Linear r-matrix algebra for systems separable in parabolic coordinates

Physics LettersA 180 (1993) 208—214 North-Holland

PHYSICS LETTERS A

Linear r-matrix algebra for systems separable in parabolic coordinates J.C. Eilbeck a b C

v.z. Enol’skii a,b

V.B. Kuznetsov

C

and D.V. Leykin

Department of Mathematics, Heriot- Watt University, Riccarton, Edinburgh EHJ4 4AS, UK Department ofTheoretical Physics, Institute of Metal Physics, Vernadsky Street 36, 252142 Kiev-680, Ukraine Department ofMathematical and Computational Physics, Institute of Physics, University of Saint Petersburg, Saint Petersburg 198904, Russian Federation

Received 21 January 1993; revised manuscript received 7 June 1993; accepted for publication 9 July 1993 Communicated by A.P. Fordy

We consider a hierarchy of many particle systems on the line with polynomial potentials separable in parabolic coordinates. Using the Lax representation, written in terms of 2x 2 matrices for the whole hierarchy, we construct the associated linear r-matrix algebra with the r-matrix dependent on the dynamical variables. A dynamical Yang—Baxter equation is discussed.

1. Introduction It is known that it is possible to provide a 2 x 2 Lax operator satisfying the standard linear r-matrix algebra for a Hamiltonian system of natural form where the potential ~//is of second degree in the coordinates (see e.g. refs. [1—5]).In recent years the study of completely integrable systems admitting a classical r-matrix Poisson structure with the r-matrix dependent on dynamical variables has attracted some attention [6—8]. It is remarkable that the celebrated Calogero—Moser system, whose complete integrability was shown a number of years ago (cf. ref. [9]), was found only recently to possess a classical r-matrix of dynamical type [10]. In this paper we study another example of a dynamical r-matrix structure. The potentials described are known as a parabolic family of potentials since they are permutationally symmetric potentials separable in generalized parabolic coordinates. They were introduced in refs. [11, 121 and their connection with restricted coupled KdV flows was studied in ref. [13] (see also ref. [14] for the isomorphism with KdY). However, the r-matrix Poisson structure associated with higher degree potentials has not been discussed. The systems represent a generalization of a known hierarchy of two-particle systems with polynomial potentials separable in generalized (n-dimensional) parabolic coordinates (cf. refs. [15,16]). The polynomial second order spectral problem associated with the Lax representation given below has been studied in a number of papers (cf. refs. [17,181). Here we reproduce all these results in the framework of a 2 x 2 Lax representation, which is a direct generalization of that given by ref. [141; this being an effective technique to describe explicitly our class of integrable systems and to investigate their classical Poisson structure. We also note that the Lax representation can be extracted from the results [191 by some limiting procedure. We consider the system within the method of separation of variables [3,5,20] which allows us to develop the classical theta-functional integration theory and to consider the associated quantum problem. The last problem is reduced to a set of multiparameter spectral problems which are a confluent form of ordinary differential equations of Fuchsian type. 208

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The central result of this Letter is the description ofthe Poisson structureof the system by a dynamical linear r-matrix algebra.

2. Lax representation Consider the hierarchy of the Hamiltonian systems of n + 1 particles defined by the Hamiltonians 1

HN(pl

a+

p~~ 1q1

(2.1)

~ 1’N

fix the member of hierarchy by the recurrence relation

where the potentials ~

(2.2)

(—l)’qNfA~2,

1=11=2

with the first trivial potentials given by ~=—2, d/1 —~7~ q~. The general 1’N at N> 2 can be written1___2q~÷1_2B, in the form of ~1i2=—2q~~1 a (N— 2) x (N— 2) determinant expression for the potentials ~ IN

g

IN-I fN2

1

g0

g1

—l 0

g1

g0

—1

°i/N=(—1)det

0 0

gNS gN-6

g1 .

.f~ f~

...

gN7

. .

(2.3)

,

0

1 0

...

g_1

1

with a

g..1=q~÷1—B, ~

2

m=0

fk= ~

1=1

and

~Wlgk,.2,

k=3

N,

(2.4)

1=0

%, ~W1,~ as given above. The first nontrivial potentials are (2.5)

~4_(~

8

q~) 11

—~q~÷1~ j~1

q~—2q~~1+ ~ Aiq~(qn+i_~Ai)+B(4q~+i_2Bq~÷i+qn÷i ~ ~).

