Additional Constants of Motion for a Discretization of the Calogero--Moser Model

T yp eset with jpsj2.cls < v er.1 .2 > Full P aper Additional Constan ts of Mot ion for a Discretization of the Calogero–Moser Mo del Hideaki Uj ino 1 ∗ , Luc Vinet 2 † , T etsu Y aj ima 3 ‡ and Haruo Yos hid a 4 § 1 Gunma National Col le g e of T e chnolo gy, Maeb ashi, Gunma 3 71–85 30 2 Universit´ e de Montr e al, Montr´ eal, Q u´ eb e c, Canada H3C 3J7 3 Dep artment of Information S cienc e, F aculty of Engine ering, Utsunomiya University, Ut sunomiya, T o chigi 321–85 85 4 National Astr onomic al Observatory of Jap an, Mitaka, T okyo 181–8 588 The maximal sup er-integrabilit y of a discretization of the Calogero –Moser mo del intro- duced b y N ijhoff and Pang is presented. An explicit formula for the additio nal constants of motion is given. KEYWORDS: Calogero–Moser model, sup er-integ rabilit y , integrable discretization 1. In tro duction When one discretizes dyn amical systems, it is h ardly p ossible to av oid mo difying the orig- inal systems. Controll ing such mo difications is thus a cen tral p r oblem in numerical an alysis. 1 It w ould b e ideal if a d iscretiza tion conserve s wh ole the structure of the original dynamical system suc h as orb its in the phase space, constan ts of motion, inte grabilit y an d so on. As an example of such ideal d iscretiza tions, a discretization of the Kep ler p roblem, which k eeps all the constan ts of motion and the o rbits in the phase space, was discov ered. 2, 3 The Kepler prob- lem is an integrable system that has a set of m utually ind ep enden t and Poi sson comm utativ e constan ts of motion, whose n um b er is the same as the degrees of fr eedom of the system. A dynamical s y s tem of N -degrees of freedom whic h has m utually indep end ent 2 N − 1 constants of motion in the f orm of single-v alued functions is called maximally sup er-in tegrable and so is the Kepler problem. The ab o ve discretization conserv es sup er-inte grabilit y of the Kepler problem. Among the family of one-dimensional integ rable systems with in verse-square int eractions called the Calogero–Mose r–Sutherland mo dels, 4 the C alog ero mo del, 5 whic h is the r oot of the family , the Calogero–Moser m od el 6 of the rational and hyp erb olic t yp es are kno wn to b e maxi- mally sup er -integrable. 7–9 What we discuss h ere is the sup er-integ rabilit y of a discretization of the rational Calogero– Moser mo del, whic h is a classical dyn amical system wh ose Hamiltonian ∗ E-mail address: ujino@nat.gunma-ct.ac.jp † E-mail address: luc.vinet@umontreal.ca ‡ E-mail address: yajimat@is.utsunomiya-u.ac.jp § E-mail address: h.yos hida@nao.ac.jp 1/7 J. Phys. So c. Jpn. Full P aper is giv en by H := 1 2 N X i =1 p 2 i − 1 2 N X i,j =1 i 6 = j γ 2 ( x i − x j ) 2 , (1.1) where γ , N , p i := p i ( t ) and x i := x i ( t ) are the coupling p arameter, the num b er of p article s, the momen tum and the co ordinate of the i -th particle at th e time t , resp ectiv ely . It will b e no exaggeration to sa y that the C alogero–Moser m od el represent s the mo dels of Calogero–M oser–Sutherland t yp e s ince the Lax formulati on and a systematic construction of the constants of motion for the mo del w as d isco v er ed earlier than those for an y other mo dels of the family. 6 Moser constructed a set of N constan ts of motion that are indep endent of eac h other. Later, m u tual Poisson comm utativit y of the constants of motion of Moser-t yp e w as pro v ed and the in tegrabilit y of th e mo del in Liouville’s sense wa s thus established. 