax_, (3) I 1

which arises in NM [ 8,9]. Here l ijk...l is a generalized Levi-Civita symbol, and I,,,& = dl,,/&k. With these initial conditions we can proceed to the lemmas.

Lemma 1. Components of the VFs (2) and (3) satisfy the following relation,

vjfi-f,Vi=Zj=O (i,j=l,...,?r). (4)

Proo$ One can view the cross products of gradient vectors of first integrals as the determinant of an IZ x II matrix L. The first row of L is composed of unit vectorsel,..., e,, and the other rows are occupied as Lij=li_l,j (i=2 ,..., n; j= l,..., n).Duringthe proof of this lemma the summation convention will be suspended. Now we proceed by resorting to Laplaces expansion theorem on determinants [lo] to get

n-l zj = C(-l)k+j+i+lM:i(a$f, + ai!Jj). (5)

k=l

Here Mi is the minor obtained from L by deleting the first and kth rows, and deleting the jth and ith COlUITltlS, a; = Ik,i and a. = Ik .

From the definition of%me fndependent first inte- grals we can write

n n-2

c fllk,l = Ik,ifi + Ik.jfj + c fk,,, fnr = 0. (6) i=l mjfi,i

Using (6) in (5) and rearranging the terms leads to

Tj = c ~(-l)k+i+i+2Mlj,lj ki km fm, (7) WCi k=l

since the sum of the products of the elements of any column by the corresponding cofactors of the elements of another column is equal to zero [lo]. Then the sum in the parentheses of (7) vanishes by proving the lemma.

Lemma 2. The Lie bracket of the VFs (2) and (3) is given by

[N, Fl = _fk,kN. (8)

ProojY Upon differentiating identity (3) with re- spect to the coordinates Xj we obtain

Vj,j fi + Vj fi,,j - fjVi.j - fj,jVi = 0.

Since the Nambu VF is divergence free [ 51, the first term vanishes. By shifting the last term to the right hand side, we obtain components of the VF in (8).

As it is noticed from the commutator (8)) if the DS given by ( 1) is conservative, then the Nambu VF commutes with the VF (2)) thus indicating that it is a symmetry VF of the DS given in ( 1) [ I]. Equipped with these results, now we are prepared to prove the main theorem of this Letter.

Theorem 1. The DS given by (1) with a VF which has a constant divergence admits the following sym- metry VF,

X = vi(x)efk,i-&. I

Proof Let us consider the VF

Acting the first prolongation of the VF ( 10) onto ( 1) and submitting to the invariant surface criterion [2] one obtains

- = TjfiJ - fjrli,j. at

(11)

Components vi should satisfy this system of linear partial differential equations (PDEs) in order for the VF (10) to be a symmetry VF of the DS given by ( 1) . Notice that PDEs ( 11) have the form of the Bel- trami equation for incompressible fluids while vi (i = 1,2,3) are the components of the vorticity vector. Now we seek a solution to ( 11) in the form

vi(x, t) = vi(x)h(t). (12)

Substituting (12) into (11) and resorting to Lemma 2 leads to the following ordinary differential equation,

h = fk,kh. (13)

The solution to Eq. (13) leads to the required result

(9).

G. &al/Physics L.&km A 223 (1996) 355-358 357

Theorem 2. If an integrating factor g(x) of the DS not. We resort to the notation used in Lemma 1, i.e., (1) is known, then the DS we set

ii = g(X).fi(X)

admits Nambu VF (3) as a symmetry VE

(14)

Proo$ Let us calculate the following Lie bracket,

lN,gF] = (vjCg_fi>,j - SfjVj,i}&* (15) I

Eijk...l~l.,l2,k * , . In-l,l z Kli, (22)

where Kii is the cofactor obtained by deleting the first row and ith column of the matrix L. In this notation PDEs ( 17) yields

We continue our proof by calculating the components in the curly braces. Due to Lemma 2 we are allowed to write

- ~l(i)fmf(i),m + Kl(i)fm,mf(i))*

Due to Lemma 1 we have

(23)

{} = VjL?.jfi + gfk,kvi* (16)

By definition, integrating the factor g satisfies [ 1 I ]

gfk.k -I- figj = 0. (17)

Using (17) in (16) we obtain

{} = (V,jfi - fji)g,j. (18)

Thus by Lemma 1, the quantities in ( 18) vanish yield- ing

[N,gFl =o, c 19)

which proves the theorem.

Klifm = Klmfi. (24)

Substituting (24) into IQ. (23) gives

(*} = $tKl(i),m.fm - Klmf(i),nr + Kl(i)ftn,m)*

(25)

Lemma 2 enables us to write

Theorem 3. An integrating factor g(x) of the DS given in ( 1) with a VF which has the property fk,k $ 0 (nonconservative) is given by

Kti,mfm - K1mfi.m = -_fm,mKli- (26)

Using (26) in (25), we see that the terms in curly braces vanish. Hence, the function given by (2 1) is indeed an integrating factor. Next we show that (20) holds under the assumptions mentioned earlier. It fol- lows from relation (24) that

Kl(i) K1( nr) -=-

f(i) f(m) thus proving the theorem.

g(x) = ~l.jk...lll,j~2.k . . .I-1,l

fl

= f2jk...I~l,jf2.k. * . ftz-1.1

=...

f2

= Enjk...lIl,jlt,k * * * In-l,1

fn *

Proo$ First we must show that

(20)

(21)

is indeed an integrating factor (indices in parentheses do not obey the su~ation convention). To achieve this, we check whether it satisfies the PDEs (17) or

Remark. According to a theorem given in Ref. [ 121, the ratio of two integrating factors yields a first inte- gral. Thus, we can find n - 1 integrating factors via Theorem 3 as follows,

t+(x) = E(i)jk...lll,jlZ,k * * * ln-l,ll

f(i) P

(p=1,2 ,..., n-l). (27)

Now we would like to illustrate our results by taking a three-dimensional DS. Nutku [ 131 showed that the Lotka-Volterra DS

i1 =xl(CX2+X3+f), k2 =x2(x1 fax3 -i-m), k3 = x3(bx1 f x2 + at) (28)

358 G. &d/Physics Letters A 223 (1996) 355-358

has a bi-Hamiltonian structure when the parameters satisfy the conditions

abc = -1, n = mb - lab. (29)

The DS given by (28) possesses two independent first integrals,

Ii = ablnxi - blnxz +lnxs,

12=abxt+x2-axs+nlnx;?-mlnxs. (30)

The Nambu VF for this problem is found to be

(31)

and the divergence of the VF defined by DS (28) is

fk,k = 1( 1 - ab) + m( 1 + b) + (1 + b)xl

+ (32)

According to Theorem 3, the DS given in (28) has an integrating factor of the form

ab

x1x2x3 (33)

The Nambu VF is not a symmetry VF of DS given in

(28)) but it is a symmetry VF of

i, = -x2 + ab(x3 + l)

, i2= XI + ax3 + m

x2x3 XIX3

i3 = bxl + x2 + b( m - la)

XIX2 (34)

The DS in (34) is obtained by multiplying the VF of the DS in (28) by the integrating factor (33). If we set a = -1 and b = - 1, then the DS in