olshanetsky m.a., perelomov a.m. classical integrable finite-dimensional systems related to lie algebras

Symmetric spaces 390 12.-Complete integrability of the abstract Hamiltonian systems 355 References 398 During the last few years many dynamical systems have been identified, that are

CLASSICAL INTEGRABLE FINITE-DIMENSIONAL SYSTEMS RELATED

TO LIE ALGEBRAS

M.A OLSHANETSKY and A.M PERELOMOV

Institute of Theoretical and Experimental Physics, 117259 Moscow, USSR

0 Introduction 315 13 Explicit formulae and moment map for the abstract Hamil-

2 Completely integrable Hamiltonian systems 320 14 Explicit integration of the equations of motion for the

3 Systems with additional integrals of motion 321 systems of type IV and VI’ (periodic Toda lattice) 364

4 Proof of complete integrability of the systems of section 3 324 14.1 The systems of type VI’ An_1 364

5 Explicit integration of the equations of motion for potentials 14.2 The systems of type IV An-1 368

6 Explicit integration of the equations of motion for potentials 15.1 Motion of the poles of nonlinear partial differential

of type II and III 331 equations and related many-body problems 371

7 Integration of the equations of motion for a system with two 15.2 Motion of the zeros of the linear evolution equa- types of particles 333 tions and related integrable many-body problems 374

8 Explicit integration of the equations of motion for the Toda 15.3 Rotation of a many-dimensional rigid body around a

(Methods of orbits) 337 Appendix A Solution to the functional equation (3.9) 382

10 Equilibrium configurations and small oscillations of some Appendix B Groups generated by reflections and root sys-

11 Abstract Hamiltonian systems related to root systems 352 Appendix C Symmetric spaces 390 12.-Complete integrability of the abstract Hamiltonian systems 355 References 398

During the last few years many dynamical systems have been identified, that are completely integrable or even such to allow an explicit solution

of the equations of motion Some of these systems have the form of classical one-dimensional many-body problems with pair interactions; others are

more general All of them are related to Lie algebras, and in all known cases the > property of f integrability results from the presence of higher

it contains some new results both of physical and mathematical interest

The main focus is on the one-dimensional models of n particles interacting pairwise via potentials V(q)= g7v(q) of the following 5 types:

01(q) = q4“ 0n(q) = a` “ sinh“(44), nm(q) = a“/sin2(a4), oyy = a“ P(aq), vv(q) = q “+ w*q* Here P(q) is the Weierstrass function, so that the first 3 cases are fe merely subcases of the fourth The system characterized d by the Toda nearest-neighbor potential, Lý hưng đ;+)Ì, is moreover

CLASSICAL INTEGRABLE FINITE- DIMENSIONAL SYSTEMS RELATED

TO LIE ALGEBRAS

M.A OLSHANETSKY and A.M PERELOMOV Institute of Theoretical and Experimental Physics, 117259 Moscow, USSR

0 Introduction

Up to now only a small number of systems, both classical and quantum systems, with two or more

degrees of freedom, which are completely integrable or, what is more, for which the explicit solution is

known, were known

During the last few years, however, a great number of such systems have been found All of them are related to Lie algebras and in all the cases known their integrability results from the presence of higher

(hidden) symmetries In specific cases such systems describe one-dimensional n-body problems with pair

interactions recently investigated by many authors

This paper is a review which presents from a general and universal viewpoint results obtained during the last few years Besides, it contains some new results both of physical and mathematical type

The interest for models of n interacting particles with known exact solutions, is associated with the fact that, while both classical and quantum problems with three and more particles with a realistic interaction acting between them (Coulomb or nuclear) do not allow one to obtain any exact solution, it can be expected that some quantitative properties of these models can be expected to be maintained in

the real case Besides, these models may be useful to estimate the accuracy of different approximate

In three-dimensional space the exact solution is known only for a system of n-particles interacting

through oscillator forces

wr Uf, , r„)= 2 Vự,—r,), — Vf{r)= 5

Sipews ATT © rs ae r9 Ø AAAs ears ly 2 A AA ov aN uae * r Py ry r Arts eis =

s > oa! LJU 0 WU 1© đ › A O LJ LJ O LJ Jal 0

of them moving independently in an overall oscillator well So it differs in an inessential way only fro

larger class of potentials

In this paper we will consider in detail the one-dimensional classical models of n particles interacting

obtained in 1969-1975 in the quantum case These are results on the form of the wave functions, on the

spectrum and the character of scattering for the systems with potentials of type I and V (Calogero

model [19, 20, 86, 47]) and with potentials of type III (Sutherland model [102, 103]) These are

316 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

complemented by refs [110, 35] ‘which studied a three-body system with the two- and three-body interations of the type

U(41, 92, 93) = 81 iD Đ(4 — 4)+ g? Pe „ 9Ö: ~ 4 — 4k) (0.6)

itk

with the function v(q) of the forms I and V Moreover, a number of classical completely integrable systems were known, i.e systems for which it is possible in principle to integrate the equations of

motion, although in practice this possibility can be realized only in few cases

Let us list some of these systems:

(1) The motion in a central potential (Newton)

U()= U(qÌ)

(2) The motion in the Coulomb or gravitational field of two fixed centers (Euler) U(q)= ailg — ai[ ` + azÌq — aa| `

(3) The free motion of a point on the surface of a three-axes ellipsoid (Jacobi [52, 54])

(4) The motion of a masspoint on a sphere under the influence of a linear force (Neumann [79])

(5) The one-dimensional motion of three particles with pair interaction of the form U(q)=

