A note on Nambu mechanics

IL NUOVO CIMENTO VOL. 106 B, N. 3 Marzo 1991

A Note on Nambu Mechanics (*).

W.-H. STEEB and N. EULER

Depar tment o f App l i ed Mathemat ics and N o n l i n e a r S~udies, R a n d A f r i k a a n s Univers i t y PO Box 52~, Johannesburg 2000, Sou th A f r i ca

(ricevuto l ' l l Maggio 1990)

Summary. - - In Nambu mechanics an autonomous system of first-order ordinary differential equations du~/dt = F i (u ) (i = 1, ..., n) is constructed with the help of ( n - 1 ) smooth functions Ii. These smooth functions are first integrals of this dynamical system. If the functions Ii are polynomials, then the system is algebraic completely integrable. We discuss the question whether the first integrals determine uniquely the autonomous system of first-order differential equations. Then we give a generalization of Nambu mechanics.

PACS 02.30.Hq - Ordinary differential equations.

1. - I n t r o d u c t i o n .

In Nambu mechanics the phase space is spanned by an n-tuple of dynamical variables u i ( i = 1, ..., n)[1-7]. The equations of motion of the Nambu mechanics ( i .e . the autonomous system of f i rs t-order ordinary differential equations) are now constructed as follows. Le t Ik: 9 ~ n ~ t T g ( k = l , . . . , n - 1 ) be smooth functions. Then

(1) du___A~ = a(ui , 11, ..., In-l)

dt a (u , , u~ , . . . , un) '

where ~(Ul , 11 , . . . , I n _ l ) / ~ ( U 1 , U2 , . . . , Un) denotes the Jacobian. Consequently, the equations of motion can also be wr i t ten as (summation convention)

(2) dui dt = eijk'"lajI1 ' " '" a l l n -1 '

(*) The authors of this paper have agreed to not receive the proofs for correction.

263

264 W.-H. STEEB a n d N. EULER

where eijk...t is the generalized Levi-Civita symbol and aj =-3/Suj. The proof that I1, ...,I~-1 are first integrals of system (1) (or system (2)) is as follows.

We have

dI__! = 0I~ duj

dt auj dt (summation convention)

= (Oj I i ) ejt,...L._, (Ol, 11) "" "(a/~_, I n _ l ) ,

= cjt,...~._~(ajli)(azI1)... ( ~ / . _ l l n - 1 ) ,

O(Ii , I1 , ..., In- 1) a(ul , ..., us)

= 0 ,

as the Jacobian matrix has two equal rows and is therefore singular. If the first integrals are polynominals, then the dynamical system (1) is algebraic

completely integrable. We discuss the following question: let a dynamical system d u i / d t = Vi (u) (i =

= 1, ..., n) be given, where the V" s are rational functions of Ul, ..., u . . Assume that the dynamical system admits n - 1 polynomial first integrals. Do the n - 1 polynomial first integrals uniquely determine the dynamical system? We have to take into account that if I1,. . . , I~_1 are first integrals, thenf( I1 , ..., In-l) is also a first integral, where f is a smooth function. Furthermore, if

(3) v = v l ( x ) ~ + . . . + V~(x) 3xl ~x~

is the vector field associated with the system d u i / d t = Vi (u) and I is a first integral, i.e.

(4) L v I = 0,

where L v denotes the Lie derivative, then I is also a first integral of the vector field fV, where f is a smooth function.

A remark is in order: the first integrals I1, I2, ..., In-1 determine the trajectories owing to I1 = Cl,/2 = c2, ..., I~-1 = C~-l, where the constants Cl, c2,--', cn-1 are given by the initial conditions. However, the speed at which one travels along these trajectories is left open. Several differential equations can have the same trajectories, but their solutions proceed along the trajectories at different speeds. The simplest nontrivial example occurs when n = 2. Let I ( p , q ) = (p2+ q2)/2. Then a possible system of differential equations is dq/dt = - p , dp/dt = q, but another one is dq/dt = = _p(p2 + q2), dp/dt = q(p2+ q2), and a third one is dq/dt = - p q , dp/dt = q2. Let us think of the trajectories as concentric racetracks, and the solutions as cars on these tracks. For the first case all cars complete a lap with identical times, while in the second case the cars on the outer track finish earlier, and in the third case the cars move both forwards and backwards on the same track.

