An integrable Hamiltonian system

29 May 1995

PHYSICS

ELSEVIER

Physics Letters A 201(1995)

LETTERS

A

306-310

An integrable Hamiltonian system F. Calogero

1

Dipartimento di Fisica, Universith di Roma “La Sapienza”, 00185 Rome, Italy, Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Rome, Italy Received 21 November

1994; revised manuscript received 9 February 1995; accepted for publication Communicated by A.P. Fordy

23 March 1995

Abstract

For any n, the dynamical system characterized by the Hamiltonian H = Eyk_ 1pjpk{h + p cos[ v(qj - qk)]} is cornpletely integrable: n constants of motion in involution are explicitly given, its iniiial-value problem is solved in completely explicit form. 1. Introduction

It has been recently pointed out that the dynamical system characterized by the Hamiltonian H=

pjIQexP[-11(4,-4k)l]

i j,k=

(1.1)

1

is completely integrable [l]. Motivated by this remarkable discovery, we have investigated the integrability of the more general Hamiltonian H=

i j,k=

PjPkf(qj-qk)’

(1.2)

1

Our main finding is that this Hamiltonian is completely integrable for f(x)=h+pCOS(VX);

plicity, that the constants h and p are real, and that v is real or imaginary; of course, by trivial resealing of the variables, one could set one of the two constants A, p to unity (provided it does not vanish), and the constant Y to 1 or i, as the case may be (again, provided it has a finite nonvanishing value). In Section 2 our main results on the integrability of (1.2) with (1.3) are reported. Section 3 contains some comments on the results reported in Section 2, as well as some findings relevant to ascertain the integrability of (1.2). All proofs are relegated to Section 4. Section 5 contains an outlook at further developments. 2. Results

(1.3)

this case, n constants of motion in involution are exhibited below, and the initial value problem is solved in cotipletely explicit form. In the following we implicitly assume, for sim-

Definition of (standard) Poisson brackets:

in

1 On leave while serving as Secretary General, Pugwash Conferences on Science and World AHairs, Geneva, London, Rome. 03759601/95/$09.50 8 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00238-3

Hamiltonian: H=

jglPjPk{A+I”

,Os[

v(qj-qk)]}e

F. Calogero / Physics Letters A 201 (1995) 306-310

Equations of motion: ej = ( qj, H]

307

{cjk,

C} =O,

(2.13a)

{cjk,

S}

(2.13b)

= affi/apj

=2k~lPt{h+P

CoS[v(qj-qk)]}9

(2.2a)

{cjk

7 “j’k’)

VS,jlPjPkPk’[Sin[ v(qj - qk)]

=

Pj= (P,, H} = -afz/aqj

=O;

+Sin[v(qk-qk’)l =2p”Pj

2

pk sin[ ‘(qj-qk)].

(2.2b)

+sin[

v(qk’-qj’)]}

+(j++k)+(j’wk’)+(j*k,

k=l

j’wk’). (2.14a)

Definition of constants of motion: {h,, cjk=PjPk(l

=

-‘Os[

2PjPk

h, =

‘(qj-qk)]]

sin*[

i

m=2,3

,*..,n,

(2.14b)

Gj = 2AP + 2~[ C cos( vqj) + S sin( uq,)] , (2.15a) Pj = 2pvpj[

tpj.

-S

cos(

vqj)

+

C sin( vg,)] ; (2.15b)

(24

j=l

Other definitions: c = t

n.

(2.3b)

j,k=l

P=

m, ml=2 ,...,

Time evolution equations: (2.3a)

u(qj-qk)/2],

cjk>

h,} =O,

(2Sa)

pi cos( vqj))

j=l

C= -ys,

(2.16a)

S=yC,

(2.16b)

y=2(A+/+P.

(2.16~)

Solutions: S = k pi sin( vqj).