Dick Bje(Gi — Qe) ? (Jacobi [53, 72])

For four and five particles see refs 186, 47

Kruskal and Miura [49] of a new "integration method for nonlinear equations —the so 5 called i inverse

scattering method or isospectral deformation method This method was first applied to nonlinear

evolution equations: the Korteweg-de Vries equation, the nonlinear Schrédinger equation and the sine-Gordon equation

The application of this technique to many-body problems [69, 43, 44] gave the possibility to establish the complete integrability of classical systems of the type I-V [73, 36, 21, 80, 87, 88, 109]; of the system with the potential U(q) = Un(q)+ g3 =; exp(2aq;) [3]; of the system with the potential energy U(q) =

Aq; + (2 q7y¥ (H Grosse; [32])*; of the Toda lattices [105, 106, 69, 43, 44, 40], i.e the systems with

_—_— P0tential energy of the form

U(a)= Ð, ø7 exp(- a(4 ~ 4+1) (0.7)

multi-dimensional spheres and ellipsoids [75, 76], n-dimensional rigid body motion [70] and its generalizations (46, 95, 96] (See in this respect the reviews [26, 75, 76, 92].)

In our paper [80] it was shown that the many-body systems under consideration possess a hidden symmetry and are a special case of a more wide class of systems connected with semisimple Lie

algebras In subsequent papers [81, 82] the reason for the complete integrability of classical systems of

such type was established In particular, it was shown that systems with potentials of type I-III are

related to free motion (motion along geodesic) in some symmetric spaces with more than n dimensions

Examples of symmetric spaces are the N-dimensional Euclidean space R™, the spheres S”, hyperboloids

H™, semisimple Lie groups This relationship enables one to find a natural generalization of the system

under consideration and to integrate completely the equations of motion using the free motion projection method

The essence of the method consists in considering the motion along a geodesic line in a symmetric

space of zero, negative or positive curvature, the dimension of the space being larger than the number

of degrees of freedom of the original system After being appropriately projected onto a space with less dimensions (n-dimensional space), one gets no longer a free motion but a motion in the potential field U(qì, , đ„)= g? >, 0(4„)

where v(q) is of the form I-III, and q, = (a, q), R= {a} is some system of vectors in n-dimensional space, the so-called root system

Let us illustrate this by simple examples when there is one degree of freedom

1 Let a particle of unit mass have a free motion with momentum p on a two-dimensional plane

X = (x1, X2) Then the radial motion is described by the Hamiltonian

3 For the free motion of a particle on the upper sheet of the two-sheet hyperboloid H?=

2_— y2_ y?_—y?2— e get

D ectio eine then def bythe muta ‘9 = cosh g

4 For the free motion of a particle on a one-sheet hyperboloid H? = {x|x?= x?— x?— x2 = — 1}, we

H = p*/2—* cosh *g, xo = sinh 4 (0.11)

318 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

5 For the free motion of a particle on a two-dimensional sphere S’ = {x|x? = x3 + x7+ x2= 1}, we

get the Hamiltonian

H=p°*/2+g”sin ”q, Xo = cosq (0.12)

6 Just as in case 3, consider now free motion on the upper sheet of a two-sheet hyperboloid but,

instead of the radial projection, consider the so-called horospherical projection

X = (Xo, X1, X2) > q = ln(xn) = In(xo + xì)

In this case we come to the motion described by the Hamiltonian

i.e we obtain the Hamiltonian for the 2-body Toda lattice It turns out that an analogous consideration

of more complicated symmetric spaces allows one to integrate effectively the equations of motion in the general n-body case as well (81, 82, 85, 83, 61]

Sections 1 to 8 and 10 are written starting at the level of a standard course in classical mechanics (see

for example [68]) Sections denoted by an asterisk are written at a higher mathematical level In particular sections 11 to 13 are based on the > theory of root t systems and symmetric spaces All the

Appendix C method of moment map) For the understanding of this section it is necessary to o be familiar with the original differential = geometric ideas related to classical mechanics [7]

The deviation of the solutions of Hamilton’s equations for systems with the Weierstrass potential and the periodic a lattice (0.7) is based on me of algebraic geometry We present in section 14 the final formulae for the solutions and omit the mathematical details The reader who is interested in the problem may find the details in the original papers and in a review [62] On the other

hand the theory of these systems as the theory of a rigid body is based on the consideration of the

1 General description Let p= (pi Dn) and q=(qi qn) be the momentum and coordinates vectors in n-dimensional

space We consider Hamiltonians of the form

and introduce six types of potentials All of them describe one-dimensional n-body problems For the

first five types all particles interact pairwise

and the function ø(£) may be one of the following form

rc

£, I a’sinh7aé, II v(f) =} a’ sin? aé€, II (1.3)

a’ P(aé), IV

The Weierstrass function P(é) = AE; w1, w2) is doubly periodic i in é in the complex plane C (w;, w2 are

WO halfperiod aS a S€CONnG OFGer Pole a e origin, whi DE iodically spread onto

whole complex plane In the limit in which one of the periods goes to infinity one gets, up to a constant,

the potentials of type [i or Hit if both periods go to infinity one obtains the potential of type 1 Thus the

systems of type IV are the most general type of systems Note also that if one replaces @ by ia in the

potential of type III, one gets the system of type II and, if one puts a = 0, one gets the system of type I

We denote by type VI the usual (nonperiodic) Toda lattice

and by type VI’ the periodic Toda lattice

_ 2 _ = | , | 1.6

320 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

It is worth noting that the type VI’ is a generalization of the type VI (g, = 0)

