Addition theorems and the Drach superintegrable systems

Addition theorems and the Dr ac h sup erin tegrable systems A V Tsig anov St.Petersbur g State University, St.Petersbur g, Russia e–mail: tsiganov@mph.ph ys.spbu.ru Abstract W e pr op ose new construction of the p olynomial int egrals of mot ion rela ted to the addition theorems. As an example w e reconstruct Drac h systems and get some new tw o-dimensional s u p erintegrable St¨ ac k el systems with third, fifth and sev en th order integ rals of motion. P A CS n umb ers: 0 2.30.Jr, 02.30.Ik, 03.65.Fd Mathematics Sub ject Classification: 7 0H06, 70H20, 35Q72 1 In tro duction The Liouville classic al theorem o n completely in tegrable Hamilto nian systems implies that almost all p oints of the manifold M a re co v ered by a system o f op en t oroidal domains with the action-angle co ordinates I = ( I 1 , . . . , I k ) and ω = ( ω 1 , . . . , ω n ): { I j , I k } = { ω i , ω k } = 0 , { I j , ω k } = δ ij . (1.1) The independen t in tegrals of motion H 1 , . . . , H n are functions of the indep enden t action v ariables I 1 , . . . , I n and the corresp onding Jacobian do es not equal to zero det J 6 = 0 , where J ij = ∂ H i ( I 1 , . . . , I n ) ∂ I j . (1.2) Let us in tro duce n functions φ j = X k  J − 1  k j ω k , (1.3) suc h that { H i , φ j } = n X k =1 J ik  J − 1  k j = δ ij . (1.4) If Hamiltonian H = H 1 then the ( n − 1) functions φ 2 , . . . , φ n are in tegrals of mot ion dφ j dt = { H 1 , φ j } = 0 , j = 2 , . . . , n, 1 whic h are functionally indep enden t on n functions H 1 ( I ) , . . . , H n ( I ). So, in classical mec hanics an y completely integrable system is sup erin tegrable system in a neigh b orho o d of an y regular p oin t of M [14]. It means that the Hamiltonian H = H 1 has 2( n − 1) integrals of motion H 2 , . . . , H n and φ 2 , . . . , φ n on a ny op en toroidal domain. If the action-angle v ariables are global v ariables o n the whole phase space M and, therefore, w e ha v e sup erin tegrable systems o n M . F or instance, the global action-angle v ariables for the op en and p erio dic T o da lat t ices are discussed in [3]. Ho w ev er, in g eneric case the angle v ariables ω k are multi-v alued functions on the whole pha se space M . If w e ha v e k a dditio nal single-v alued algebraic inte gra ls of motion K t he tra jectories a re closed (more generally , t hey a re constrained to an n − k dimensional manifold in phase space). An y additional in tegral is a function on the actio n- angle v ariables. Since we hav e to understand ho w to get single-v alued additional inte gra ls of motio n from the m ulti- v alued action-a ngle v ariables. In this pap er we discuss a p ossibilit y to get p o lynomial in tegrals of mot io n from the m ulti-v alued angle v ariables by using simplest addition theorem. 2 The St¨ ac k e l systems. The system asso ciated with the name of St¨ ac k el [9, 10] is a holonomic system on the phase space M = R 2 n , with the canonical v a riables q = ( q 1 , . . . , q n ) and p = ( p 1 , . . . , p n ): Ω = n X j =1 dp j ∧ dq j , { p j , q k } = δ j k . (2.1) The nondegenerate n × n St¨ ac k el matrix S , whose j column dep ends on t he co ordinate q j only , defines n functionally indep enden t in tegrals of motion H k = n X j =1 ( S − 1 ) j k  p 2 j + U j ( q j )  . (2.2) F rom this definition one immediately gets the separated relations p 2 j = n X k =1 H k S k j − U j ( q j ) (2.3) and the angle v a riables ω i = X j =1 Z S ij d q j p j = X j =1 Z S ij d q j p P n k =1 H k S k j − U j ( q j ) . It allow s reducing solution of the equations of motion to a problem in algebraic geom- etry [10]. Namely , let us supp ose that there are functions µ j and λ j on the canonical separated v ariables µ j = u j ( q j ) p j , λ j = v j ( q j ) , { q i , p j } = δ ij , (2.4) 2 whic h allows us to rewrite separated equations (2.3) a s equations defining the h yp erel- liptic curv es C j : µ 2 j = P j ( λ j ) ≡ u 2 j ( λ j ) n X k =1 H k S k j ( λ j ) − U j ( λ j ) ! , (2.5) where P j ( λ j ) a r e polynomials on λ j . In this case the action v ariables I k = H k (2.2) ha v e the canonical P oisson brac k ets (1.1) with the angle v ariables ω i = n X j =1 Z A j S ij ( λ j ) p P j ( λ j ) d λ j = n X j =1 ϑ ij ( p j , q j ) , (2.6) whic h are the sums of in tegrals ϑ ij of the first kind Ab elian differen tials on the hyper- elliptic curv es C j (2.5) [10 , 14], i.e. they are sums of the mu lti- v alued functions o n the whole phase space. 