A NOTE ON NAMBU MECHANICS

2. - T h r e e - d i m e n s i o n a l c a s e .

W e re s t r i c t ou r se lves in the fol lowing to the case motion a r e g iven b y

dUl 311 312 311 312

dt ~u2 3u3 3u3 3 u 2 '

du2 311 312 3I~ 312 (5b) - - -- e2jk Oj 11 Dk 12 --

dt 3u3 3Ul 3Ul 3 u ~ '

du3 311 312 3I~ 312 (5c) - ~sjk aj 11 a~/2 --

dr 3u~ Sue 3u2 3Ul

I f I1 a n d / 2 a r e quadra t i c polynomials , i.e.

265

n = 3. T h e n the equa t ions of

(6a) I i ( u ) = a l l U ~ / 2 +a12u lu2+. . . +a33u2/2 + a l u l + a 2 u 2 + a s u 3 ,

(6b) I2(u) = bll U~ /2 + b12ui wz + ... + b~u~ /2 + bi ui + b2u2 + bsu3 ,

then the equa t ions of mot ion a r e g iven b y

dUl (7a) dt - (a12 bla - a13 b12) Ul 2 + (a12 b~ + am b13 - a13 b~ - a ~ b12) u 1 ?A~ 2 -{-

+(a12 b~ + a ~ bl3 - a13 b~ - a ~ b12) u l u3 + (a22 b~ - a ~ b~) u~ +

+ ( a ~ b~ - a33 b~) ue u3 + ( a ~ b~ - a33 bzs) u J +

+(a12 b3 + ae bls - als be - a3 b12) Ul + (a22 b3 + a2 b~ - a ~ be - a3 b~) ue +

+ ( a ~ b3 + a2 b33 - a ~ b2 - a3 b~) u3 + a2 b3 - a3 b2,

du2 (7b) dt - (al3 bll - a l l b13) Ul 2 + (al3 b12 + a23 bll - a l l b23 - a12 bls) Ul u2 -~-

§ bll - a l l b~) Ul u3 + ( a ~ b12 - ale b~) u~ +

+ ( a ~ b13 + a ~ b12 - a12 b~ - a13 b22)u2 u3 + (a33 bl3 - al3 b ~ ) u ~ +

+(al3 bl + a3 bll - a l l b3 - a l b13) Ul + (a23 51 + a3 b12 - a12 b3 - a l b~ )u r +

+(a33 bl + a3 b13 - ala b3 - al b33) u3 + a3 bl - al b3,

du3 (7c) d-T =(ai lb12-a12b11)u2+(al lb22-a~2bl l )UlU2+

+(a l l 523 + al3 b12 - a12 bl3 - a ~ b11) u 1 u 3 +

+(a12 b~ - a ~ b12 ) u22 + (a12 b~ + a13 b~ - a ~ b13 - a23 b12) u2 u3 +

+ ( a l 3 b~ - a ~ bls) u~ + ( a l l b2 + a l ble - a12 bl - a2 b l l ) Ul -}-

+(a12 b2 + a l b~ - a ~ bl - ae b12) u2 + (a13 b2 + al b~ - a ~ bl - as bls) wa +

+ a l b2 - ae bl �9

266 W.-H. STEEB and N. EULER

3. -- E x a m p l e s .

Three important applications in physics are the Euler rigid-body motion, the motion of a charged particle in a constant electric and magnetic field and a special case of the Lorenz model. The first two examples are well known in the literature [1-4]. For the first case the equations of motion are given by

dut (8a) ~t - ~ - = (~8- ~2)u2u3,

du2 (8b) ~ ~ = (~1 - s u~ul ,

du3 (8c) ~3 - ~ - = (~2 - ~t) ut u2.

The equations of motion can be derived within Nambu mechanics from the first integrals

1 (Dtu~ +t~2u~ + D3u~) (9a) 11 (u) = ~ ,

1 ( D t u12+ t)2 t)3 u~) (9b) I2(u) = ~ ~ - - ~ ~ u~ + ~ .