Consider now the configuration space for the systems I-VI Note that the potentials of type I-V, (1.3), become infinite at q, = g; (k#j) Thus the ordering of the particles in the course of the motion cannot change, and we put q, > q, if j >k The configuration space for the systems I, II and V in the c.m.s is therefore of the form

and for the systems III and IV it has up to periodicity, the form

£

where d is a period of V(Z)

The configuration spaces A and A, for n = 3 are respectively the interior of the angle 7/3, and the

regular triangle displayed in fig 1:

the particles are equal to 2aq, and the potential is equal to g*a’/r? where r is the distance between

particles along the circle Similarly for the system of type II we can consider the motion along a

hyperbola with potential g’a7/r’ instead of the motion along a straight line with potential

g’a* sinh”* a(q, - 4x)

The potentials VI and VI’ (1.5) are nonsingular So the corresponding configuration space is the

Here the dots stand for time differentiation Such a system is called completely integrable if there exist

variables Ij(p,q), ¢(p, q) of “‘action-angle” type defined globally over the whole phase space of this system Such variables need not always exist As a rule, the variables of action-angle type are of local character and cannot be defined in the whole phase space (see, e.g [7])

They have a simple time-dependence

A criterion for complete integrability of a dynamical system with n degrees of freedom is given by

the Liouville theorem (see e.g ref [7]):

If there are n functionally-independent global integrals of motion Ij(p, q) in involution, i.e such that the Poisson brackets of any two integrals are zero:

ol ) —_ ol aise i} <9

{I, l1}= > {ee ôq ôq; ap; ’ (2 3)

then the system is completely integrable

Thus, completely integrable systems possess at least n global integrals of motion and, consequently, a certain hidden symmetry

To find the form of a potential such that the systems under consideration possess integrals of motion,

let us use, following refs [73, 21, 36] the isospectral deformation method often called the Lax trick [67]

Namely, let us s try to find a pair of Hermitian n Xn matrices L and M (the s SO called L-M pair) whose

322 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

Therefore, the eigenvalues of the matrix L(¢) are time-independent or, equivalently, the matrix L(t)

is isospectrally deformed with time Thus the eigenvalues of the matrix L, or, equivalently, but more conveniently, the symmetric functions of the eigenvalues, namely

I, = z sp(L*) (3.4)

are integrals of motion

If in such a way one can find n independent integrals of motion and show that they are in involution, i.e that their Poisson brackets are zero, then the system under consideration is completely integrable

Let us realize this program for systems of type I-IV, 1.e characterized by the Hamiltonian

Substituting L and M into the Lax equation (3.2), and requiring this equation to be equivalent to

Hamilton’s equations we get an explicit expression for the function

A little modification of this method [86] allows one to treat also potentials of type V To this end one

considers the equation |

324 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

The matrix L is determined by the formula (3.6) where

For this it is sufficient to show that the integrals of motion i, (k = 2, 7) (3.4) are functionally

independent, and in involution: {J,, Jj} = 0

I, =~ > pj + terms of lower degree in the momenta (4.1)

* The integrals for the Toda lattice were obtained earlier without using the Lax method [50]

** The Lax equation for the matrices is of the form L = [L, M] (cp (3.2)).

So the functional independence of the quantities J, follows from that of the quantities S, = 27_, pj, that can be easily proved

The proof of involutivity of the integrals J, is a more complicated problem

As was shown by Moser [73] for systems of type I this follows immediately from the fact that as

t> +0, the distance between any two particles diverges, |q;(t)— q.(t)| > + Then

systems with interaction not of the pair type considered in [110, 35]

Let us consider the systems of section 3, with the function V(q) of types L-IV (1.3) Let L=

P +iX(g = 1) be an Hermitian n x n matrix constructed according to the formulae (3.6) where the function x(é) satisfies the functional equation (3.9) Let ¢ = (¢1, @,) and = (ứa, „) be eigen- vectors of the matrix L, corresponding to eigenvalues A and pw (A# pz)

This method of proving relation (4.3) is in its basic idea, close to that of ref [69] in which the

involutivity of the integrals of motion for the nonlinear Schrédinger equation and for the Toda lattice

was proved Note, first of all, that

326 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

where

On the other hand, we find from eq (4.2)

Prt, =i(A - py? > x(qu — 91) Re (4.8)

Substituting the expressions for o,y and ¢,, into eq (4.6) we get

{A, w}= iu >> RuRej{x'(Qj — Ge) X (Qe — 9) — (4G — đ) x (4 — 4i) (4.9)

Making use of the functional equation (3.9) we transform this equation into the form

A w}= 7 RaRulz(a- 4) 2 - a] x(a) a) (4.10)

Eq (4.10) contains two sums In the first one we first perform the summation over /, and in the

second one over j

We now use the relation

It can be readily seen that the expressions in the first and in the second brackets are antisymmetrical

Consequently, {A, ~}= 0 and the systems under consideration are completely integrable

A similar but more simple proof is valid for the Toda lattice [69] Another proof is given in [43, 44]

It is worth noting that we can consider another set of integrals, namely, the coefficients in the characteristic polynomial of the matrix L

J, = exp{- -2 >3 x!(q; — q+1) 2p, 2p 5} [> (4.14)

(VI) For the other integrals, the following recurrence formula holds:

1 n

in [109] the involutivity is proved using the form of these integrals

5 Explicit integration of the equations of motion for potentials V(q) of type I and V

In the preceding section it was shown that the systems under consideration are completely integrable for potentials of all five types However, the Liouville theorem from which this statement follows, does not provide a constructive integration procedure to solve the equations of motion; indeed this is generally a complicated problem

In this section we show how to integrate explicitly the equations of motion for the systems with potentials of types I and V using a new method ~ the so called projection method.* This is done following [81]

The idea of the method consists in the use of a space with a larger number of dimensions N= nỶ~ 1,

An ˆ matian take ne | Arr Va a & nteacrea we no a2 nsnrovnr

———————————and have the general sOitH—on

With the help of a unitary transformation U the Hermitian matrix x can be reduced to a diagonal form,

* A different approach was discussed in [10].