2.1 Addition theorems and algebrai cally sup erin tegrable sys- tems In g eneric case the action v a riables (2 .6) ar e the sum of the m ulti-v alued functions ϑ ij . Ho w ev er, if w e are able to apply some additio n theorem to the calculation of ω i (2.6) ω i = n X j =1 ϑ ij ( p j , q j ) = Θ i  K i  + const, (2.7) where Θ i is a multi-v alued func tio n on the algebraic argument K i ( p, q ) then one will get algebraic in tegrals of motion K i ( p, q ) b ecause { H 1 , ω i } = { H 1 , Θ i  K i  } = Θ ′ i · { H 1 , K i } = 0 . So, the addition theorems (2.7) could help us to classify algebraically sup erintegrable systems a nd vice vers a. Plane curve s with the gen us g ≥ 1 are related to elliptic and Ab elian inte gra ls. Additio n theorems of these functions are the con tent of Ab el’s theorem [15]. The main result of this paper is that almost all the kno wn examples of algebraically sup erin tegrable system s relate with the one of the simplest addition theorem e x e y = e x + y , or ln( x 1 ) + ln( x 2 ) = ln ( x 1 x 2 ) (2.8) asso ciated with the zero-gen us h yp erelliptic curv es C j C j : µ 2 j = P j ( λ j ) = f j λ 2 j + g j λ j + h j , j = 1 , . . . , n, (2.9) where f j , g j , h j are linear functions o n n integrals of motio n H 1 , . . . , H n . In f a ct, if S ij ( λ ) = 1 then after substitution (2 .9) into (2.6) one g ets the sum of the rational functions ϑ j = Z 1 p g j λ j + h j d λ j = µ j g j 3 or logarithmic functions ϑ j = Z 1 q f j λ 2 j + g j λ j + h j d λ = f − 1 / 2 j ln µ j + 2 f j λ j + g j 2 p f j ! . F or the h yp erelliptic curv es of higher gen us one g ets elliptic functions, whic h ha v e more complicated addition la w [15]. In order to use addition low (2.8) w e hav e to mak e the follo wing steps: • W e hav e to apply canonical transformatio n o f the time that reduce n -th row of the St¨ ac k el matrix to the canonical Brill-No ether form [11, 13] S nj = 1 , j = 1 , . . . n , suc h that ω n = 1 N n X j =1 Z v j ( q j ) 1 p f j λ 2 + g j λ + h j d λ, (2.10) where N is norma lizat io n, whic h is restored from { ω n , H n } = 1 . • Then we hav e to use addition la w (2.8) for construction of the p olynomial in momen ta integrals of motion • W e ha ve to mak e inv erse transformation of the time, whic h preserv es the p o lyno- mial form of integrals suc h us it dep ends on q v ariables only [1 1, 13]. Let us consider construction of p olynomial in the momen ta integrals of motion at n = 2 . 2.1.1 Case f 1 = f 2 = 0 If f 1 , 2 = 0 w e ha v e ω 2 = 1 4 2 X j =1 Z v j ( q j ) 1 p g j λ + h j d λ = p 1 u 1 g 1 + p 2 u 2 g 2 = K g 1 g 2 , (2.11) where K = g 2 p 1 u 1 + g 1 p 2 u 2 is the p olynomial in the momen ta inte gra l of motion of the first or the t hird order. Remind that g 1 , 2 are linear f unctions on the St¨ ack el in tegrals H 1 , 2 , whic h are the sec ond order p olynomials on p 1 , 2 . Example 1 Let us consid er the t w o-dimensional St¨ ac k el system defined b y t w o Rie- mann surfaces C 1 , 2 : µ 2 = P 1 , 2 ( λ ) = ( H 1 ± H 2 ) λ + α 1 , 2 , and substitutions (2.4 ) µ j = q j p j , λ j = q 2 j , 4 whic h give rise to the following separated equations p 2 1 , 2 = H 1 ± H 2 + α 1 , 2 q 2 1 , 2 . In tegrals of mot ion H 1 and H 2 are solutions of these separated equations H 1 , 2 = p 2 1 ± p 2 2 2 − α 1 2 q 2 1 ∓ α 2 2 q 2 2 , (2.12) whic h coincide with in tegrals of motion fo r the t w o- particle Calogero system after an ob vious p oin t tra nsformation x = q 1 − q 2 2 , p x = p 1 − p 2 , y = q 1 + q 2 2 , p y = p 1 + p 2 . (2.13) The angle v aria bles (2.11) read as ω 1 , 2 = 1 4 Z q 2 1 1 p ( H 1 + H 2 ) λ + α 1 d λ ± 1 4 Z q 2 2 1 p ( H 1 − H 2 ) λ + α 2 d λ = − p 1 q 1 ( H 1 − H 2 ) ∓ p 2 q 2 ( H 1 + H 2 ) 2( H 1 + H 2 )( H 1 − H 2 ) . The corresp onding cubic in tegral o f motion K (2.11) is equal t o K = 2( H 1 + H 2 )( H 1 − H 2 ) ω 2 = − p 1 q 1 ( H 1 − H 2 ) + p 2 q 2 ( H 1 + H 2 ) . The brac ket { K , H 2 } = 2( H 2 2 − H 2 1 ) is easily restored from the canonical brac ke ts (1.1). 