On the quantization of the Kowalevskaya top

Volume 101A, number 2

PHYSICS LETTERS

12 March 1984

ON THE QUANTIZATION OF THE KOWALEVSKAYA TOP A. RAMANI Centre de Physique Th~orique, Eeole Polytechnique, F 91128 Palaiseau, France and

B. GRAMMATICOS and B. DORIZZI Dkpartement de Math~matiques (M. T.1], Centre National d'Etudes des T~l~communications, F 92131 lssy les Moulineaux, France Received 3 January 1984

The equations of motion of a heavy top can be integrated for three different combinations of the parameters of the system. Historically, the discovery of these three integrable cases is attributed to Euler, Lagrange and Kowalevskaya, respectively. While the quantization of the first two cases can be performed in a straightforward way, the quantum integrability of the Kowalevskaya top is far from trivial. We show here how one can recover quantum integrability for this case as well.

The equations o f motion of a rigid b o d y around a fixed point constitute one of the oldest and commonest examples of generally non-integrable systems. Moreover the study of the motion o f a top is the first case where the singularity structure o f the solutions o f the equations o f m o t i o n in the complex-time plane was used as a detector of integrability. The equations of motion o f an asymmetric top around a fixed point under the influence o f its own weight are traditionally written as Ad~21/dt=(B-

C)~2~ 3 +Zot3- y07,

Cd~23/dt = (.4 - B) ~21~22 + y ~ - x#0,

d T / d t = f22a - f21 t3,

-1 2 + Bf22 + C f 2 2 ) + XoOt + y 0 /3 + z 0 7 , E--~(A~21

(2)

and the vertical projection o f the angular m o m e n t u m vector: m z = A ~ l U + B ~ 2 / 3 + C~23 7.

(3)

However, generally the system is not integrable as no third integral exists. Two cases o f integrability were known before the end o f the 19th century. The case o f the free top is due to Euler, where we have x 0 = Y0 = z0 i.e. the fixed point of the top coincides with its center of mass. The third integral is just the amplitude o f the angular momentum:

Bd~22/dt = (C - A ) f23~21 + x 0 7 - Zoa ,

d~/dt = ~23fl - ~227 ,

its physical content, although this cannot be assessed at a glance from the equations o f motion. Two integrals o f motion exist always, namely, the energy:

d/3/dt = ~217 - ~ 3 a ,

(1)

where ~21, ~ 2 , f23 are the components of the angular velocity; a,/3, 3' the direction cosines (a 2 + 132 + 72 = 1);A, B, C the moments o f inertia o f the top and x0, Y0, z0 constants related to the position o f the fixed point with respect to the center o f mass o f the top. That the system is hamiltonian is obvious from 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

A2~22 + B 2 ~ 2 + C 2 ~ 2 = M 2.

(4)

The second case, due to Lagrange, is the case of the symmetric top A = B and x 0 = Y0 = 0. The third integral is the third projection of the angular m o m e n t u m (with respect to the body-fLxed axes). M 3 = C~23 .

(5) 69

In 1888, Kowalevskaya [1] analyzed the problem from the point of view of the analytic structure of the solutions in the complex time plane. Applying the ideas developed by Fuchs for first-order nonlinear differential equations, she was able to identify the parameter values for which the system's movable (i.e. initial condition dependent) singularities are simply poles. Apart from the Euler and Lagrange cases, she obtained a third parameter-combination (which came to be known as the Kowalevskaya top): A=B=2C

and

z0 = 0 ,

with pole-like singularities. In this case, she was able to exhibit the third integral ( a l2 -- a 2 - x0a) 2 + (2a1~2 2 - x0/3) 2 = K

(6)

(where Y0 has been taken equal to zero, without loss of generality as A = B), and she completed the integration in terms of elliptic functions. No other cases of integrability for the top motion are known to date and the system generally presents natural-boundary-type singularities in the complex-time plane [2]. The singularity analysis, associated to the name of Painlev~ who, together with Gambier, applied it systematically to second-order differential equations, has been proven a most powerful tool as a detector of integrability. The recent applications in the domain of integrable dynamical systems described by ordinary and partial differential equations are numerous. New integrable systems have thus been discovered due to the Painlev6 approach [3], whereas the singularity structure of well-known non-integrable systems such as the classical three-body problem [4] or the Van der Pol oscillator [5,6] was proven to be definitely not pole-like. In the present paper, we are going to address the question of the quantum integrability of the Kowalevskaya top motion. In the hamiltonian formulation, in terms of the Euler angles 0, ~, ~, one has: 1 2 + I .~pqj _ 2 - ~pCpq~ 1 a 2 H = (~p¢ cos 0 ) / s i n 2 0 + l"~P,2 + "~Po

+x 0sinCsin0,

A =xp ,

(8)

the corresponding Weyl operator is ~t = px2,~x = .2l)2 - ihf~.