10, 11 F urthermore, it tur ned out that the mo del had N − 1 additional constan ts of m otion wh ic h are indep endent of the Moser-t yp e ones and indep endent of eac h other as w ell. 8 This conclud es the maximal sup er-in tegrabilit y of the Calogero–Moser m od el. A time-discretizatio n of th e Caloge ro–Moser mod el that conserv es the Moser-t yp e con- stan ts of motion was present ed by Nijhoff and Pang, 12 whic h wa s reformulated int o a more con- v enient form b y Suris. 13 The aim of th e pap er is to sho w that the maximal su p er-in tegrabilit y of the Calogero– Moser mo del holds goo d even after the ab o ve time-discretiza tion: in other w ords, the time-discretization of the C alog ero–Moser mo del has N − 1 additional constants of motion, wh ic h are indep en den t of the Moser-t yp e ones and indep end en t of eac h other at the same time. I n § 2, we sh all giv e a br ief summary of the discretization of the C alog ero–Moser mo del and its d iscrete Lax form. In § 3, we shall explicitly construct N − 1 additional constants of motion of the discrete Calogero–Moser m od el. C oncluding remarks are sum m arized in § 4. 2. The Discrete C alogero–Mo ser Mo del Throughout the pap er, w e emplo y Suris’ form ulation of the discrete Calogero–Mo ser mo del, w hic h is giv en by the f ollo wing discrete symp lectic m ap ( x i,n , p i,n ) → ( x i,n +1 , p i,n +1 ), i = 1 , 2 , . . . , N , 1 − ∆ tc − 1 0 p i,n = N X j =1 c 0 x j,n +1 − x i,n + c 0 − N X j =1 j 6 = i c 0 x j,n − x i,n , 1 − ∆ tc − 1 0 p i,n +1 = N X j =1 c 0 x i,n +1 − x j,n + c 0 − N X j =1 j 6 = i c 0 x i,n +1 − x j,n +1 , (2.1) where ∆ t , x i,n := x i ( n ∆ t ) and p i,n := p i ( n ∆ t ) den ote the d iscr ete time-step, th e co ordinate and the m omen tum of the i -th particle at the n -th discrete time n ∆ t . 13 The constant c 0 is 2/7 J. Phys. So c. Jpn. Full P aper defined by c 2 0 := − γ ∆ t . In terms of the Lax pair, whic h consists of tw o N × N matrices b elow,  L n  ij = p i,n δ ij + γ x i,n − x j,n (1 − δ ij ) ,  M n  ij = c 0 x i,n +1 − x j,n + c 0 , (2.2) the discrete symp lecti c m ap (2.1) is expressed by the discrete Lax equation, L n +1 M n = M n L n , (2.3) whic h is equiv alen t to L n +1 = M n L n M − 1 n . (2.4) The companion matrix M n th u s p la ys a role of the time-evo lution op erator of the Lax matrix L n . With the aid of the trace iden tit y T r AB = T r B A where A and B are arbitrary N × N matrices as well as the discrete Lax equation (2.3), one confirms that the trace of the p o wer of the Lax matrix L n satisfies T r  L n +1  m = T r  M n L n M − 1 n  m = T r  L n  m . Th us the discrete Calogero–M oser mod el (2.1) as w ell conserves the Moser-t yp e quan tities, whic h are exactly the same as the N constan ts of motion of Moser-t yp e in the con tinuous time case, 6 I ( m ) n := T r  L n  m , m = 1 , 2 , . . . , N . (2.5) The Moser-t yp e quan tities (2.5) are single-v alued for they are rational functions of p i,n ’s and x i,n ’s. In order to confirm th e m u tual in d ep endence of the Moser-t yp e quantitie s, all one has to do is to chec k their exp licit form s when γ = 0, I ( m ) n    γ =0 = N X i =1  p i,n  m , (2.6) whic h is nothing but the p ow er sums of p i,n ’s that are indeed in dep enden t of eac h other. Note th at the Hamiltonian (1.1) corresp ond s to the second constant of motion of Moser-t yp e, H    t = n ∆ t = I (2) n / 2. The companion matrix M n of the Lax p air (2.2 ) satisfies another Lax equation, D n +1 M n = M n D n + M n ∆ tL n  I − ∆ tc − 1 0 L n  − 1 , (2.7) where I is the identit y matrix and D n := d iag( x 1 ,n , x 2 ,n , . . . , x N ,n ). Th e ab o v e relation (2.7) w as the crucial key to the solution of the initial v alue pr oblem of the discrete symp lectic map (2.1). In the n ext s ecti on, we shall sho w ho w the relation (2.7) wo rks in a sys tematic con- struction of N − 1 additional constan ts of m otio n of the discrete Calogero– Moser mo del (2.1). 