2.1.2 Case f 1 = k 2 1 f and f 2 = k 2 2 f with in teger k 1 , 2 In this case, we hav e ω 2 = 1 2 2 X j =1 Z v j ( q j ) 1 p f j λ 2 + g j λ + h j d λ = 2 X j =1 1 2 k j √ f ln  P ′ j 2 k j √ f + p j u j  , = 1 2 k 1 k 2 √ f ln "  P ′ 1 2 k 1 √ f + p 1 u 1  k 2  P ′ 2 2 k 2 √ f + p 2 u 2  k 1 # , where P ′ j = dP j ( λ ) dλ     λ = v j ( q j ) = 2 f j v j ( q j ) + g j . So function Φ 2 = exp(2 k 1 k 2 p f ω 2 ) =  P ′ 1 2 k 1 √ f + p 1 u 1  k 2  P ′ 2 2 k 2 √ f + p 2 u 2  k 1 = = 1 f ( k 1 + k 2 ) / 2  K ℓ + p f K m  (2.14) 5 is the generating function of p olynomial in tegrals of motion K m and K ℓ of the m -th and ℓ -th order, respectiv ely . The v alues of m and ℓ dep end on v alues of k 1 , 2 . As ab o v e, the algebra of integrals H 1 , 2 and K m ma y b e restored from the canonical brac k ets (1.1). Example 2 Let us consider the t w o- dimensional St¨ ac k el system defined b y the t w o Riemann surfaces C 1 , 2 : µ 2 1 , 2 = P 1 , 2 ( λ 1 , 2 ) = k 2 1 , 2 λ 2 1 , 2 + β 1 , 2 λ 1 , 2 + ( H 1 ± H 2 ) and substitutions (2.4 ) µ j = p j , λ j = q j , whic h give rise to the following separated equations p 2 1 , 2 = k 2 1 , 2 q 2 1 , 2 + β 1 , 2 q 1 , 2 + ( H 1 ± H 2 ) . In tegrals of mot ion H 1 and H 2 are solutions of these separated equations H 1 , 2 = p 2 1 ± p 2 2 2 − k 2 1 q 2 1 ± k 2 2 q 2 2 2 − β 1 q 1 ± β 2 q 2 2 , (2.15) whic h coincide with in tegrals of motion for t he harmonic o scillator. The same St¨ ac k el system coincides with the Kepler problem after w ell-known canonical transformation of the time, whic h c hanges the row o f t he St¨ ac ke l matrix [11]. The angle v ariable (2.11) reads as ω 2 = − 1 2 Z q j 1 p k 2 1 λ 2 + β 1 λ + ( H 1 + H 2 ) d λ − Z q 2 1 p k 2 2 λ 2 + β 2 λ + ( H 1 − H 2 ) d λ ! = = 1 2 p k 2 1 k 2 2 " q k 2 2 ln p 1 + P ′ 1 2 p k 2 1 ! − q k 2 1 ln p 2 + P ′ 2 2 p k 2 2 !# . Using suitable branches of q k 2 1 , one gets function (2 .14) Φ = exp(2 k 1 k 2 ω 2 ) =  p 1 − 2 k 2 1 q 1 + β 1 2 k 1  k 2  p 2 + 2 k 2 2 q 2 + β 2 2 k 2  k 1 , whic h generates p olynomial in tegrals of motion K m and K ℓ . As an example, if k 1 = 1 and k 2 = 3 one gets third and fourth order in tegrals of motion K m and K ℓ resp ectiv ely . Of course, oscillator is one of the well-studie d superintegrable systems, whic h was earmark ed to illustrate generic construction only . Additional theorem (2.8 ) a llo ws us to get a h uge family of the n - dimensional sup erin tegrable systems , whic h ha v e to b e classified and studied. 6 2.2 Classification In o rder to classify sup erin tegrable system s a sso ciated with the addition theorem (2.8) w e hav e to start with a pair o f the Riemann surfaces C j : µ 2 = P j ( λ ) = f λ 2 + g j λ + h j , j = 1 , 2 , (2.16) where f = α H 1 + β H 2 + γ , g j = α g j H 1 + β g j H 2 + γ g j , h j = α h j H 1 + β h j H 2 + γ h j , and α , β and γ are real or complex n umbers. In order to use addition theorem (2.8 ) we fix last row of the St¨ ac k el matrix. Namely , substituting S 2 j ( λ ) = κ j (2.17) in to the (2.3-2.6) one gets ϑ 2 j = Z S 2 j ( q j ) d q j p j = Z κ j d λ j p P j = κ j f − 1 / 2 ln  µ j + 2 f λ j + g j 2 √ f  , (2.18) so the angle v ar iable ω 2 = 1 √ f ln  p 1 u 1 + P ′ 1 2 √ f  κ 1  p 2 u 2 + P ′ 2 2 √ f  κ 2  , P ′ j = dP j ( λ ) dλ     λ = v j ( q j ) , is the m ulti-v alued function on the desired algebraic argument K =  p 1 u 1 + P ′ 1 2 √ f  κ 1  p 2 u 2 + P ′ 2 2 √ f  κ 2 . If κ 1 , 2 are p ositiv e in teger, then K =  1 2 √ f  κ 1 + κ 2  K ℓ + p f K m  (2.19) is the generating function of p olynomial in tegrals of motion K m and K ℓ of the m -th and m ± 1-th order in the momen ta , resp ectiv ely . As an example we ha v e K m = 2 ( p 1 u 1 P ′ 2 + p 2 u 2 P ′ 1 ) , κ 1 = 1 , κ 2 = 1 , K m = 2 P ′ 2  2 p 2 u 2 P ′ 1 + p 1 u 1 P ′ 2  + 8 p 1 u 1 p 2 2 u 2 2 f , κ 1 = 1 , κ 2 = 2 , K m = 2 P ′ 2 2  3 p 2 u 2 P ′ 1 + p 1 u 1 P ′ 2  + 8 p 2 2 u 2 2  p 2 u 2 P ′ 1 + 3 p 1 u 1 P ′ 2  f , κ 1 = 1 , κ 2 = 3 , (2.20) where m = 1 , 3, m = 3 , 5 m = 3 , 7, b ecause P ′ 1 , 2 and f are linear functions on H 1 , 2 , whic h are the second order p olynomials on momen ta. 7 The corresp onding expres sions for the K ℓ lo ok like K ℓ = P ′ 1 P ′ 2 + 4 p 1 p 2 u 1 u 2 f , κ 1 = 1 , κ 2 = 1 , K ℓ = P ′ 1 P ′ 2 2 + 4 f p 2 u 2 ( p 2 u 2 P ′ 1 + 2 p 1 u 1 P ′ 2 ) , κ 1 = 1 , κ 2 = 2 , K ℓ = P ′ 1 P ′ 2 3 + 12 P ′ 2 p 2 u 2 ( P ′ 2 p 1 u 1 + P ′ 1 p 2 u 2 ) f + 16 p 1 u 1 p 3 2 u 3 2 f 2 . κ 1 = 1 , κ 2 = 3 . (2.21) The imp o sed condition (2.17) leads to some restrictions on the functions v j ( q j ) and u j ( q j ). In fact, substituting canonical v a riables (2.4 ) into the equations (2.16) we obtain the follow ing express ion for the St¨ ac ke l matrix S =       αv 2 1 + α g 1 v 1 + α h 1 u 2 1 αv 2 2 + α g 2 v 2 + α h 2 u 2 2 β v 2 1 + β g 1 v 1 + β h 1 u 2 1 β v 2 2 + β g 2 v 2 + β h 2 u 2 2       , det S 6 = 0 . (2.22) So, f or a giv en κ 1 , 2 expressions for ϑ 2 j (2.18) yield tw o differen tial equations on functions u, v and parameters β : S 2 j ( q j ) = κ j v ′ j ( q j ) u j ( q j ) = ⇒ κ j u j v ′ j = β v j + β g j v j + β h j , j = 1 , 2 . (2.23) F or the St¨ ac k el systems with rational or tr ig onometric metrics, w e ha v e to solv e these equations in the space of the truncated Lauren t or F ourier p olynomials, respective ly . Prop osition 1 I f κ j 6 = 0 e quations (2.23) h ave the fol lowing thr e e monom ial solutions I β = 0 , β h j = 0 , u j = q j , v j = q β g j κ j j , I I β g j = 0 , β h j = 0 , u j = 1 , v j = − κ j ( β q j ) − 1 , I I I β = 0 , β g j = 0 , u j = 1 , v j = κ − 1 j β h j q j , (2.24) up to c an o n ic al tr ansf o rmations. The fourth solution ( IV) is the c ombination of the first and thir d solutions for the differ ent j ’s. In order to prov e this fact w e can substitute u = aq m and v = bq k in to the (2.23) and divide resulting equation on q m + k − 1 abk κ = b 2 β q k +1 − m + q 1 − m bβ g + q 1 − k − m β h . Finally , w e differen tiate it by q and mu ltiply on q m 0 = − ( m − k − 1) b 2 β q k − ( m − 1) bβ g − ( m + k − 1) β h q − k . Suc h a s b 6 = 0 and k 6 = 0 one gets three solutions (2.24) o nly . 8 Then w e suppose that after some p oin t transformation x = z 1 ( q ) , y = z 2 ( q ) , p x = w 11 ( q ) p 1 + w 12 ( q ) p 2 , p y = w 21 ( q ) p 1 + w 22 ( q ) p 2 , (2.25) where w ij 6 = 0, kinetic part of the Hamilton function H 1 = T + V has a sp ecial form T = X  S − 1  1 j p 2 j = g 11 ( x, y ) p 2 x + g 12 ( x, y ) p x p y + g 22 ( x, y ) p 2 y , where g is a metric on a configuratio na l manifold. F or instance, if w e supp ose tha t T = X  S − 1  1 j p 2 j = p x p y , then one gets the f o llo wing alg ebraic equations w 11 w 21 =  S − 1  11 , w 12 w 21 + w 11 w 22 = 0 , w 12 w 22 =  S − 1  12 (2.26) and the partia l differential equations { x, p x } = { y , p y } = 1 , { p x , y } = { p y , x } = { p x , p y } = 0 . (2.27) on parameters α and functions z 1 , 2 ( q 1 , q 2 ), w k j ( q 1 , q 2 ). The remaining free parameters γ , γ h j , γ g j determine the corr esp o nding p oten tial part of the Hamiltonia n V ( x, y ). In fact, since integrals H 1 , 2 is defined up to the trivial shifts H k → H k + c k , our p ot ential V ( x, y ) dep ends on three arbitra r y parameters only . Summing up, in order to get all the sup erintegrable systems on a complex Eu- clidean space E 2 ( C ) asso ciated with the addition theorem (2.8) w e hav e to solve equa- tions (2.23 ,2.26,2.27) with resp ect to functions u j ( q j ), v j ( q j ), z 1 , 2 ( q 1 , q 2 ), w k j ( q 1 , q 2 ) and parameters α and β . Example 3 Let us consider second solution from the list (2.24) at κ 1 = 1 and κ 2 = 2. In this case equations (2 .26,2.27) hav e the follow ing partial solution S =  1 0 1 q 2 1 4 q 2 2  , x = √ q 1 q 2 , p x = √ q 1 q 2 p 1 + 1 2 √ q 1 p 2 , y = √ q 1 q 2 , p y = √ q 1 q 2 p 1 − q 2 2 2 √ q 1 p 2 . Adding p otential t erms o ne gets tw o Riemann surfaces C 1 : µ 2 = H 2 λ 2 − H 1 λ − γ 1 , and C 2 : µ 2 = H 2 λ 2 − 2 γ 2 λ + 4 γ 3 where µ j = p j , λ j = q − 1 j . Solutions of the corres p onding separated v aria bles a r e the second order St¨ ak el integrals of motion, whic h in ph ysical v ariables lo ok lik e H 1 = p x p y + γ 1 xy + γ 2 p xy 3 + γ 3 y 2 , H 2 = ( p x x − p y y ) 2 4 − γ 2 r x y − γ 3 x y . (2.28) It is new in tegrable systems, whic h is missed in the kno wn lists of sup erin tegrable systems [1, 6, 7, 8]. 