328 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

Q(t) = diag(qi - dn) (5.5)

and, without loss of generality, we may assume the values q; to be ordered (see (1.7)) as follows

Note that in the simplest case n=2, x=Zix; (0; are the Pauli matrices), Q = diag(—q, q) and

q = |x| so that transforming from x to Q in accord with (5.4) corresponds to a “spherical” projection

Let us now try to derive equations for the q,(t) and

L and M are Hermitian n Xx n matrices

Differentiating eq (5.6) with respect to í, we get

i.e the Lax equation

Thus, the pair of L and M matrices must satisfy eqs (5.7) and (5.9) As was shown in (1.3), for matrices L and M having the form (3.6) and (3.7), eq (5.9) is satisfied It can be easily seen that eq (5.7)

is also satisfied by a dire bstitution o 6) and into 7) for the ems of type I i.e for the case V(q)=q°’

cannot be arbitrary since (n — 1) of its eigenvalue coincide

As a matter of fact, the matrix {O, L] has the form (using dyadic notation)

[(O,.LJ=g'’Wv-], v=(1 0

Without loss of generality we can assume u(0) = I Then the matrices a and 5 in (5.3) are given in terms of the initial values by the formulae

where the matrix L is given by the formula (3.6)

So, we have obtained the final result: the coordinates q,(#)— the solutions of the equations of motion for the system of type I—are the eigenvalues of the matrix

Let us discuss now the scattering process The potential u(q) in (1.2) vanishes as |g; — g.| > ~, so that

q$()~p7t†+q7, t>+e (5.13)

Thus the scattering process is characterized by the canonical transformation from the variables (p; , qx)

to the variables (p7, qi)

But it can be easily seen that

Hence it follows that the values p; differ from the values p, only by a permutation

330 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

Using now the equality |

Q(t)= u7'(t) x(t) u(t) = Pt+i[M, Q]t+ u(t) bu(t)

it follows that

Q*=u"'(+~) bu(t~), Q*= SQ"'S”' (5.21)

Thus, relation (5.17) is proved

An alternate proof of relation (5.17) is given in ref [73]

The result (5.16) was first discovered in the quantum case by Marchioro (N = 3) [72] and Calogero (for arbitrary N) [20] The natural conjecture that it holds also in the classical case was first proved by Moser, who also proved (5.17) [73]

Relations (5.16) and (5.17) imply that the outcome of the scattering process in the problem under

consideration coincides with that resulting from the sequential scattering of separate pair interactions

But Sp[Q(t)] is a polynomial in g; of degree k invariant relative to permutations

Hence we get:

Corollary I The polynomial of degree k in q; invariant relative to permutations is a polynomial of degree

k in t (w = 0) or sin wt and cos wt (w# 0)

Note that explicit solution of the equations of motion for the systems of type I and V (see formulae (5.12) and (5.25)) allows one to establish a simple relationship between these solutions

Let q,(t) be the solution of the equations of motion for the system of type I (w = 0) Then from (5.12) and (5.25) it follows that the quantities

are the solution of the corresponding system of type V (œz 0) The reciprocal statement is of course also valid

Analogous relationship for systems of more general type are given in [90]

Note also that the quantities

have a simple time-dependence Namely, such a quantity is a polynomial of degree k= 2k, in ¢ if

œ = 0 or in cos wt and sin wt if #0 The Poisson brackets algebra for such quantities was studied in

[15]

To integrate analogously the equations of motion for the systems of type II v(€)= a’ sinh"? ag and

II ø(€) = a2 sin ? a£, one should note, first of all, that the set X° considered above — namely, the set of

Hermitian n Xn matrices with zero trace —1s the Riemann symmetric space of zero curvature with the

natural choice of metric, ds* = Sp(dx dx) (see Appendix)

A wo Habe › ry c c D aa er C Tr H FY D B wate manhìa =@ tì Ẻ C a e HCtC men

is s the homogeneous space X= = S(t CVSUG) In other words 2 an biter matrix x€X can be

Let x(t) be a curve in X” Then the matrices x *(t) #0) and 2) Mt) < can ‘be considered as

wo-vecto Plds © e proup G= ›€} aS al Ol, +8 ally speaking, v 0 Ũ

space X"; their halfsum, however, will be, as it can be easily seen, a vector field on the space X_ If now

332 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

Note that this equation can be obtained from that for the geodesic lines for the metric invariant relative to the action of the group G= SL(n, C): ds? = Sp(x"* dx x" dx) Indeed, from the condition

5 ds = 0 we get X—xx7'x = 0*

hence there immediately follows (6.1)

It is clear that the following curve is a geodesic line on X™

x(t) = b exp{2%:}b”, b € SL(n, C), A" = A, Sp W = 0

Let us represent now the Hermitian positive-determined matrix x(t) as x(t) = u(t) exp{2a Q@)} uˆˆ(0),

where u(t) is a unitary matrix

Calculating with the use of (6.4) the values xx~* and x"'X we get

ites = 2a u(t) L(t) u(t),

where the matrices L and M are given by the formulae (6.6) and (6.7)