3/7 J. Phys. So c. Jpn. Full P aper 3. Additional Constants of Motion Our main pu rp ose is to confirm that th e N − 1 quant ities b elo w K ( m ) n :=T r D n  I − ∆ tc − 1 0 L n  L n  m − 1 T r L n − T r  L n  m T r D n  I − ∆ tc − 1 0 L n  , m = 2 , 3 , · · · , N , (3.1) are conserve d by the discrete time ev olution of the discrete Calogero–Mo ser m o del (2.1 ) and that they are ind ep enden t not only of the Moser-t yp e quantitie s (2.5) but also of eac h other. Note that the case m = 1 is omitted in eq. (3.1) b ecause K (1) n = 0. The discrete s y m plectic map (2.1) is equiv alen t to the discrete Lax equations (2.3) and (2.7). F rom the discrete Lax equations (2.3) and (2.7), one obtains D n +1  I − ∆ tc − 1 0 L n +1  M n = M n D n  I − ∆ tc − 1 0 L n  + M n ∆ tL n , (3.2) whic h is rewritten as D n +1  I − ∆ tc − 1 0 L n +1  = M n D n  I − ∆ tc − 1 0 L n  M − 1 n + M n ∆ tL n M − 1 n . (3.3) The r elat ion (3.3) giv es the time-ev olution of the matrix D n  I − ∆ t c − 1 0 L n  . Using eqs. (2.4) and (3.3) as well as the trace iden tit y , one can p erform the calculation b elo w, K ( m ) n +1 =T r D n +1  I − ∆ tc − 1 0 L n +1  L n +1  m − 1 T r L n +1 − T r  L n +1  m T r D n +1  I − ∆ tc − 1 0 L n +1  =T r M n  D n  I − ∆ tc − 1 0 L n  + ∆ tL n  M − 1 n  M n L n M − 1 n  m − 1 T r M n L n M − 1 n − T r  M n L n M − 1 n  m T r M n  D n  I − ∆ tc − 1 0 L n  + ∆ tL n  M − 1 n =T r D n  I − ∆ tc − 1 0 L n  L n  m − 1 T r L n − T r  L n  m T r D n  I − ∆ tc − 1 0 L n  + ∆ t  T r  L n  m T r L n − T r  L n  m T r L n  = K ( m ) n , (3.4) whic h pro v es the conserv ation of K ( m ) n . As one can observe in the third line of eq. (3.4), cancellat ion of the u n wan ted terms d eriv ed from the second term in the r .h.s. of eq. (3.3) is cru cial . The additional constan ts of m otio n (3.1), whic h w e call the W o jciec ho wski-type quan tities, are rational functions of p i,n ’s and x i,n ’s. When the coupling parameter γ and the time-step ∆ t are zero, the N − 1 constan ts of motion { K ( m ) n } (3.1) r ed uces to symmetric p olynomials of p i,n ’s and x i,n ’s, lim γ → 0 lim ∆ t → 0 K ( m ) n = N X i =1 x i,n  p i,n  m − 1 N X j =1 p j,n − N X i =1  p i,n  m N X j =1 x j,n . (3.5) Though it is less trivial than the m u tual indep en dence of I ( m ) n    γ =0 (2.6), the quant ities (3.5) are indep endent of those in eq. (2.6) and ind ep enden t of eac h other, to o. Its v erification is essen tially the same as that for the additional constan ts of motion of W o jciec ho wski-t yp e in 4/7 J. Phys. So c. Jpn. Full P aper the contin uous time case. 8 Th us we fi nd that the discrete sym p lectic map (2.1) h as 2 N − 1 constan ts of motion { I ( m ) n , K ( m ) n } , wh ic h are indep endent of eac h other an d single-v alued as w ell. This concludes that the discrete symplectic m ap (2.1) giv es not only an in tegrable, b ut a maximally sup er-in tegrable d iscretiza tion of the Calogero–Mose r mod el (1.1). T h is p rop ert y of the d iscrete symplectic map (2.1) corresp ond s to the maximal sup er-int egrabilit y of the Calogero–M oser