(2.6)

The potential (2.5) is a many particle generalization of one of the known integrable cases of the Hénon—Heiles system for which n= 1 (cf. refs. [14,211). Analogously the potential (2.6) is a many particle generalization of the system “(1: 12: 16)”, known to be separable in parabolic coordinates (cf. refs. [15,161). The system with the potential (2.5) possesses two remarkable reductions: (a) at q~~1 =const, it reduces to the Neumann system which describes the motion of a particle over a sphere in the field of a second order potential and (b) at q~+ = ~ q~ it reduces to an anisotropic oscillator with a fourth order potential (see e.g. ref. [16]). The same reduction can be carried out for other members of the hierarchy. We look for a Lax representation in the form LN(z)= [MN(z), LN(z)], where fV(z) U(z)\ LN(z)__~w() _V(z))’

/ 0

MN(Z)=~Q()

l’\ o)’

(2.7)

with 209

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U(z)=4z—4q~÷

(2.8)

1+4B—~—~-,

(2.9) (2.10)

WN(z)=—~C(z)+U(z)QN(z),

where QN(z) is a polynomial of degree N— 2. The ansatz for the functions U(z), V(z), W(z) is a generalization of the corresponding ansatz constructed by Newell et al. [14] to give the Lax representation for the integrable Hénon—Heiles system. Here we introduce additional degrees of freedom n> 1, and higher degrees of the polynomial QN are considered. See also ref. [3] for the link to the su(l, l)-Gaudin magnet which corresponds to a free n-dimensional particle separable in parabolic coordinates. It is possible to show that the Lax representation (2.7) is valid for all the hierarchy of Hamiltonian systems (2.1), (2.3) with the polynomial QN( z) and the function WN( z) given by the recurrence relations l8~WN.I(qI,...,qfl±I)

(2.11)

L

(2.12) (2.13)

WN(z)=WJ~(z)+W(z), 2~1’N.I,

N=2

W~(z)=zW~.1 — W(z)=~

(2.14)

1~~-,

where ~‘N—I is the potential fixing the (N— 1 )th member of the hierarchy and (2.14) to obtain a formula for the functions WN(z) and QN(Z), 2—— 21 N-3 ~ k=0

N-k-I

QN(z)=z~

~

~‘N_k_IZ+

k=O

(2.15)

zk,

n

N—2

WN(z)=C4Z—2

Q2= 1. We can easily solve (2.11)—

2

(2.16)

~___1__, ,1

z+A 1

For example, for the first nontrivial cases we have (2.17)

Q3(z)=z+2q~+1, 2+4zq~+ W3(z)=4z

a

(2.18)

1+4Bz+4q~+1+~ q~+~ k=1

for the many particle Hénon—Heiles system and (2.19) W4(z)4z+4Bz+4zq~~1+4zq~+1+z~ q~+4q~~1+2q~+1 ~ q~— ~ A1q~—4Bq~÷1+ ~ 1=1

i= I

—i-—

1=1 Z+

(2.20) for the system with N= 4. The Lax representation yields the hyperelliptic curve C~= (w, z), Det(L~(z)~wfl=0 210

(2.21)

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generating the integrals of motion HN, ~

i= 1,

aF~(j) 2= 16zA~_2(z+B)2+8HN+~=I ~

N=3

w

...,

6 September 1993

n. We have from (2.21) and (2.8)(2.10) (2.22)

where 12

N—I

F~P=2q~~ (—1)’ A~4?/~_j+4pfl+Ipjqj—p,2(Aj+4q~+j—4B)+ ~

(2.23)

i=l,...,n

,

kmA,n

j=I

k

with l 0=q1p~—q~p1, i,j=l, ishing Poisson brackets.

...,

n. The integrals of motion HN, F$~),i=l

n, are independent and have van-

3. Separation of variables To define the separation variables, i.e. the canonically conjugated variables ir1, functions cb~such that

~,

i= 1

n+ 1, and n+ 1

where HN, ~ are the integrals of motion in the involution, we use the scheme (cf. refs. [3,22,23]) according to which these variables are defined in terms of the Lax matrix as i~= V(u1), UCu1) =0. The set of zeros ~ j=1 n+ 1, of the function U(z) in the Lax representation (2.7) defines the parabolic coordinates given by the formulae a

n+l

+A

(

~

1m Ak) ~ A,+B+ ~ ji q2=_4~ ml m=l i=I j1 llkm (‘ The momentum lrm canonically conjugated to lUm has the form q~+1=

\Pj

fl

~m~(lti)’l~m

~

I

I~n

i=I

m=1

,

n.