9 In this case in tegrals of motion K 5 (2.20) and K 6 (2.21) are the fifth and sixth order p olynomials in the momen ta, respective ly . Of course, w e can try to get quartic, cubic and quadratic integrals of mot ion K 4 , K 3 and K 2 from the recurrence r elat io ns K 5 = { K 4 , H 2 } , K 4 = { K 3 , H 2 } , K 3 = { K 2 , H 2 } (2.29) and the equation { H 1 , K j } = 0, j = 4 , 3 , 2. Solving first recurrence equation one gets quartic integral of motion K 4 = − 4( xp x − y p y ) 2 ( p 2 x + γ 1 y 2 ) − 4 γ 2 2 xy + 8 p x ( xp x − y p y ) γ 2 √ xy + 16 x ( p 2 x + γ 1 y 2 ) γ 3 y . The other recurrence relations ( 2.29) hav e not p olynomial solutions. Of course, w e can try mo dify recurrence relatio ns { K 3 , H 2 } = K 4 + F 4 ( H 1 , H 2 ) , { K 2 , H 2 } = K 3 + F 3 ( H 1 , H 2 ) in order to get cubic and quadratic inte gra ls of motio n. How ev er, using ansatz K 2 = h 1 ( x, y ) p 2 x + h 2 ( x, y ) p x p y + h 3 ( x, y ) p 2 y + h 4 ( x, y ) w e can directly prov e that there is not the additional quadra t ic in tegral of motion, whic h commute with H 1 (2.28). 3 The Drac h systems In 193 5 Jules Drac h published t wo articles on the Hamiltonian systems with the third order in tegra ls of mot io n on a complex Euclidean space E 2 ( C ) with the following Hamil- ton function [2] H 1 = p x p y + U ( x, y ) . (3.1) 10 Up to canonical transformations x → ax and y → by the corresp onding p o ten tials lo ok lik e [7, 1 2 ]: ( a ) U = α xy + β x r 1 y r 2 + γ x r 2 y r 1 , where r 2 j + 3 r j + 3 = 0 , ( b ) U = α √ xy + β ( y − x ) 2 + γ ( y + x ) √ xy ( y − x ) 2 , ( c ) U = α xy + β ( y − x ) 2 + γ ( y + x ) 2 , ( d ) U = α p y ( x − 1) + β p y ( x + 1) + γ x √ x 2 − 1 2 , ( e ) U = α √ xy + β √ x + γ √ y , ( f ) U = α xy + β y 2 x 2 + 1 √ x 2 + 1 + γ x √ x 2 + 1 , ( g ) U = α ( y + x ) 2 + β ( y − x ) + γ (3 y − x )( y − 3 x ) 3 , ( h ) U = ( y + mx 3 ) − 2 / 3  α + β ( y − mx/ 3) + γ ( y 2 − 14 mxy 3 + m 2 x 2 9 )  , ( k ) U = αy − 1 / 2 + β xy − 1 / 2 + γ x , ( l ) U = α  y − ρx 3  + β x − 1 / 2 + γ x − 1 / 2 ( y − ρx ) . Suc h as Drac h made some assumptions on the fo r m of the third order integrals of motion K 3 (3.6) in the calculation it is not immediately clear whether the o bta ined list is complete. 3.1 Non-separable systems. Let us discuss the Drac h systems, whic h can not b e reduced to the St¨ ac k el syste ms b y an y p oin t transformatio n of v ariables. The first system (a) is non-St¨ ac k el system related to the three-particle p erio dic T o da lattice in the cente r- of-mass frame [1 2] and there are global action-angle v ariables [3]. The (h) system is reduced to the St¨ ac k el system by non-p o int cano nical transforma- tion and, therefore, existenc e of the third order inte gra l of motion is related with this non-p oint transformation [12]. Later this system ha s b een redisco ve red by Holt [4]. F or the (k) case in the Drac h pap ers [2] w e can find Hamiltonian H ( k ) 1 = p x p y + αy − 3 / 2 + β xy − 3 / 2 + γ x (3.2) and the following cubic integral of motion K ( k ) 3 = 6 w ( x, y )  ∂ H ∂ x p y − p x ∂ H ∂ y  − P ( p x , p y , x, y ) , (3.3) 11 where P = 3 p 2 x p y , w = − y . It is easy to prov e that { H ( k ) 1 , K ( k ) 3 } 6 = 0 and, therefore, we ha ve to suggest the p ossibilit y of a small mistake in the Drac h pap ers [2]. F ollow ing to [12], if w e solv e equation { p x p y + U ( x, y ) , K ( k ) 3 } = 0 with res p ect to U ( x, y ), then one gets our case ( k) H 1 = p x p y + αy − 1 / 2 + β xy − 1 / 2 + γ x. On the other hand, w e hav e pr ov en directly that the Ha milto nian H ( k ) 1 (3.2) has one second order integral of motion H ( k ) 2 = p 2 x − 4 β y 1 / 2 + 2 γ y (3.4) and has not cubic integral of motion. Moreo v er, it is easy to see that quadratic in te- grals of motion H ( k ) 1 , 2 (3.2,3.4) can not b e reduced to the St¨ ac k el integrals b y a ny p oint transformation of v a riables. Belo w w e do not consider (a) and (h) systems and consider (k) case in our no t ation only . 