Different ordering rules would lead to: A ' =2f~2 + XihlJx "

However the operator ,~" = f~x21) x + h 2 f ( x )

cannot be recovered by any ordering rule although its classical limit is still A. Given two c-number functions A, B in involution i.e. {.4, B}p = 0 where { , }p is the Poisson bracket, is it guaranteed that their Weyl operators .4,/~ commute? The transform of the commutator [A, B] (divided by ih) is the c-number function called the Moyal bracket: (A, B} M = (2/h) A (sin ½h•) B 1 5 = A I ~ B - - ~ h 2 A I ~ 3 B + 7-~6AX B + ...,

(9)

where

i=1

- 2x 0 {-14[(p¢ - pc cos 0)2/sin20 - p2] sin ~ sin 0 + -~cos ~Po(P¢~ - PC cos 0)},

where we have taken C = 1 and Y0 = 0 without loss of generality. What we mean by quantization of the Kowalevskaya top is the construction of three commuting operators/t, M,/(, the classical limit of which is respectively H, M, K. The difficulty is that a given classical quantity does not uniquely define a quantum operator because of the non-commutativity. However there exist ordering rules which establish one-to-one correspondence between operators and c-number functions. In the following we will use the Weyl rule [7] that associates to any c-number function A ( x i , Pxi) an operator A, while its converse is the Wigner transform. The Weyl rule is not a unique ordering. However the freedom allowed by different prescriptions is not very wide. Consider for instance the classical quantity:

N

M =p~,

K = ~6 [p2 + (p~ _ PC cos 0)2/sin20] +x02 sin20

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12 March 1984

PHYSICS LETTERS

Volume 101A, number 2

(7)

~xi~Pi

~Pi ~xi "

Note that the Poisson bracket is just the first term of this expansion. Therefore, in general, the vanishing of the Poisson bracket does not guarantee the vanishing

Volume 101A, number 2

PHYSICS LETTERS

of the Moyal bracket, i.e. of the commutator of the Weyl operators. I f A is now the hamiltonian H and B the second constant K of an integrable, classical 2-D hamiltonian system, we see that [/),/~] need not vanish. Therefore for the system described by Weyl-quantized /~, /~ does not ensure integrability in general, although there are some cases where H and K do commute. When they do not, one first tries to obtain commuting operators/t' and/~' through different quantization rules from H and K. There are few cases where this works. Most often however, this is not sufficient [8]. Still, one can modify K through the addition of purely "quantum" terms i.e. terms explicitly depending on h and therefore disappearing in the classical limit but which do not stem from a simple reordering as described above in eq. (8). There are numerous cases where one can find such a modified constant that ensures the integrability of the quantum system described by the Weyl transform of the original hamiltonian. For the case of the Kowalevskaya top, we must find three quantum operators. The choice o f / l l = pC is quite obvious and will never lead to any difficulty as no sensible/) or/~ will ever depend explicitly on ¢~ and commutators with pC will trivially vanish. The only remaining problem is that of the commutation o f / ) w i t h K. The difficulty stems from the nonvanishing of the Moyal bracket of H and K. One can easily check that no reordering will solve this difficulty. What is more, we have checked that no new quantum term added to K will ensure the vanishing of the Moyal bracket. Reordering/t in addition will not help either. However, in the same way as one can add to K purely quantum terms that do not stem from reordering, one can envision to do the same to H. And, indeed, this has been done already [ 9 - 1 1 ] . We have tried the following ansatz: H'=H 0+h2H2'

K'=KO+h2K2+h4K4.

(11)

Due to the special form of H and K, the only nonvanishing terms in (H, K) are of order h 2 and h 4 and one can easily convince oneself that the extra terms in (11) are the only ones one needs to consider. We have computed (H', K 3 using the formal computer language AMP [12] and this led to a set of dif-

12 March 1984

ferential equations for H2, K 2 and K 4. This set was highly overdetermined but it turned out that these equations are indeed compatible and their solution reads: H 2 = - ( 4 sin20) -1 , K 2 = - ~ 7 (P~ - p c cos 0) 2 (sin20) -1 + (2p 2 - p¢p¢ cos 0 - pC2 1 i p 02) ( 4 sin20) - 1, K 4 = 13(16 sin40) -1 - (2 sin20) -1.

(12)

These corrective terms ensure the quantum integrability, in the Weyl representation, o f the system described by H', the second constant being K'. At the classical limit, one recovers the Kowalevskaya top. However, the physical interpretation of this sytem for finite h is not clear. Note that this was also a problem for the other systems treated in refs. [9-11]. In conclusion, let us state that for classically hategrable systems, one can find a quantum integrable analog, either by a proper choice of the ordering rules or by adding quantal corrections to the classical constant of motions, and, if need be, to the hamiltonian itself. The Kowalevskaya top falls in the latter class. There does not exist to date (at least to our knowledge) any counterexample to the above statement. The authors acknowledge many interesting discus° sions with J. Hietarinta. [1] S. Kowalevskaya, Acta Math. 14 (1890) 81. [2] Y.F. Chang, J.M. Greene, M. Tabor and J. Weiss, Physica 8D (1983) 183. [3] A. Ramani, B. Dorizzi and B. Grammaticos, Phys. Rev. Lett. 49 (1982) 1539. [4] H. Yoshida, Necessary condition for the existence of algebraic first integrals I and II, submitted to Celest. Mech. (1983). [5] A. Ramani, B. Grammaticos and B. Dorizzi, Phys. Lett. 97A (1983) 87. [6] T. Bountis, Phys. Lett. 97A (1983) 85. [7] J. Shewell, Am. J. Phys. 27 (1959) 16. [8] J. Hietarinta, Phys. Lett. 96A (1983) 273. [9] J. Hietarinta, preprint 1983, Classical versus quantum integrability. [10] J. Hietarinta, Preprint Turku-FTL-R44. [11] J. Hietarinta, B. Dorizzi, B. Grammaticos and A. Ramani, C.R. Acad. Sci. (1983), to be published. [12] J.M. Drouffe, AMP User's manual (1981). 71