Take now the matrices L and M as (3.6) and (3.7) where x(é) = a coth ag Then, as we know from section 2, the Lax equation is fulfilled Substituting L and M into eq (6.6) we see that this equation for

L and M is fulfilled as well

Thus, choosing the matrices L and M as (3.6) and (3.7) and x(é) = a coth aé, all the self consistency conditions are fulfilled, and we arrive at the final result: the quantities exp(2aq,(t)), where q,(t) are the solutions of the equation of motion for a system of type II, are the eigenvalues of the matrix

where

b=expaO0)) Q=diag(q: qn) (6.14) and the matrix X can be found from the equation

Note in conclusion that all the results for the system of type III are obtained from the corresponding

results for the system II by the replacement a — ia In the limit a > 0 one reobtains of course the results for the system I

is worthwhile to note the ne rs relation fo he scattering process 2 6) or equivalen [

holds also for systems of f type | II | But i in spite of (5.17) or (5.21) there 1 is a Phase shift now It is evident

334 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

Then the potential

U@) & sinh* a(q; — 4x) (7.2)

is transformed into the potential

particular, the quantities exp{2aq,(t)} are eigenvalues of the matrix

However, this formula does not answer the question whether bound states exist for the system with the

potential (7.3) The answer is given in [85]

Let us note first that the formula (7.4) does not determine a geodesic in the space of Hermitian positively defined matrices, as it was the case for the systems of type II It appears that the potential (7.3) corresponds | to the space of Hermitian matrices Xan with the signature (m, na) Thus the problem

of the odounc eđ to the elucidation of the beh aviour of the øeode Dã Xninz

is evident that the bound states correspond to geodesics contained i ina finite domain of Xmm (geodesics

aa Fhe set of ø ¬e42e c 5 ale n >@ actefe ".7-sễamne

UO UJ BÚ UJ SI LJ COG nịn2 a! PCÓC Jd

Xu =X" corresponding to the systems II (7.2) And, naturally, the dynamics of the system with the

potential (7.3) is richer than the dynamics of the system with the potential (7.2) Every geodesic in X,,,,,,

is determined (up to a transformation using the constant matrix b) by the matrix exp{24t} (see (7.4)) that

“compact” parameters and r+ k “noncompact” ones Accordingly, all types of motions of the system may be divided into n2+ 1 classes Note here, that, for the system II, we have only one class (n2= 0)

The geodesic x(t) € X.,,,, lies in a finite domain of X,,,,,if a1 = + + = a, = By = +++ = By = 0 (see (7.5))

Generally a geodesic depends on n—1 parameters (a1 @p B1 - Bx: 91. 9%) and the “bounded”

geodesic is exclusive It depends only on k <n, parameters It is this “bounded” geodesic that corresponds to finite motion of the system But since it is not in a general position, some particles run to infinity when one changes the initial conditions

8 Explicit integration of the equations of motion for the Toda lattice*

Let us consider the equations of motion for the system of type VI (the nonperiodic Toda lattice)

It is more convenient to consider the space X~ of real symmetric positive-definite n x n matrices with unity determinant, instead of the space of Hermitian matrices, It is the homogeneous space X™ = SL(n, R)/SO(n)

Let Z be a subgroup in SL(n, R) of the upper triangular matrices with units on the diagonal and H be

a subgroup of diagonal matrices Then any symmetric matrix x € X” can be uniquely represented as

x=2z(x)h(x)z(x), zEZ, AEH ` (8.2)

The pair (h(x), z(x)) is the so called horospheric coordinates of the matrix x and h(x) is the horospheric

UI LJ LJ — ,# ` Äïc Ả()DJƠ ~VSAY Dlalic, > aic c- UsUuUad Ui USD

coordinates If A;(x) are the lower principal minors of order j of the matrix x and h7(x)=

336 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

We will prove the following propositions

Proposition 8.1 On applying the horospheric projection h(x) = exp Q(t) (8.3) of the geodesic (8.4)

we obtain an equivalent problem of motion with the potential (8.1)

Proposition 8.2 Let (q°, p°) be the initial values in Hamilton’s equations for the Toda lattice and 2% be

the matrix Mie = PFO je + g;_ ©XP(4;_+— đ7)Ôjk.: + 6y ©XP(đ) — đ;+)Ô;k— (8.6) and A; be the lower principal minor of order j of the matrix exp(2at) Then we have

Considering the expression x(t) x(t) we find Xx"!1=2z|2z”!¿+ P+3exp(O)Z'(z) Jz,

Let us denote

z !‡¿=M (8.10)

The matrix M belongs to the Lie algebra of the subgroup Z We assume M to be equal to (3.26) Thus

we consider only special geodesics, namely, those with the special forms of the matrix M (8.10) It follows from (3.25) and (3.26) that the matrices L and M satisfy the relation

But x(¢) is a geodesic and therefore it satisfies equation (8.5) Thus, the left-hand side of (8.13) is zero

and the Lax equation L =[L, M] which is equivalent to the Hamilton’s equations for the Toda lattice is

fulfilled Thus we have proved the first proposition

It is clear that for equation (8.8) we have 'z(x(0))=1

without any restrictions Comparing (8.4) and (8.8) we get the following formula

b = exp(Q”)= diag[exp(4¡) , cxp(đn.)Ì- (8.14)

We obtain also the expression (8.6) for the matrix 3 Besides, we have exp2O60)) = z7'b exp{2%}b'z7'!