3.2 Classification. F or the Dr a c h systems κ 1 = κ 2 = ± 1 , 1 / 2 a nd Φ 2 = exp(2 p f ω 2 ) =  P ′ 1 2 √ f + p 1 u 1   P ′ 2 2 √ f + p 2 u 2  = = 1 4 f  K ℓ + p f K m  (3.5) ma y b e considered as the generating function of the p olynomial in tegrals of motion (2.20-2.21) K m = 2  p 1 u 1 P ′ 2 + p 2 u 2 P ′ 1  , m = 1 , 3 , (3.6) K ℓ = P ′ 1 P ′ 2 + 4 p 1 p 2 u 1 u 2 f , ℓ = 2 , 4 , of the m -th and ℓ -th order, resp ectiv ely . It’s clear that m = 1 , 3 a nd ℓ = 2 , 4, b ecause P ′ 1 , 2 and f are linear functions on H 1 , 2 , which are second order p olynomials on momen ta. W e hav e to underline, tha t w e use differen t κ 1 = κ 2 2 = ± 1 , 1 / 2 for the agreemen t of K m (3.6) with the initial Drac h integrals o f mot ion [2] only . One gets thir d order p olynomial in tegral of motion K 3 (3.6) if and only if P ′ 1 ( λ ) or P ′ 2 ( λ ) dep ends on H 1 or H 2 . It leads to the additional restrictions on α ’s and β ’s 2 X k =1 ∂ 2 P j ( λ ) ∂ H k ∂ λ 6 = 0 for j = 1 or j = 2 . (3.7) In order to get all the superintegrable systems on a complex Euclidean space E 2 ( C ) asso ciated with addition theorem (2.8) w e hav e to solv e equations (2.2 3,2.26,2.27) and (3.7) at κ 1 , 2 = 1 with resp ect to the functions u j ( q j ), v j ( q j ), z 1 , 2 ( q 1 , q 2 ), w k j ( q 1 , q 2 ) and parameters α and β . 12 Prop osition 2 Th e D r ach lis t of the St¨ ackel systems with the cubic inte gr a l of motion (3.6) asso ciate d with the addition the or em (2.8) is c omplete up to c anonic al tr ansfo r- mations of the extende d phase sp ac e. The results of corr esp o nding calculations ma y b e jo ined in to the table: C 1 , 2 (2.16) subs. (2.4) z 1 , 2 (2.25) S b µ 2 = H 1 λ 2 + ( H 2 + 2 α ) λ − β + 2 γ µ 2 = H 1 λ 2 + ( H 2 − 2 α ) λ − β − 2 γ µ j = p j q j z 1 , 2 = ( q 1 ± q 2 ) 2 4  q 2 1 q 2 2 1 1  c µ 2 = α 4 λ 2 +  H 2 + H 1 2  λ + γ µ 2 = α 4 λ 2 +  H 2 − H 1 2  λ − β λ j = q 2 j κ j = 1 2 z 1 , 2 = q 1 ± q 2 2  1 2 − 1 2 1 1  d µ 2 = H 2 λ 2 − √ 8( α + β ) λ + H 1 − 2 γ µ 2 = H 2 λ 2 − √ 8( α − β ) λ + H 1 + 2 γ µ j = p j z 1 = q 2 1 + q 2 2 2 q 1 q 2 z 2 = q 1 q 2  1 1 1 q 2 1 1 q 2 2  f µ 2 = H 2 λ 2 −  γ 2 − H 1  λ − α 4 − β 2 µ 2 = H 2 λ 2 −  γ 2 + H 1  λ − α 4 + β 2 λ j = q − 1 j κ j = − 1 z 1 = q 1 − q 2 2 2 √ q 1 q 2 z 2 = √ q 1 q 2  1 q 1 − 1 q 2 1 q 2 1 1 q 2 2  e µ 2 = H 1 λ 2 + 2( β + γ ) λ + H 2 + 2 α µ 2 = H 1 λ 2 − 2( β − γ ) λ + H 2 − 2 α µ j = p j z 1 , 2 = ( q 1 ± q 2 ) 2 4  q 2 1 q 2 2 1 1  k µ 2 = γ 2 λ 2 + ( β + H 1 ) λ + H 2 + α µ 2 = γ 2 λ 2 + ( β − H 1 ) λ + H 2 − α λ j = q j κ j = 1 z 1 = q 1 − q 2 2 z 2 = ( q 1 + q 2 ) 2 4 ( q 1 − q 2 1 1 ) g µ 2 = − γ 3 λ 2 +  H 1 2 + H 2  λ + α µ 2 = − γ 3 λ 2 − β 4 λ + H 2 4 − H 1 8 µ 1 = p 1 q 1 , λ 1 = q 2 1 µ 2 =2 p 1 , λ 2 = q 2 κ j = 1 2 z 1 , 2 = q 1 ± q 2 2  1 2 − 1 2 1 1  Similar to the o scillator a nd the Kepler problem, the Kepler c hange o f the time t → e t , 13 where d e t = v ( q ) dt, v ( q ) = det S det e S , (3.8) relates the Drac h systems (b),(d) and (e) with t he sys tems (c),(f ) and (k), resp ectiv ely . Here S are the St¨ ac ke l matrices for (b),(d) and (e) systems and e S are the St¨ ac k el matrices for (c),(f ) and (k) systems . Remind, tha t these mat r ices S and e S hav e differen t first row only , see [11, 1 3]. 3.3 In tegrals of motion Using definitions (3.6) we can prov e that integral of motio n K 4 (3.6) is the function on H 1 , H 2 and K 3 K 2 4 = 16 h 1 h 2 f 2 + ( K 2 3 − 4 h 1 g 2 2 − 4 h 2 g 2 1 ) f + g 2 1 g 2 2 . (3.9) Substituting this expression into the defin ition of Φ 2 (3.5) we can get in tegral K 3 as function on the action-angle v ariables I 1 , 2 = H 1 , 2 and ω 2 . As usual, p olynomial algebra of integrals of motion H 1 , 2 and K 3 follo ws from the canonical brac ke ts (1.1) { H 1 , H 2 } = { H 1 , K 3 } = { H 1 , K 4 } = 0 , { H 2 , K 3 } = δ K 4 , { H 2 , K 4 } = δ f K 3 , (3.10) { K 3 , K 4 } = F Z ( H 1 , H 2 , K 3 ) , where • δ = 4, F I = 16 f ( g 1 h 2 + g 2 h 1 ) − 4 g 1 g 2 ( g 1 + g 2 ) for b,c cases; • δ = − 2, F I I = K 3 + 32 f h 1 h 2 − 4( g 2 1 h 2 + g 2 2 h 1 ) for d,f cases; • δ = 2, F I I I = 4 f ( g 2 1 + g 2 2 ) − 16 f 2 ( h 1 + h 2 ) for e,k cases; • δ = 4, F I V = 2 f ( g 2 1 + 8 g 1 h 2 ) − 8 f 2 h 1 − 4 g 1 g 2 2 for g case. The difference in the v alues of δ is related with the difference in κ ’s whic h has b een defined by the Drac h in tegrals of motion [2]. Of course, w e can put κ 1 = κ 2 = 1 in all the cases suc h that δ = 2 and p o lynomials F lo ok as in (4.1). It reduces p olynomials P 1 , 2 in the table only . As ab ov e (2.29), we can try to find a no t her second order p olynomial in tegral of motion K 2 from the equations K 3 = { H 2 , K 2 } , { H 1 , K 2 } = 0 . Solutions of these equations K 2 = ±  2 p 1 p 2 u 1 u 2 + 2 v 1 v 2 f + v 1 g 2 + v 2 g 1  = ± K 4 − g 1 g 2 2 f (3.11) 14 ha v e b een found in [7] in f ramew ork of the Lagrangian