The first integral It represents the total kinetic energy. The nonrelativistic motion of a charged particle with mass m and charge q in a

constant electric field E = (El , E2, E3) and constant magnetic field B = (Bt, B2, B3) is given by

(10a)

(10b)

(10c)

dVl dt

dv2 dt

dvz dt

where v = (vt , v2 , va) denotes E 2 = E a = 0 , E r e 0 , then we

q El+ q ( B s v 2 - B2va) - - = ~ ~

- - = --~q Ee + - ~ q (Bl v s - B s v t )

= m q E3 + - ~ q ( B 2 v l - B t v2) ,

the velocity of the particle. If B1 • B2 = 0, B3 r 0 and obtain

dvl q q ( l la) dt = -m E1 + - ~ B3ve ,

dv2 q B3 vt (llb) dt = - ~ '

(11c) dvs dt = O.

This dynamical system can be derived from the first integrals

�9 (12a) I i ( v ) = - v 3 ,

A NOTE ON NAMBU MECHANICS

1 B2v2+ 1 B2v2+ElV2 (125) I2(v)= ~ 3 1 ~ �9

The Lorenz model is given by

(13a) dUl dt = ue - ~r ,

(13b) du2 dt = -UlU3 + ul - ~u2,

(13c) du3 dt - UlU2 - b~3

If ~ + 0 these equations reduce to an integrable system

(14a) dUl dt = u s ,

(14b) du2 dt = - u l u s + u l '

(14c) dua d~t = u l u 2 "

These equations can be derived from the first integrals

(15a) I1 (u) = - u ~ + u~ + u ~ ,

1 u~+u3. (15b) 12 (u) = -

These two in, st integrals define a two-parameter family of period solutions.

267

4. - C o u n t e r e x a m p l e .

The example we use to answer the question plays an important role in the motion of energy levels for a Hamilton operator of the form H = H 0 + 2 V with a nondegenerate discrete spectrum. Various authors[8-17] discussed the ,,motiom> energy levels En (2) where ~(. plays the role of the time. They derived the following autonomous system of first-order ordinary differential equations:

dEn (16a) d~ - Pn ,

dp~ V ~ Vnm (165) d--~- = 2

m(r En - E m '

(16c) d---~ k ( , , ) E r a - E------~ S~ - E-----~ E ~ - E~ '

where Pn(~)= (~bn(~)[Vkbn(~)) and Vmn(~): = (~m(~)[Vl~bn(~)) (m :/: n). I f we have a finite-dimensional system with N energy levels, then the number of differential equations n is given by n = N(3/2 + N/2).


We consider the case wi th two simplifies to

(17a) dEj d~

(17b)

(17c) dpl d~

(17d) dVo~ d~

where j = 0, 1 and Uol -- Vlo. The following quantities: E: = E 1 - Eo,

(18a)

energy levels Eo(~) and E1 (~). Then system (16)

(18b)

(18c)

On inspection, we find that

m _ _ p j ,

dpo d~ -2Eo-- - E l

vow1 _ _ __-- 2 ~1 - E 0 ,

Vol (Po - Pl) E o - E 1 '

form of system (17) motivates to introduce the P: =Pl-Po and V: = Vol. We obtain

dE d)~ = P '

dp _ 4~-- d)~ ' dV pV d)~ E

p2 (19a) 11 (E, p, V) = -~- + 2V 2 ,

(19b) 12 (E, p, V) = EV

are first integrals. IfI~ and 12 are first integrals, thenf(I1,12) and g(I1,12) ( f a n d g are smooth functions) are also first integrals, provided that a(f, g)/a(I1 12) ~ 0. Inser t ingf and g into system (5) (t (--)~) yields

(20a) dE =Epl 3f 3g 3f 3g ] d~ 311 312 312 aI1 '

dp [ 3f 3g 3f 3g ] (20b) d--~ =4V2 311 312 312 311 '

(20c) dV -pV[ 3f 3g ~f 3g ] d~ = 311 312 312 ~ "

In order to reconstruct system (18) from the first integrals I1 and I2, we have to impose the following condition on f and g, namely

3f 3g 3f 3g 1 (21) 311 312 312 3 1 1 - E "

A NOTE ON NAMBU MECHANICS 269

We see that condition (21) cannot be satisfied. Thus the answer to our question is not affirmative.