The subgroup H of diagonal matrices is a normalizer of the subgroup Z of triangular matrices or, in

other words, z~*b = bz,, b E H (see (8.14)), z, z: € Z Hence bˆ' exp{2O0)} (b' = z¡exp{29)z

Now, formula (8.7) follows from (8.3) and (8.14)

Here we give the formulae for Q(t) only For the expression of the momentum P(t) the explicit form of the second horospherical coordinate z(x) is necessary The coordinate z(x) is expressed by other onprincipal) mi tr xi

9.* Reduction of Hamiltonian systems with symmetries (Methods of orbits)

It is well-known that the integrals of motion of a dynamical system are closely connected with the

group of symmetries (E Noether’s theorem) The existence of the integrals permits to carry out the

reduction, as usual in classical mechanics, and to consider a system with fewer degrees of freedom (the reduced system) As we have seen in a number of examples in the Introduction the dynamics of the

reduced system becomes more complicated as compared with the original one The abstract scheme

Here we briefly describe the moment map and the reduction procedure Our review is based on the

work [75] We here discuss this approach for our systems

338 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-diemnsional systems related to Lie algebras

Hamiltonian group action and the moment map Let M be a manifold with closed non-degenerate two-form 2, therefore M is a symplectic manifold If the form 2 is exact, 2 = d@, we call M an exact symplectic manifold For example the cotangent bundle M = T*X to the manifold X with the canonical one-form 6 = p dq is an exact symplectic manifold

Let G be a Lie group and/ (g) a representation of G in the group of the diffeomorphisms of M The action /(g) is called a symplectic transformation (or a canonical transformation in the physical literature) if it preserves 2 (/*(g) 2 = 2) If moreover it preserves 6, we call it an exact symplectic action, For example, an exact symplectic action may be obtained in, the following way Let f (g) be a diffeomorphism of a manifold X Then the induced transformation ⁄ of the cotangent bundle T*X,

{ar)=Y@.4/*0)

is an exact symplectic action The symplectic vector field Y on M may be defined by the relation

OY @)D| = YOU),

where g = exptg,g € G is the Lie algebra of G, ® is a smooth function on M Let us introduce the

one-form dF that can be written with the help of local coordinates as follows:

_ for the symplectic action ion fC (g) we define the Hamiltonian by the formula F= F FG 8)- Iti is evident

ì© HãNn onia ' ion ong (5 and ĐPTCIO de ợ E dưa

space (*, by the formula

The map 9: M> @* is called the moment map

For example, if G is the three dimensional transition group in R®, then G* = R* If M = T*R*® = Rý, then

i dby ¢ in € &* Thus, o is the linear momentum If G= SO acts on M, then g € &= 00) correspond to the angular momentum

Tf_a dynamic stem eccribed 6 Hami ny Vv 1 arian nae ETOUP i

fe i i.e H /0)= H the the function (2), ‘valued in the space (Ø*, is invariant under the

is called coadjoint representation For a given element » € ©* let us consider the orbit defined by the action of Ad*

O, = {vy € G*|p = Adétu, g € G} (9.2) The main point is that the orbits are symplectic manifolds (see [59]), with the symplectic form given by:

NE, &) = (% [&:, &)), vEeG*, @c 6 (9.3)

The form is nondegenerate, skew-symmetric and closed

Let the action of G on M be exact symplectic If ¢ is a moment map, then o(flg)x) = AdZo(x) In other words the image of the set {o(flg) - x)|g € G} for fixed x € M under the moment map coincides with the orbit Ad*g(x) Thus the following diagram is commutative

by / (g) then M is realized as an orbit of the coadjoint representation of the group G

The two-form 2 (9.3) and the Hamiltonian H determine the flow on the orbit O, (9.2) Indeed, let

Thus, this subset of M is characterized by fixed values of the integrals determined by wy It is evident from the diagram (9.4) that ø “(w) is invariant under the action of the subgroup G, of G given by the

340 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

formula

To eliminate the redundant degrees of freedom we construct the set of orbits of G, in 9” "(u)

Thus M is the base of the bundle 97 '(u) with the projection 7: g '(4)—>M Under the appropriate

assumptions about g~*(4) and G, the set M acquires the structure of a symplectic manifold which we

call the reduced phase space Let i be the injection g~'(u)—M and Zø be the projection ø~!{w)—>M then i*2 denotes the pullback of 2 to g~'(z) and the two-form on M is defined by the relation

a*) = i* 2

In other words for two tangent vectors Y', Y?€ Tg *(z) we denote Y* = daY*, where dz: Tp” '(u)>

TM Then 92 is defined as follows

Consequently the dimension of the reduced phase space is equal to

The simplest example of reduction is given in the Introduction In all cases the group G is an Abelian subgroup of the transitive transformation group of two dimensional homogenous manifolds From this point of view the projection used in sections 5-9 is the map 7 from the space ¿_ '(w)C T*X to the

reduced phase space Now we proceed to _explain the results obtained there in the spirit of this =

approach

The systems of type | and V [57] The cotangent bundle T*X° to the space X° of Hermitian n x n matrices

with -zero trace introduced in section 5S is determined_as follows

(9-12) (7-15)

€ space is the phase spac introdu i i invari ,

6 = Spy dx)

The symplectic structure is defined by the two-forms

2 = dé = Sp(dy a dx) (9.14)

The Hamiltonian system (H, 2), where

describes the geodesic flow (5.2), (5.3) on X° This system is invariant under the exact symplectic action of

the group K = SU(n)

(x, y)> (xg, 8° "yg*), g=g””'€K (9.16)

The transformation (9.16) generates the vector field

(x, y)> (a, x], (a, y)), (9.17)

where g € R= su(7) is the Lie algebra of the group SU(m) Then one can determine the Hamiltonian (9.1) of the vector field (9.17)