(3.1)

n+l.

(3.2)

a+I

The separation equations are of the form ~r~=w2Cu 1),

i= 1

n+ 1

(3.3)

,

2(z) is given by (2.22).

where functionequations w The the separation have two uses to integrate the equations of motion in terms of theta functions and to quantize the systems. We mention here that canonical quantization in the space of separation variables leads to the following multiparameter spectral problem for the wave function of the system P= H5’~ 1!1~, —

1

~

A~÷

(3.4)

1)_—0,

withj= 1 intervals

—

n+ 1 and the spectral parameters .~l “permitted zones”.

2~+I. Problem

(3.4) has to be solved on n+ 1 different

4. r-matrix representation It follows from the results of section 2 and ref. [61 that the system admits an r-matrix algebra. Let { } be the standard Poisson bracket and { ~ } be the standard Poisson bracket in the product of two linear spaces ,

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V2®V2, Then the classical Poisson structure for the hierarchy of dynamical systems described by the Lax operator L(z) with the entries (2.8)—(2.l0) can be written in the form {L(x)®L~(v)}= [r(x—y), L~(x) +L~(y)] + [s~(x,

y), L~~(x) —L~(y) 1’

(4.1)

where L~”~(x)=I®L ~(x), L~N)=L~~(x)®I, I is the 2x2 unit matrix and the matrices r(x—y) and s~(x, y) are given by the formulae 1000 r(x—y)= ~P,

P=

(4.2)

,

0001 s~(x,y)=2aN(x,y)S,

S=a_®~,

a_=(?

(4,3)

~),

with QN(x)—QN(y) x—y

XI_YN_I x—y

—

lN~3XN_k_I_yN_k_I k=0 X—Y

2

8~1~N_k_l,

(44)

The equality (4.1) contains all the information concerning the hierarchy of dynamical systems, in particular one can obtain a simple proof of the involutivity of the integrals of motion. The representation (2.7) can also be derived from (4.1) (cf. refs. [4,6]). We write relation (4. 1) in the form {L~(x)®L~(y)} = [d~(x,

y), L~(x)]

—

[d~

(x,

y), L~(y)],

(45)

with d~/”~=r, 1+s,~, dj s~.N)r0at i
[d1~(x,

y), d~’~(x, z)] + [d~~(x,

y), d~~(y,z) ] + [d~’~(z, y), d~’~(x, z)]

+{L~(y)®d~(x, z)}—{L~(z)®d~(x, y)}+ [c(x, y, z), L~(y)—L~(z)] =0

(4.6)

and cyclic permutations. In this context t’~(x)®I®I, L~(y)=I®Lt’~(y)®I, L~(z) =I®I®L~(z), L~(x)=L and c(x, y, z) is some matrix dependent on the dynamical variables. If we denote S 12 = S®!, S23 =I®S, S13 = a.. ®I®o, where the matrix S is defined in (4.3), then the following equality is valid for each member of the hierarchy of dynamical systems, {L~(y)®s~(x,z)}—{L~(z)®s~(x,y)}

=2$N(x, y, z) [F23,S13 +S12]

—

y, z) [s,L~N)(y) _L~N)(z)]

(4.7)

with cyclic permutations. In (4.7) the matrix s=a_®a_®a_ and PN(x, y, z) =

QN(X)

(y—z) + QN(Y) (z—x) +QN(z) (x—y) (x—y)(y—z)(z--x)

(4.8)

Therefore the content of this Letter can be interpreted as finding a solution for the dynamical Yang—Baxter equation (4.6) which describes the evolution of the hierarchy of a many particle one-dimensional Hamiltonian system separable in parabolic coordinates. 212

