HUEPHAH 0H3HKA, 2007, moM 70, № 3, c. 595-599

ELEMENTARY PARTICLES AND FIELDS. THEORY

NAMBU-POISSON DYNAMICS OF SUPERINTEGRABLE SYSTEMS

© 2007 N. V. Makhaldiani*

Laboratory of Information Technologies, Joint Institute for Nuclear Research, Russia

Received May 16, 2006

After an introduction in the Nambu—Poisson dynamics (NPD) some applications of NPD in the finite-dimensional (superintegrable) and infinite-dimensional (extended quantum mechanics and hydrodynamics) systems are considered.

PACS numbers: 02.30.Ik

1. INTRODUCTION

The Hamiltonian mechanics (HM) is in the ground of mathematical description of the physical theories [1]. But HM is in a sense blind, e.g., it does not make difference between two opposites: the ergodic Hamiltonian systems (with just one integral of motion) [2] and (super)integrable Hamiltonian systems (with maximal number of the integrals of motion).

By our proposal [3], Nambu's mechanics (NM) [4] is proper generalization of the HM, which makes difference between dynamical systems with different numbers of integrals of motion explicit.

In Section 2 we review the results of the papers [3, 5], we consider a system of nonlinear ordinary differential equations which in a particular case reduces to Volterra's system [6] and integrate this system using Nambu—Poisson formalism [4, 7]. In Section 3 we review some results of the paper [8].

2. FINITE-DIMENSIONAL DYNAMICAL SYSTEM

In this section we consider the following dynamical system [3]:

X n

YnY, (e

Xn+m - eXn-m), 1 < n < N, (1)

m=1

Vn = Vn(Vn+1 - Vn-l).

(2)

E-mail: mnv@jinr.ru

It is connected also to the Toda's lattice system [9]

yn — gyn+i-y™ + eyn-y"-1

Indeed, if

then

Xn - yn Vn-l

rf - pXn + 1 _ pXn-1

n-

If Yn = 1 and p > 1, the system (1) reduces to the Bogoiavlensky's lattice system [10]

p

Vn — vn ^ ^ (vn+m vn-m). m=1

(3)

For N = 3, p = 1, and arbitrary Yn, (1) is connected to the system of three vortices of two-dimensional ideal hydrodynamics [5, 11].

2.1. System of Three Vortexes

The system of N vortexes can be described by the following system of differential equations [11, 12]:

N

zn - 1

Ym

m=n Zn Zm

1 < n < N,

(4)

1 < p < [(N - 1)/2], 3 < N, Xn+N = xn,

where Yn are real numbers, and [a] means the integer part of a.

The system, (1) for jn = 1, p = 1, and xn = ln vn becomes Volterra's system [6]

where zn = xn + iyn are complex coordinate of the centre of nth vortex.

For N = 3, it is easy to verify that the quantities

X1 = ln\Z2 - z3\2, X2 = ln |Z3 - Z112, X3 = ln \zi - Z2 \2 satisfy the following system:

x 1 = Y1(eX2 - ex3), x2 = y2(ex3 - eXl), x3 = Y3(eXl - eX2),

(5)

(6)

595

10*

after change of the time parameter as

g(xi +X2+X3)

dt =

4S

dr = e (xi+X2+X3)/2Rdr, (7)

where S is the area of the triangle with vertexes in the centres of the vortexes and R is the radius of the circle with the vortexes on it.

The system (6) has two integrals of motion

Hi

^ Yi '

i=i n

H2 = E

i=i

Yi

(8)

and can be presented in the Nambu—Poisson form [5]:

dHi dH2 , w eXj 1

xi = uijk^--^- = \xi, Hi,H2} = uijk--,

dxj dxk Yj Yk

where

Uijk = tijk P, P = Y1Y2 Y3, (10)

and the Nambu—Poisson bracket of the functions A, B, C on the three-dimensional phase space is

r * T, ™ dAdB dC

{A,B,C} = uijkdXAl dX!~Wk- (11)

The fundamental bracket is

{Xi ,X2, X3 } = Uijk-

Then we can again change the time parameter as du = pdr (13)

and obtain NM [5]:

dHi dH2

x i

£ijk

dxj dxk

xi x2 Y3 H2----

Yi Y2

x3 = Y3 l h 2 ----— I, (14)

insert it into H1, find x2 as an implicit function of x1,

e

JX1 + eT3(H2-^ ) Y2 Y3

Dxi

Hi -

Yi

integrate motion equation of x1:

x 1 = Y1(eX2 - eX3) = n(x1),

X1

dx n(x)

= T - To-

X10

For

Y3 Y2

= ±1,± 2,± 3,- 4,

n(x) is a superposition of elementary functions.