F(x, y; g) = Sp(y-: [, x]) = Sp([x, y] - 8), and the moment map from T*X° to the dual space ®*:

342 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

(9.21) are equal to each other Therefore we get

The relation (9.20) allows us to describe the reduced phase space M=ø !{)/G„ The subset

@ " (w)CT*X° 0 7) is the subset of all geodesic flows with the special fixed matrices a and b (5.11) The projection 7: g~'(%)—M is defined by the action of the isotropy subgroup G,, It is given explicitly by the relation (9.20) Thus to any pair (x, y) € T*X° we can associate a unique pair (Q, L)

a(x, y) = (Q, L) = (uxu"', uyu') (9.23)

As it follows from (9.12), (9.22), the dimension of the reduced phase space is equal to 2n — 2

The projection 7 takes the symplectic structure (9.14) into the standard form (see (9.10))

2 = Sp(uyu-? a uxu-") = Sp(L A Q) = = dp, ^ đạ, (9.24)

=1 The Hamiltonian H of the geodesic flow (9.15) is transformed by the projection into the Hamiltonian of the system of type I

The space T?X™ is dual to the tangent space to X” in the point x

The group G = SL(n, C) acts on T*X™ as

which follow from the second equation 0 31)

moment map g from T*X™ to the dual space R* = @ Because of the relation g = —g* "for gce8 and

344 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

Thereby we see that really the second Lax equation in (6.12) is trivial

Thus the unitary transformation (6.4) and (6.8) x= uexp{2aQ}u"', (9.37)

belongs to the isotropy subgroup G, C K The reduced phase space M= 9 '()/G,, is described by the

pair of matrices Q and L (6.6), (6.5)

em a O ems of tyne IT and 6b 1@ ØTFOUD O ne 2DD€C Fianeuiar matrice 3 he vroun G oO

hidden symmetries So we have a common original cotangent bundle and two different projections into

the -reduced-spaces of the systems-of type H-and-VI

However, it is natural to consider the cotangent bundle T*X™ to the homogeneous space X" =

L{n, F ê D e real symmetric positively defined-matrices with unit-determinant-as we had done

in section 8 The reason of it will be clarified in section 13 We thus consider the Hamiltonian system (9.26}-(9.27) and the exact symplectic action

(x, y)—>(zxz', z”}yz”}) (9.39)

One must only remember that all matrices considered now are real and z € Z which is the group of the upper triangular matrices with units on the diagonal Let Z’ be the group of the transposed matrices (the lower triangular matrices), 8 and 4’ be the Lie algebras of the Lie groups Z and Z’ The spaces 8 and

8 are dual in correspondence with the Killing form Sp(3132)

The Hamiltonian F of the vector field generated by the action (9.39) has a form

But L’ and pw’ are the constant matrices (9.45) and (9.44) So the triangular transformation z

diagonalizing the geodesic (8.8) ,

346 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

and specializing the matrix L (8.9)

belongs to the isotropy group G„: (9.8) The set of matrices x, y ¡in (9.47), (9.48) describes the space

@~'(w') Thereby the reduced phase space M= ø~!')/G,„ is the set of pairs (C, L) or equivalently

2, P):

m: (x, y)>(O,L)G6M

We obtain straightforwardly from the original Hamiltonian system (9.26), (9.28) the Hamiltonian of the

unperiodic Toda lattice as šSp L7 and the standard symplectic structure 7_, dp; dq;

Now we shall show that the reduced space M can be realized as an orbit of the coadjoint representation This is an example of the general scheme presented earlier Let B be the group of real upper triangular matrices with unit determinant The group B acts transitively on the space X™ i.e any symmetric positive-definite matrix x can be represented as gg’ (g € B) Moreover it acts transitively on the phase space of the geodesic flow T*X™, or, more rigorously, on the space of the unit cotangent vectors The moment map ø for the group B transforms the pair (x, y) into (xy)_ where the minus sign denotes now the projection into the space 8’ of the lower triangular matrices with zero trace (the diagonal elements need not vanish) Then it follows from (9.4) that ¢ maps T*X™ into a single orbit of B acting on %’ To find the orbit it is necessary to point out the stabilizator (the isotropy subgroup G,,.), for some element uz’ € 8’ Let us take the element y’ as

This is a subgroup of Z with zero second diagonal Thereby the reduced space M is canonically

isomorphic to the orbit O,, = B/Z,,: It is evident that

dim O, = dim B — dim Z„ = 2n — 2

We can give such a parametrization of the orbit that the canonical two-form (9.3) coincides with the

———————— §tandard symplectic structure One can fñnd ¡in detail this construction in [4]

10 Equilibrium configurations and small oscillations of some dynamical systems

It is worth noting that _ Systems of type III, V and VI’ have equilibrium configurations These

Ù Pura 1O are Ô 2U] Cad DY da Ù UC U C ar KaVIC Caturcs wit O d C CŨ vuTa 1O

positions are closely connected with the zeroes of classical polynomials Furthermore there are matrices

determining the frequencies and normal modes o € small osci S$ near uilibrium nfigura-

tions with eigenvalues and eigenvectors of simple forms Here we treat only part of them One can find more detailed discussions in refs (26, 2, 28] and in a number of other works, published mainly in Lettere al Nuovo Cimento

Let us start with consideration of the model of type V (for simplicity with g = 1), i.e the model, characterized by the Hamiltonian

Hereafter a prime appended to a sum indicates that the singular term must be omitted

As it is shown in [25], the quantities x; satisfy also the equations

So the equilibrium configurations for the systems, characterized by Hamiltonians (10.1) and (10.5), incide