(2.5b)

j=l

If A + p = 0, or P = 0, the quantities C and S are also time-independent; otherwise they evolve simply in time (see below). In any case the quantity C* + S* is a constant of motion, since there holds the identity

{h,,

H} =O,

{P,

(2.7a) ,...,

n,

H} =O,

(2.W (2.8)

{C, H} = -2( A+ p)vPS,

(2.9a)

{S, H) =2(h+/.+PC;

(2.9b)

{cjk,

P}

{h,,

P} =0,

(2.10a)

=O,

m=2

,...,

n;

(2.1Ob)

{C, P} = - VS,

(2.11a)

{S, P} = vc,

(2.11b)

[C, S} = - UP;

(2.17a)

S(t) = S, cos( rt) + C, sin( yt) =A sin(yt+

= [(H-

=O,

m=2

(Y),

(Y);

(2.17b)

(2.6)

Poisson brackets: H}

=A cos(yt+

sin( yt)

A = (C; + S,f)l’* = (C” + S*f’*

C* + S* = (H - AP*)//.L

{cjk,

C(t) = C, cos( -yt) -S,

(2.12)

AP*)/#*,

(Y= arctan( S,/C,)

(2.18a) ;

(2.18b)

qj(t)=(cr/v)+2(A+p)Pt +(2/v)

arctan [(P-A)/(P+A)]“* (

Xtan[ pv(P*

-A*)l’*(t,

- t,]).

(2.19)

Pj( l) = Pi(O) x(P+A

c0s[2~~(P*-A*)~~*(t-t~)])*

x(P+A

~0s[2p24P*-A’)~~*1,]}-*. (2.20)

308

F’. Calogero / Physics Letters A 201 (I 995) 306-310

3. Comments

the (even> function f(x) equation

For n = 2, P and H are two independent constants of motion in involution; for n > 2, the quantities P, H and h,, m = 2,. . . , n - 1 are n constants of motion (see (2.7b) and (2.8)) in involution (see (2.7b), (2.8) and (2.14b), and they are clearly independent (note, however, that h, = [(A + p)P* HI/d. The explicit solution of the initial-value problem (i.e., to compute qj(t), p,(t), j = 1, 2,. . . , n from given q&O), pk(0), k = 1, 2,. . . , n), is given by the formulas (2.19), (2.20); the constants CX,A, P appearing there are given, in terms of qk(0), pk(0), k= 1, 2,..., II, by (2.18b), (2.18a) and (2.4) (at t = 0; note that the constants C, and S,, are the values of C and S at t = 0, see (2.17) and (2.5)); the constants tj are then given, in terms of qj(0), by (2.19) (at t = 0). Eqs. (2.19), (2.20) are written in the manner most appropriate to the case of real v and initial data such that P* > A’; in this case the time evolution of the qj is a linear superposition of a linear drift (common to all qj), and an oscillatory motion with the common period T = ?~/pv(P* A*)‘/*; as for the pj, they all oscillate with the common period T. Of course the integrability of Hamiltonian (2.1) continues to hold in the limiting case in which the function A + p cos( vx) is replaced by A + /3x2 (limit of small v).

Proposition. The requirement that the Hamilto-

nian H=

I?

PjPkf(qj-qk)

(3.1)

j,k=l

o(x+y)[

-f’(x)

+ 2o’(x+y)Mx)

-P(y)

+f’(y)l

-f(r)1

= o(x)y(y) -a(y)?(x), with /3(x> also even,

(3.4)

P( -x) = P(x), (3.5) and (Y(X), /3(x), y(x) and f(x) finite at x = 0. Note that (3.3a) does indeed satisfy this functional equation, with p(x) = y(x) = 0 and (Y(X) =c sin(vx/2).

(3.6) A more general solution of this functional equation, corresponding however to the same choice (3.3a) for f(x), reads as follows, o(x)

=u sin[v(x-2)/2],

y(x)

= -21.~~ sin( vz/2)

p(x)

=O,

cos[ v( x - z)/2], P-7)

with a and z arbitrary constants. An even more general solution, but again corresponding essentially to the same choice (3.3a) for f(x), can be obtained using the obvious fact that, if a(x), p(x), y(x) and f(x) solve (3.4), so do G(x) =a exp(cx)a(x),

-wwl9 f(x) =bf(x), T(x)=b exp(~)y(x), Rx)

=0(x)

(3.8)

with a, b and c arbitrary constants. It is also easily seen that a more general solution than (3.7) with (3.3a) reads as follows, a(x)

=u sin[v(x-z)/2],

y(x)

= 21~~ sin[ v(2u - z)/2]

possess constants of integration of the form Cjk=pjpkg(qj-~k)

P(x)

satisfies the functional

p(x)

=O,

xcc+(x-z)/2)1, (3.2)

implies that f(x)

=h+/_L cos(vx),

(3.3a)

g(x)

= c[ 1 - cos( vx)] .