Iff(I1 , I2) - - I1 and g(I1 , / 2 ) = / 2 , then we obtain the system

(22a) dE d)~ - E p ,

(22b) d_p_p = 4V 2 d~

(22c) dV d)~ = - p V .

System (22) is a special case of system (7).

5 . - T r a n s f o r m a t i o n s .

It is obvious tha t the system (18) can be transformed into a sys tem such tha t the answer to our question is affirmative. For example, let E ' ( E , p , V ) = E , p' (E,p, V ) = p E and V' (E, p, V ) = V. Then system (18) takes the form

d E ' P ' (23a) - - = - - d~ E ' '

(23b) dp ' = 4V,2 + p'2 d2 E '2 '

(23c) dV..__.~: = _ p ' V_...~' d~ E ,2

The first integrals are given by

(24a) 11 ( E ' , p ' , V' ) 1 P'2 = + 2V '2 , 2 E 'u

(24b) 1 1 ( E ' , p ' , V ' ) = E ' V ' .

The first integrals given by eqs. (24a) and (24b) lead to eq. (23) within the Nambu mechanics. Consequently, for this transformed system the answer is affu~native. Another possible transformation is E ' (E,p , V) = E, p ' (E ,p , V) = p and F ( E , p , V) = = EV. We arrive at

(25a) d E ' = p ' d)~

dp ' F 2 (25b) - - = 4

d~ E ,3 ,

(25c) dF = 0. d~


The first integrals are given by

1 ,2 F 2 (26a) I~ ( E ' , p ' , F ) = ~ p + 2 E ' 2 ,

(26b) I~ (E ' , p ' , F ) = F .

From these two first integrals we obtain the equations of motion (25).

6. - G e n e r a l i z a t i o n o f N a m b u m e c h a n i c s .

As mentioned above the Nambu mechanics is an algorithm to generate completely integrable systems. Now we give another algorithm to generate completely integrable systems. This algorithm is as follows: we consider n = 3. The extension to higher n is straightforward. Let it = V(u) be an autonomous system of first-order ordinary differential equations, where V: t~s--* ~ 3 are smooth functions. The right-hand side defines the vector field

(27) V= V1(u) ~-~x + V2(u) ~-~22 + V3(u) 3 3u3 "

Let I1 and /2 be two smooth functions (Ik: t~ 3__. ~ ) . Assume that

(28a) Lv I1 = O,

(28b) LvI2 = O,

where Lv (') denotes the Lie derivative. This means that I1 and/2 are first integrals. It can easily be seen that 1/'1, Vz and V3 from system (7) fulfil eqs. (28) identically. Thus Nambu mechanics is a special case of this algorithm. Let V~ r 0 and

311 (29) D - 3u2

Then the solution to eqs. (28a) and

(30a) it1 = V1 (u),

3Iz 312 311 ] 3u3 3 u 2 3 u 3

4=0.

(28b) is given by

(30b) it2 = D-l ( 3123ul 3u3311 3u1311 ~/u~)Vx(u) '

(30c) it3 = D-1 ( 3Ia 312 311 312 1 3u~ 3u~ 3u2 3ul ] VI(u),

where V1 is an arbitrary smooth function. Let us now consider explicitly time-dependent first integrals[18, 19] of the

autonomous system it = V(u) with n = 3. Then the vector field V must be replaced by W = V + 3/3t. We assume that the explicitly time-dependent first integrals are of the form

(31a) 11 (u, t) = f lu ) exp [ct],

A NOTE ON N A M B U M E C H A N I C S

(31b) I2 (u, t) = g(u) exp [ct] ,

where c is a real constant. The conditions Lw11 = 0 and LwI2 = 0 yields

(32a) V2 = D-1 [ Sg Sf SUl 3U~

/ (32b) V3 = D -1 ~ Sf Sg

\ SUl Su2

where V1 is arbitrary and

(33) D -- ( Sf \ Su2

If c = 0, then we find system (30).