It is known also that the quantities x;, satisfying equations (10.4), are the zeroes of the Hermite polynomial Hy ()

yy ¢ 0

This result is rather old; namely it was discovered by Stieltjes almost a hundred years ago (see e.g

[104], subsection 6.7) The result concerning the coincidence of the equilibrium positions for the systems (10.1) and (10.5) is obtained in [25] and is the consequence of a general theorem [90]

348 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

Let us consider the sequence of dynamical systems with Hamiltonians of type (10.5) and (10.1) and potential energy of the type

where 0,U, = 0U,/dq; and s is the degree of U; in U, Suppose that the system (10.5) has a stable

isolated equilibrium configuration q° = (qi, ., Qn) = (1, ., Xn)-

Theorem 1 [90] The system with potential energy (10.8) for any s has a stable isolated equilibrium configuration which coincides with the equilibrium configuration of the system (10.5) The eigen- frequencies of the small oscillations around the equilibrium configuration for the system with potential energy U, are equal to the sth degree of the corresponding eigenfrequencies of the small oscillations for the system (10.5) The normal modes of the small oscillations for all these systems coincide

The proof of the theorem is elementary Let us now apply the theorem to the system (10.5) with the generalized potential connected with the root system (see Appendix B)

1 U9) =5 D497 - ¬ 8 In|đa| (10.9)

Here R = {a} is the root system, R, is the subsystem of positive roots, gq, = (a,q), ga are positive constants which are equal for equivalent roots, i.e for roots which are related by transformations of the

(10.10) Corollary 1 The absolute minimum of the potential energy

whose eigenvalues give the squares of the frequencies of the small oscillations near the equilibrium configuration Namely

ay = 8, + 2 8aœœ/(qa) ” (10.15)

It can be proved the

Theorem 2 [90] The eigenvalues of the matrix a are equal to the orders 1, ,», of the basic invariants of the group W (Note that in the case of the system of type* A,,-, (10.6) v; = j + 1 See table

in Appendix B.)

Corollary 2 The frequencies of the small oscillations of the system with the potential (10.9) are

V1, ., WY, while those for the system (10.8) are z‡⁄2, , w‡⁄2

Remark 1 For the systems of type* A,-1 and B, this theorem may be verified by direct calculation of the matrix a and by comparison with the matrices A and B of ref [24]

Remark 2 It follows from the relation (10.15) that

đụ; = 243 (10.16)

Hence the vector q° = (q$, .,q,) provides the normal modes of the small oscillations of the system

with the potential U2 with frequency w = v; = 2

Remark 3 From formula (10.11) we get the explicit expression for the matrix a

It follows that the eigenvalues of the matrix a® are equal to the squares vj, ,”> of the basic

invariants of the group W

350 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

Let us give now the method for computing the normal modes Let us consider only the case A,_,

(about the other cases see [90]) We introduce the matrices

X„ = (L~ ôx)( — Xx)”,

Mx = dp (> (% x.)?) ~(l— 8x} - a), (10.21)

On = ÔXk; Q*=+X

Note that the matrix iX is equal to the matrix L for equilibrium configuration, which plays the role of

momenta Hence the matrices Q* introduced in [87] play the role of raising and lowering operators

Indeed, the result of ref [87] implies for the equilibrium configuration

[M,Q*]=+Q* (but[Q™, O*] 4AM) (10.22)

Therefore if u“? are eigenvectors of the matrix M with eigenvalues j,, then from (10.22) we get

O*tu® = Atu&*», O-u,=0, Otu,=0 (10.23)

and Mest = be + 1

It is easy to see that

u=(1, ,1) and pi=0

From this we obtain*

=0 an quantities B, = tr vanis quilibrium Thereby igenv the

matrix M, that identify normal modes of small oscillations, may be obtained from the vector w®),

where P,(x) is a polynomial of degree k, having definite parity The explicit expression for the

normalized vector u“ was obtained in ref [12] We mention that analogous results exist also for

Laguerre and Jacobi polynomials and Bessel functions (see [2, 28, 9])

* This result was obtained firstly in [24] using another method.

Let us note also that the eigenvalues of the matrix

5X; COS Ø + (1— ô»„)i(x; — x„) ” sin Ø (10.28)

where x; are the zeros of Hermite polynomials H„(x), coincide with the x; themselves (for all values of 6), see refs [2, 28]

As for the results for systems of type VI’ see the papers [45, 77], and for systems of type III the papers [38, 39]

Let us give some results The equilibrium configuration of the system with Hamiltonian (discussed in

where xo is some constant, and, as it was proved in [38], the circular frequencies of the normal modes

of the small oscillations of this system near the equilibrium configuration are given by the simple

a by product we obtain also [39 2 e off-diagonal Hermitian matrix of rank whose ele

are given by the formula

352 M.A Olshanetsky and A.M Perelomov, Classical integrable finite-dimensional systems related to Lie algebras

are related to the matrix A defined above by the equations

= -&(B?— 2(2 + ơ{)B - ơ?]) (10.38)

where I is the unit matrix and

11.* Abstract Hamiltonian systems related to root systems

The results of the Previous sections are valid for a more general class of Hamiltonian systems In this

An re Coatn oe c CC ` Aw reners G h^ mee o ac ° ha

U(@)= 23 gz0(4) (11.2)

eGR,

where functions v(q ) are defined by (1.3) -

The systems of type VI are constructed as follows Let A = {a} be an admissible root system, ga be

an arbitrary constant The potential in this case is of the form

La aEA

U(q)=