formalism. The a lgebras of quadratic in tegrals of motion H 1 , 2 and K 2 ha v e b een considered in [1]. There is some opinion that all sup erinte gra ble systems with quadratic (linear) in te- grals o f mo t ion a re multise para ble, i.e allow s the separation of v ariables in the Hamilton- Jacobi equation in at least t w o differen t co ordinate systems on the configuration space [6]. The some of the Drac h system s ma y b e considered as coun terexamples asso ciated with the Lie surfaces [1]. Namely , three of the superintegrable Dra c h systems with quadratic in tegrals H 1 , H 2 , K 2 are separable in the one co ordinate system on the configuration space only . Prop osition 3 F or the (b) and (c) D r ach systems inte gr als of motion H 1 , K 2 ar e sep- ar able in the c o or dinates x = q 2 2 q 1 , y = q 1 q 2 , and the c orr esp ondi n g sep ar ate d r elations do not al lows us to get cubic inte gr als of motion. F or the (e ) and (g) c ases inte gr als of m otion H 1 , K 2 ar e sep a r able in the c o or dinates x = q 1 − q 2 2 , y = − ( q 1 + q 2 ) 2 4 and we c a n use the c orr es p onding sep ar a te d r elations to the c onstruction of the cubic inte gr als of motion. F or the (d ), (f ) and (k) c ases quadr atic in te gr als o f motion H 1 and K 2 do n ot sep ar able by the p oint tr ansformations. W e can prov e this Prop osition by using computer program from [5]. 3.4 The (l) system. Without lost of g eneralit y we can put ρ = − 3 in the (l) case. Substituting this Hamil- tonian in to the computer program from [5] one gets the separated v ar ia bles x = ( q 1 − q 2 ) 2 2 , y = ( q 1 + q 2 ) 2 2 and the corresp onding separated relations p 2 j = P j ( q j ) = − 4 αq 4 j ∓ 8 √ 2 γ q 3 j + 4 H 1 q 2 1 ∓ 4 √ 2 β q j + H 2 , j = 1 , 2 . (3.12) whic h give rise to one h yp erelliptic curv e µ 2 = P ( λ ) at µ = p j and λ = ± q j . The angle v ariable ω 2 = 1 2 Z q 1 d λ p P ( λ ) + 1 2 Z q 2 d λ p P ( − λ ) = 1 2 Z q 1 d λ p P ( λ ) − 1 2 Z − q 2 d λ p P ( λ ) (3.13) 15 is a sum of the incomplete elliptic in tegrals of the first kind on the common hyperelliptic curv e. According to [15, 2] there is addition theorem and additional cubic in tegra l of motion K 3 = 2( e P ′ 1 p 2 + e P ′ 2 p 1 ) , whic h lo oks lik e a s the Drac h in tegral (3.6), but in this case functions e P ′ 1 , 2 = ( q 1 + q 2 ) 2 ∂ ∂ q 1 , 2 P ( ± q 1 , 2 ) ( q 1 + q 2 ) 4 ha v e completely another algebro-geometric explanation. All the details will be pub- lished in the f o rthcoming publications. As sequ ence, the alg ebra o f in tegrals of motion H 1 , 2 and K 3 differs from the corre- sp onding algebras for other Drac h system s relat ed with another additio n theorem. As an example, the recurrence c hain K j +1 = { H 2 , K j } terminates on the fo ur t hs step only K 7 = { H 2 , K 6 } = − 480 K 4 K 3 + 256 H 2 1 K 3 − 768 α H 2 K 3 − 3072 β γ K 3 . The solution of t he in v erse recurrence chain lo oks like K 2 = p 2 y + 2 α x − 4 γ √ x. It is in teresting, that the algebra of quadratic in tegrals o f motion H 1 , 2 and K 2 is one of the standard cubic algebras [1] and w e do not explanation of this fact. 4 New s up erin tegr abl e systems on z ero-ge nus h y- p erelliptic curv es at κ 1 = 1 and κ 2 = 2 , 3 Let us put κ 1 = 1 and κ 2 = 2 in (2.23) and try to solv e equations (2.26)-(2.27). Here is one sup erin tegra ble sy stem w ith cubic additional in tegral K m (2.20) and quadratic in tegral K ℓ (2.21) V I I I = γ 1 (3 x + y )( x + 3 y ) + γ 2 ( x + y ) + γ 3 ( x − y ) , S = ( a b 1 1 ) , and sev en systems with the real p o t en tials V I = γ 1 (3 x + y )( x + 3 y ) + γ 2 ( x + y ) 2 + γ 3 ( x − y ) 2 , S =  a/q 1 b/q 2 1 /q 1 2 /q 2  , V (1) I I = γ 1 xy + γ 2 p x 3 y + γ 3 x 2 , S =  1 /q 1 1 1 /q 2 1 4 /q 2 2  , V (2) I I = γ 1 √ xy + γ 2 x 2 + γ 3 y x 3 , S =  0 1 1 /q 2 1 4 /q 2 2  , V (3) I I = γ 1 √ xy + γ 2 p x 3 y + γ 3 x 5 / 4 y 3 / 4 , S =  1 0 1 /q 2 1 4 /q 2 2  , V (4) I I = γ 1 xy + γ 2 y x 3 + γ 3 y 3 x 5 , S =  0 4 /q 2 1 /q 2 1 4 /q 2 2  , V (1) I V = γ 1 √ xy + γ 2 ( √ x − √ y ) √ xy + γ 3 √ xy ( √ x + √ y ) 2 , S =  1 q 2 2 / 4 1 /q 1 1  , V (2) I V = γ 1 xy + γ 2 ( x − y ) + γ 3 ( x + y ) 2 , S =  a/q 1 b 1 /q 1 1  16 for whic h in tegra ls of motion K m and K ℓ (2.20-2.21) are fifth and sixth or der p olyno- mials