(3.3b)

Remark. The possibility that Hamiltonian (3.1) be completely integrable also for other choices of fix) than (3.3a) is not excluded by the above Proposition; indeed Hamiltonian (3.1) is presumably integrable if

f(x)

=h+p

cos[v(x--)I;

(3.9) but the requirement that f(x) be even forces u to vanish mod(rr), yielding again (up to trivial changes) the solution (3.7) with the choice (3.3a) for f(x).

4. Proofs The proofs of (2.1)-(2.8) require little ingenuity (especially once the results are known); we feel no

F. Calogero / Physics Letters A 201 (1995) 306-310

need to provide details here. Likewise, the integration of (2.15a), and then (2.15b) (in both cases, with C and S given by (2.171, and of course P, C,,, So, A, a all time-independent), is an easy task; moreover, the reader who does not wish to undertake this amusing exercise needs only to verify that (2.191, (2.20) satisfy (2.15) (with (2.17)). To prove the Proposition of Section 3, note first of all that the ansatz (3.1) implies that f(x) is even hence f’(x) odd,

(of course the constant v2 need not be positive). And it is now clear that this equation, together with (4.11, implies (3.3a). Finally, to justify the Remark at the end of Section 3, we note that the equations of motion (4.2) correspond, under the hypotheses stated there, to the Lax matrix equation i = [L, A],

(4.8)

+

‘jk

t

Pm

P(qj-qm)*

(4.9)

m=l

k=l

-2Pj

(4.7)

with

f(-x) =f(x), f’(-x) = -f’(x), (4.1) so that the equations of motion entailed by Hamiltonian (3.1) read as follows, n (4.2a) 4jc2 C Pkf(Cij_qk)7

i)j=

309

2

(4.2b)

Pkf(qj-qk).

k=l

Assume now that the quantities C+ see (3.21, are time-independent:

+PjPk(9j-~k)g’(qj_qk)=‘.

(4.3)

Using (4.2) and (4.1) it is easily seen that a necessary and sufficient condition for this equation to hold, is validity of the following functional equation, g’(x+y)

=

dx+y)

f’(x)

--f'(Y)

f(x)

--f(Y)

.

(44

It is now trivial to verify that (3.3) provides a solution of this equation. It is also easy to prove that this is the most general solution of (4.4) and (4.1). Indeed, the application of the operator a/%x - a/ay, that clearly annihilates the left-hand-side of (4.41, yields f”(X)f(Y)

-f”(Y)+)

=Y(x)f(x)

-

Lfw12

-f”(Y>f(Y) + [f’(Y)12.

f”‘( x)/f’(

=f”‘(y)/f’(y), x) = - Y2

Outlook

It is clear that the model treated in this paper remains completely solvable if Hamiltonian (2.11, H = AP2 + p( c2 + 9)

(5.1)

(see (2.41, (2.5) and (2.6)), is replaced by H=hF(P)+p(C2+S2),

(5.2)

with F(x) an essentially arbitrary function. Then clearly the solution given above still applies, provided P is replaced everywhere by $F’(P). The complete solvability of these Hamiltonians, and the simple character of the motion they entail, suggests that there should exist quantal versions of these models, which are also solvable. Moreover, a geometrical interpretation of these results is likely to be of interest. We plan to address these questions in future publications, where we shall also provide a more detailed analysis of the solutions of these models than has been presented here.

(4.5)

Then the application of the operator a2/axay, that clearly annihilates the right-hand-side of this equation, yields f”‘(x)/f’(x) namely

5.

(4.6a) (4.6b)

Acknowledgement It is a pleasure to acknowledge useful conversations with M. Bruschi, 0. Rag&co and S. RauchWojciechowski. In particular I owe to the latter the idea of introducing the constants of motion in involution (2.3b).

310

F. Calogero/Physics

Letters A 201 (1995) 306310

Note added

References

We have now found, together with J.-P. Franpoise, that Hamiltonian (1.2) is integrable in the more general case f(x) = A + p cos(ux) + $ sin(v I x I), with r_L another arbitrary constant. This includes both (1.1) and (1.3) as special cases, and entails the integrability of (1.2) with f(x) = h + (YI x I +/3x2.

[l] R. Camassa and D.D. Helm, Phys. Rev. Lett. 71(1993) 1661; R. Camassa, D.D. Holm and J.M. Hyman, Adv. Appl. Mech. 31 (1994) 1.