mo del in the conti n uous time case. 6, 8 4. Concluding Remarks The main result of the pap er is the constru ctio n of the N − 1 additional constan ts of motion (3.1) b esides the kno wn N constants of motion (2.5) of the discrete symp lecti c map (2.1). The result concludes th e maximal sup er-in tegrabilit y of the discrete Calogero– Moser mo del (2.1). It shou ld b e remark ed that the N − 1 additional constan ts of motion { K ( m ) n } are not ex- actly the same as those f or the Calogero–M oser mo del in the con tin uou s time case, 8 b ecause of the additional term prop ortional to ∆ t in their construction (3.1). In the con tin u ous time limit ∆ t → 0, ho w ever, th e additional constants of motion (3.1) redu ces to exactly the same additional constant s of motion for the non-discrete Calogero– Moser mo del disco v ered by W o- jciec ho wski. 8 In other words, K ( m ) n is a one-parameter deformation of th e additional constants of motion of W o jciec howski-t yp e in the con tinuous time th eory . Since th e orbit in the 2 N - dimensional phase sp ace of the maximally sup er-inte grable mo del of N degrees of freedom is uniquely determined by its 2 N − 1 constan ts of motion, the orbit of the d iscr ete sym plectic map (2.1) in the 2 N -dimens ional p hase space differs fr om that of the Calogero–Mo ser mo del in the con tinuous time case, even though b oth ev olve from the same initial v alues. The former giv es a one-parameter deformation of the latter. When one deals with the Calogero–Mo ser mo del, its pairwise inte ractions are u sually repulsive . The discr ete symplectic map (2.1) with a pure imaginary γ conserves the Calogero– Moser Hamiltonian (1.1) with repulsive interact ions. In this case, h o w ev er, its solution b e- comes complex in general. Th u s in the p h ysical sense, the d iscrete symplectic map cannot describ e a d iscrete v ersion of the Calogero–Moser mo del with repu lsive interacti ons. O n the other hand, another sup er-integ rable d iscr etization of the Caloge ro–Moser mo del is given from the sup er-in tegrable d iscretiza tion of the Calogero–Moser mo del with an external h ar- monic confinement. 14 This d iscretiz ation conserves exactly the same constan ts of motion of the Calogero– Moser mo del in the con tin uous time case and hence repr o du ces exactly th e same orb it in the p hase space. Repulsive in teractio ns can b e dealt with as well. Details on the comparison of the tw o different discretizations will b e presented in a separate pap er. 5/7 J. Phys. So c. Jpn. Full P aper Ac knowledgemen ts Most of the w ork wa s carried out du ring th e s hort sta y of H.U. at CRM, Un iv ersit ´ e de Mon tr´ eal hosted b y L.V. H.U. is grateful to the warm hospitalit y of the institute. This author is also supp orted by the Grant-i n-Aid for Y oun g S cien tists (B) (No. 177 40259 ) from the Ministry of Education, Culture, Sp orts, Science and T echnolog y of J apan. The w ork of H.Y. is partially su pp orted by a Grant -in-Aid for S cien tific Researc h of J SPS, No. 185402 26. 6/7 J. Phys. So c. Jpn. Full P aper References 1) E. Hairer , C. Lubich and G. W anner: Ge ometric Numeric al Int e gr ation, Structu re -Pr eserving Al- gorithms for Or dinary Differ ential Equations , Springer Series of Computational Mathematics 3 1 (Springer, 200 4) 2nd ed. 2) Y. Minesak i a nd Y. Nak amura: Phys. Lett. A 306 (2002) 12 7. 3) Y. Minesak i a nd Y. Nak amura: Phys. Lett. A 324 (2004) 28 2. 4) J. F. v an Diejen and L. 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