in the momenta. Solution of the equations K m = ±{ H 2 , K m − 1 } and { H 1 , K m − 1 } = 0 lo oks lik e K m − 1 = 4 p 1 p 2 u 1 u 2 (2 f v 2 + g 2 ) + 4 v 2 (2 f v 1 + g 1 )( f v 2 + g 2 ) + (4 f h 2 + g 2 2 ) v 1 = 4 µ 2 ( µ 1 P ′ 2 + P ′ 1 ) − (4 f h 2 − g 2 2 ) λ 1 − 4 h 2 g 1 . It is additional in tegral of motion, whic h is second o r der p o lynomial for the system with p oten tial V I I I and fourth order po lynomial in the momen ta for the other sys tems. No w w e presen t some sup erin tegrable St¨ ac k el systems at κ 1 = 1 and κ 2 = 3. Here is one system with cubic additio na l in tegra l K m (2.20) V (1) I I I = γ 1 ( x + 2 y )(2 x + y ) + γ 2 ( x + 2 y ) + γ 3 (2 x + y ) , S = ( a b 1 1 ) , and sev en systems with the real p o t en tials V I = γ 1 ( x + 2 y )(2 x + y ) + γ 2 ( x + y ) 2 + γ 3 ( x − y ) 2 , S =  a/q 1 b/q 2 1 /q 1 3 /q 2  , V (1) I I = γ 1 xy + γ 2 x 2 / 3 y 4 / 3 + γ 3 x 1 / 3 y 5 / 3 , S =  1 /q 1 0 1 /q 2 1 9 /q 2 2  , V (2) I I = γ 1 √ xy + γ 2 √ y x 5 / 2 + γ 3 y 2 x 4 , S =  0 1 1 /q 2 1 9 /q 2 2  V (3) I I = γ 1 √ xy + γ 2 x 4 / 3 y 2 / 3 + γ 3 x 7 / 6 y 5 / 6 , S =  1 0 1 /q 2 1 9 /q 2 2  , V (4) I I = γ 1 xy + γ 2 y 2 x 4 + γ 3 y 5 x 7 , S =  0 1 /q 2 1 /q 2 1 9 /q 2 2  , V (2) I I I = γ 1 ( x 2 − 5 x √ y + 4 y ) + γ 2 x √ y + γ 3 √ y , S =  2 q 1 2 q 2 1 1  , V I V = γ 1 ( x + 5 y )(5 x + y ) + γ 2 ( x − y ) + γ 3 ( x − y ) 2 , S =  a/q 1 b 1 /q 1 1  for whic h in tegrals of motion K m and K ℓ (2.20-2.21) are sev en th and eigh ts order p olynomials in t he momen t a . The a lgebra of integrals of motion H 1 , 2 and K m (2.20) is the fifth or sev en th order p olynomial alg ebra in terms of t he co efficien t s of t he h yp erelliptic curv es { H 2 , K m } = 2 K ℓ , { H 2 , K ℓ } = 2 f K m , { K m , K ℓ } = ± F Z , where p olynomial F Z dep ends on the type o f solution (2.24) o nly: F I = 2(4 f h 2 − g 2 2 ) κ 2 − κ 1  4 f ( κ 2 1 h 2 g 1 + κ 2 2 h 1 g 2 ) − g 1 g 2 ( κ 2 2 g 1 + κ 2 1 g 2 )  , F I I = 4(4 f h 2 − g 2 2 ) κ 2 − κ 1  4 f ( κ 2 + κ 1 ) h 2 h 1 − κ 1 h 1 g 2 2 − κ 2 h 2 g 2 1  ∓ K 2 m , F I I I = 4(4 f h 2 − g 2 2 ) κ 2 − κ 1  4 f ( κ 1 h 2 + κ 2 h 1 ) − κ 1 g 2 2 − κ 2 g 2 1  f , (4.1) F I V = 2(4 f h 2 − g 2 2 ) κ 2 − κ 1  4 f (2 κ 2 f h 1 − κ 1 h 2 g 1 ) − 2 κ 2 f g 2 1 + κ 1 g 1 g 2 2  . Here c hoice o f sign + or − dep ends on κ ’s. 17 As ab ov e, the St¨ ac k el transformatio ns (3.8) relate systems asso ciated with one ty p e of the solutions (2.24), whereas algebra o f integrals of motion is inv ar ian t with resp ect to suc h tra nsformations. The complete classification of suc h sup erintegrable systems r equires further in v esti- gations. 5 Conclus ion W e discuss an application of the addition theorem to construction o f a lgebraic integrals of motion fro m the multi-v alued actio n- angle v ariables. W e prop ose new a lg orithm to construction of t he superintegrable St¨ ac k el sys tems asso ciated with ze ro- gen us h yp erelliptic curv es. It allows us to prov e that the Drach classification of the St¨ ac ke l sys tems with cubic in tegral of motion ( 3.6) a sso ciated with the addition theorem (2.8) is complete. Moreo v er, we presen t some new t wo-dimens ional sup erin tegrable system s with third, fifth and sev en th order integrals of mot io n. The prop osed metho d may b e a pplied t o construction of the higher order additional p olynomial integrals of motion fo r the n -dimensional sup erin tegrable St¨ ac ke l systems on the differen t manifolds. On the o ther hand, we prov e t ha t there are some sup erin- tegrable systems, whic h miss out of this construction. It will b e in teresting to study a mathematical mec hanism of the app earance such sup erin tegrable systems. The researc h w as partia lly supp ort ed b y the RF BR grant 06- 01-001 4 0. References [1] C. Da sk alo y annis, K . Ypsilan tis, Unifie d tr e atment and classific ation o f sup erinte- gr able systems with inte gr als quadr atic in momenta on a two-dim ensional manifo l d , J. Mat h. Ph ys., v.47, 2 0 06, 042904 , 38 pag es, math-ph/04120 5 5. [2] J. Drac h. Sur l’int ´ egr ation lo gique des ´ equations de la d ynamique ` a deux varia b les: F or c e s c onservatives. I n t ˆ egr ales cubiques. Mouvem ents dans le plan. 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