(34a) dUl dt

(34b) dur dt

(34c) du3 dt

with the first integrals

(35a)

(35b)

SUl SU3 V I ( U ) - } - D - l c g ~ u 3 - f ' ~ u 3 '

Due SUl VI(U)+ D - l c f -~u2 -g-~u2 '

Sg Og sf ) Su3 Su2 S--~8 r

I

An example is the system

-- CU 1 .'{- C23 U2 U3 ,

-- CU 2 -{- Cl3UsUl ,

_ _ _-- CU 3 ~ - c 1 2 u l u 2

1 I 1 (U, t) =- -~ (ClsU 2 -- C23U22 ) exp [-2ct] ,

1 /2 (u, t) -- ~ (cl2 Ul 2 - c~ u~ ) exp [-2ct] .

271

procedure nambu (I, u, n);

% Assumes operator I, u with I(1) . . . (n- 1) % assigned expressions in u(1)...(n) <<matrix J(n, n); % Jacobian

% Establish Jacobian elements independent of the index ii: for row : = l : n d o for c o l : = 2 : n d o J(row, col) := df(I(col- 1), u(row)); % Establish Jacobian elements dependent on the index ii % and write out system of ODEs: f o r i i : = l : n do <<for row : = 1 : n do J(row, 1) : = df(u(ii), u(row));

The algebraic manipulations necessary for finding the Nambu equations of motion, when the first integrals are given, are straightforward but often tedious. Such calculations are better done by computer using an algebraic language. Let us give the implementation of Nambu mechanics using REDUCE, where I1 (u) = ul + + u2 + u3 and/2 (u) = Ul u2 us. The extended approaches described above can also easily be implemented with the help of REDUCE.


>>; nambu

write "duC,ii,")/dt = ", det J; >>; clear J; % Tidy up!

operator I, u$

I(1) : = u(1) + u(2) + u(3)$ I(2) : = u(1)*u(2)*u(3)$

nambu (I, u, 3)$

du(1)/dt = u(1)*( - u(3) + u(2))

du(2)/dt = u(2)*(u(3) - u(1))

du(3)/dt = u(3)*( - u(2) + u(1))

R E F E R E N C E S

[1] Y. NAMBU: Phys. Rev. D, 7, 2405 (1973). [2] M. RAZAVY and F. J. KENNEDY: Can J. Phys., 52, 1532 (1974). [3] G. W. KENTWELL: Phys. Lett. A, 114, 55 (1986). [4] W.-H. STEEB: Problems in Mathematical Physics (Bibliographisches Institut, Mannheim,

1990). [5] W.-H. STEEB and A. J. VAN WONDER: Phys. Scr., 38, 782 (1988). [6] W.-H. STEEB and N. EULER: Prog. Theor. Phys., 80, 607 (1988). [7] W.-H. STEEB and N. EULER: Nonlinear Evolution Equations and PainlevO Test (World

' Scientific, Singapore, 1988), p. 114. [8] P. PECHUKAS: Phys. Rev. Lett., 51, 943 (1983). [9] T. YUKAWA: Phys. Rev. Lea., 54, 1883 (1985).

[10] T. YUKhWA: Phys. Rev. Lett. A, 116, 227 (1986). [11] W.-H. STEEB and J. A. Louw: Chaos and Quantum Chaos (World Scientific, Singapore,

1986), p. 122. [12] K. N.CK.~IURA and M. LAKSItMANAN: Phys. Rev. Left., 57, 1661 (1986). [13] W.-H. STEEB and A. J. VAN TONDER: Z. Naturforsch., 42a, 819 (1987). [14] W.-H. STEEB and J. A. LOUW: J. Phys. Soc. Jpn., 56, 3082 (1987). [15] W.-H. STEEB, J. VAN TONDER, C. M. VILLET and S. J. M. BRITS: Found. Phys. Left., 1, 147

(1988). [16] W.-H. STEEB, A. J. VAN TONDER, L. LOUW and S. J. M. BRITS: Hel. Phys. Acta, 61, 979

(1988). [17] W.-H. STEEB: in Finite Dimensional Integrable Nonlinear Dynamical Systems, edited by

P. LEACH and W.-H STEEB (World Scientific, Singapore, 1988). [18] W.-H. STEEB: J. Phys. A, 15, L389 (1982). [19] M. KU~: J. Phys. A, 16, L689 (1983).

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A note on Nambu mechanics