2.2. Four-Dimensional System The next important case is N = 4 and p = 1,

x 1 = Yi(eX2 - eX4), x2 = Y2(eX3 - eX1 ), (18) x3 = Y3(eX4 - eX2), x4 = y4(eX1 - eX3).

Like as N = 3,p = 1 case, for (18) we have two integrals of motion

eX1 eX2 eX3 eX4

Hi = — + — + — + —, (19)

Yi Y2 Y3 Y4

xi x2 , x3 x4

H 2 =--1---1---1--.

Yi Y2 Y3 Y4

(20)

For the superintegrability of the system (18), we need one more integral of motion, H3. To find that integral (9) let us suppose Nambu's form of the system (18)

xn = {xn,Hi,H2,H3} = (21)

dHi dH2 dH3

= YlY2Y3Y4(-umM-

dxm dxk dxi We found from (21) a solution for H3:

1 xi x2 x3 x4 H3 = -ô----1----

2 Yi Y2 Y3 Y4

(22)

Because we already have three integrals of motion, (12) we can integrate the system (18). From (20) and (22) we get

(H2 + 2H3 X2

X4 = Y4 -o---

Y2

Note that this system is superintegrable, for N = = 3 degrees of freedom, we have maximal number of the integrals of motion N — 1 = 2. Now we define x3 from H2,

x3 = Y3

and (19) gives us

2

H2 - 2H3

(23)

xi Yi

e

X1

+

e

X2

Yi Y2 +

+

e ti

-T3(H2 /2-H3)

Y3e

- ^X2

e Y2 2

+

(24)

Hi.

Y4e-Y4(H2/2+H3)

So x2 is an implicit function of xi, x2 = U2(xi,Hi,H2,H3) (15) when

Y = ±1, ±2, ±3, -4,

Y± Y2

(25)

(26)

(16) the function n2 reduces to the composition of the elementary functions. Now we can solve the equation for x1 ,

x 1 = Yi(e"2 — ex4) = ni(xi), (27) by one quadrature

xi

(17)

N (xi) =

dx ni(x)

= t - to-

3

x

NAMBU—POISSON DYNAMICS

597

Note that, from the motion equations (18) or (23) it is easy to see that we have a separation of the odd and even degrees of freedom,

Xl +Хз H H2 и

--1--= H13 = -7;--H3,

Yi Y3 2

x2 + x4 H H2 + H --1--= H24 = + H3.

Y2 Y4 2

(29)

N

Hi

e

¿1 Yn

N

H2=

n=1

Xn

Yn

For even N, N = 2M, we found the third integral of motion

2M

H 1 ^ (~1)nXn H3 = 2^-

n=1

Yn

In this case, we have a separation of even and odd degrees of freedom,

M

x2n— 1

£

n=1

- = H1m = HT — H3 ,

Y2n-1 2

M

= {Xn,Hi}1 = W

1 dH1

dxm xk

= {Xn,H2}2 = W

2

dH2

<dx Ln 2)2 nmdx dXm dXm

Corresponding Poisson structures, u1 and u2 are degenerate, because they are 3 x 3 antisymmetric tensors. The eigenvectors with vanishing eigenvalue are

dH2 1 dHi

dxr,

Yn

dxr,

(36)

(30)

Now we can put the system in the form

x 1 = YI^2 - eY4H24e"%X2) = fi(x2), x2 = Y2(e73Hl3e"^X1 - eXl) = f2(xi).

For numerical solution this system may be more convenient.

2.3. General Finite-Dimensional System

For the generalcase we have two integrals of motion for the system (1)

respectively.