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Details of all these results will be published elsewhere [24]. In conclusion we remark that the dependence of the r-matrix on dynamical variables can in principle be avoided by embedding the system into a completely integrable system with more degrees of freedom. This systern can be described by the standard linear r-matrix algebra (4.1) with s=0. For example, for the case N=3, set the functions U(z), V(z), l’i’(z), to be the entries of the Lax operator r(z), + l6~2q~÷ U(z)=U(z), V(z)= V(z)—8~z—8B1~, ~i’(z)= W(z)— l6~12x+Q—81?l3p~÷~ 1, (4.9) where the functions U(z), V(z), W(z) are given by formulae (2.8)—(2.l0) at N=3 and .~, .~ are new canonically conjugated coordinates with respect to the standard Poisson bracket. It is easy to see that the corresponding system satisfies algebra (4.1) with the matrix s = 0. The corresponding enlarged dynamical system has integrals of motion ~, ~ .~,i= 1, n, given by the formulae ...,

~

(4.10) (4.11)

~

+~

(~_~n+i~

~

p~q,— ~A1p1q1+~p~~1

~

q~),

(4.12)

where in (4.11), (4.12) H3 and FS’~are the integrals of motions of the many particle Hénon—Heiles system calculated by formula (2.23). We can see that at ~ 0, f~= 0 the system reduces to the many-particle Hénon— Heiles system. We expect that the analogous apparatus can also be developed for systems separable in elliptic coordinates, in which case we have to change the ansatz for U(z) to U(z)=l+~7=I q~/(z+A1).

Acknowledgement The authors are grateful to E.K. Sklyanin and M.A. Semenov-Tyan-Shanskii for valuable discussions. We also would like to acknowledge the EC for funding under the Science programme SCI-0229-C89-100079/JU1. One of us (JCE) is grateful to the NATO Special Programme Panel on Chaos, Order and Patterns for support for a collaborative programme, and to the SERC for research funding under the Nonlinear System Initiative.

References [1] L.D. Faddeev, in: Les Houches lectures, eds. J.B. Zuber and R. Stora (North-Holland, Amsterdam, 1984) pp. 719—756. [2] M. Gaudin, La fonction d’onde de Bethe (Masson, Paris, 1983). [3] YB. Kuznetsov, J. Math. Phys. 33 (1992) 3240. [4] M. Semenov-Tian-Shanskii, Funct. Anal. App!. 17 (1983) 259. [5]E.K. Sklyanin, Soy. Math. 47 (1989) 2473. [610. Babelon and CM. Viallet, Phys. Lett. B 237 (1990) 411. [7] L. Freidel and J.M. Maillet, Phys. Lett. B 263 (1991) 403. [8] J.M. Maillet, Phys. Lett. B 162 (1985) 137.

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191 MA. Olshanetski and A.M. Perelomov, Phys. Rep. 71(1981) 313. [10] J. Avan and M. Talon, Classical r-matrix structure of Calogero model, preprint (1992). [11] S. Wojciechowski, Review of recent results on integrability of natural Hamiltonian systems, in: Proc. SMS systémes dynamiques non Iineaires: integrabilité et comportament qualitatif, ed. P. Winternitz (Press Université de Montreal, Montreal, 1986). [12] S. Wojciechowski, Three families of integrable one-particle potentials, Dipartimento di Fisica, Università di Roma, preprint 440 (1985). [13] S. Rauch-Wojciechowski, Phys. Lett. A 160 (199!) 241. [14]A.C. Newell, M. Tabor and YB. Zeng, Physica D29 (1987) 349. [l5lJ.Hietarinta,Phys.Rep. 147 (1987) 87. [161 A.M. Perelomov, Integrable system of classical mechanics and Lie algebras (Birkhauser, Basel, 1991). [171 A.P. Fordy and M. Antonowicz, Common. Math. Phys. 124 (1989) 465. [18]Y.ZengandY.Li,J.Math.Phys. 31(1990)2835. [19] M. Antonowicz and S. Rauch-Wojciechowski, J. Math. Phys. 33 (1992) 2115. [20] E.G. Kalnins, W. Miller Jr. and P. Winternitz, SIAM J. App!. Math. 30 (1976) 630. [211 A.P. Fordy, Physica D 52 (1991) 204. [22] E.K. Sklyanin,J. Soy. Math. 31(1985)3417. [23] E.K. Sklyanin, Commun. Math. Phys. 142 (1992), to be published. [24] J.C. Eilbeck, V.Z. Enol’skii, V.B. Kuznetsov and D.V. Leykin, Classical Poisson structure for a hierarchy of one-dimensional particle systems separable in parabolic coordinates, preprint (1993).

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