If we define, e.g., x3 from H2, on the restricted "phase space" (x1 ,x2) with Hamiltonian H(x1,x2) = H1(x1,x2,x3(x1,x2)), we find the following regular symplectic dynamics:

dH

xn = {xn,H} = Y1Y2£nm-,—, n,m = 1, 2. (37)

' dxn

Indeed, e.g.,

x 1 = Y1Y2

ex2 + dx3 ex3 Y2 dx2 Y3

= Y1 (ex2 - ex3). (38)

(31)

(32)

Motion equations take canonical Hamiltonian form

x =

dH

dp '

P = -

dH

dx '

(39)

(33)

after change of the variables as x1 = y1 x, x2 = Y2P. We can quantize this system introducing coordinate, momentum, and Hamiltonian operators X, p, and H,

x = x, p = —ihd, xp — px = \x,p] = ih,

dx

H

eYiX eY2p eY3(H2-x-p) +-+ ■

Y1

(34)

Y2 +

Y3

(40)

eYix eY2p

+-+

Y1

Y2

eY3H2-f Y2

Y3

-Y3 xe-Y3 p

STx2n H H2 + H

; y - = H2m = — + H3.

n=1 Y2n 2

When N > 5, for integrability, we need extra integrals of motion.

2.4. Simplectic Reduction of Three-Dimensional System to Two-Dimensional One and Quantization

Numbu—Poisson dynamics (NPD) (9) reduces to two Hamiltonian dynamics (HD),

г и и 1 dH1 dH2 xn = {xn, H1H2 } = WnmkZ--= (35)

2.5. Simplectic Reduction of Four-Dimensional System to Two-Dimensional One and Quantization

It is easy to see that the motion equations on the restricted phase space (x1,x2) can be put in the regular simplectic form

xn = {xn ,H} = (41)

dH(x1 ,x2 ,x3(x1 ),x4(x2))

= Y1Y2^nm-

dxn n,m = 1,2.

Motion equations take canonical Hamiltonian form

x

dH

dp '

dH

dx '

after change of the variables as x1 = Y1x, x2 = Y2p. We can quantize this system introducing coordinate, momentum, and Hamiltonian operators x, p, and H,

H

eYiX eY2P e~f3 (Hi3-x) + — +-+

71

72

73

(42)

+

eT4(fl24 -p) eYix g73 Hi3

+

-Y3x

73

+

eY2P

74 Yi

eY4H24

+ e-e-Y4 p = K (p) + V (x).

Y4

Y2

+

2.6. Parametric Contraction of Three-Dimensional System to Two-Dimensional One

Let us put the second integral of (8) in the following form:

Y3H2 = yJ - + + x3 (43)

Y1 Y2

and take limit y3 ^ 0, for finite Y1, Y2, H2, x1, x2, and x3. As a result we have x3 = 0. To this values, for consistency, corresponds H1 = 1/y3. From the motion equations we have x1 = x2 = x, Y1 = —Y2 = Y and

X = Y(ex - 1),

with general solution

x(t) = -Yt - ln(e-x0 - 1 + e-Yt).

(44)

(45)

2.7. Reduction of Four-Dimensional System

For y4 ^ 0, from the integral H2 (20) we have x4 = 0 and from (29), x2 = const. From motion equations (18) then we obtain

Y3

x1 = n1t + x10, x3 =--x1 + const. (46)

Y1

3. INFINITE-DIMENSIONAL SYSTEM

As an example of the infinite-dimensional NPD, let me conside the following extension of the Schrodinger quantum mechanics [8]:

V2

iVt = AV - —, = -A^ + V

(47)

(48)

An interesting solution of the equation for the potential (48) is

4(4 — d)

V=

(49)

The variational formulation of the extended quantum theory (47), (48) is given by the following La-grangian:

L = [iVt - AV + 2 V2 ) 4.

The momentum variables are dL

Pv = ^77 = iP* = 0. dVt

(50)

(51)

As a Hamiltonians of the Nambu theoretic formulation we take the following integrals of motion:

Hi = J ddx ^AV — 1V^ ^, (52)

H2 = J ddx(Pv — i^), H3 = j ddxPi,.

We invent unifying vector notation, 0 = (01,02,03, 04) = (^, P^, V, Pv). Then it may be verified that the equations of the extended quantum theory can be put in the following Nambu theoretic form:

Mx) = {0(x),Hi,H2,H3} = (53)

. f S(0(x), Hi,H2,H3)

J S(0i(y)A2(y),fa(y),^4(y))

where the bracket is defined as

{Ai, A2, A3, A4} = r A SA2 SA3 5A4

dy,

(54)

i^ijkl

fyi (y) 60 j (y) 60k (y) Sfa (y) 5(Ai,A2, A3, A4)

dy =

6(0i (y),My),fo (y),My)) 6Ak

dy =

= idet

60i

where d is the dimension of the spase. In the case of d = 1 we have the potential of the conformal quantum

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