Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces

Necessa r y conditio n s for classica l super-integrabilit y of a c ertain family of potent i als in constan t cur v ature spaces Andrzej J. Maciejewsk i 1 , Maria Przyb ylska 2,3 and Haruo Y oshida 4 1 J. Kepler Institute of A strono my , Univ ersity of Zielona Góra, Licealna 9, PL-65–4 17 Zielo na Góra, Poland (e -mail: maciejka@astro.ia.uz.zgora.pl) 2 T oru ´ n Centre for Astronomy , N. Copernicus Univ ersity , Gagarina 11, PL-87–1 00 T oru ´ n, Poland, (e-mail: Maria .Przyb ylska@a stri.uni.torun.pl) 3 Institute of Physics, U niversity of Zielona Góra, Licealna 9, PL-65–4 17 Zielo na Góra, Poland 4 National Astronomi cal Observatory , 2-21- 1 Osa wa, Mitaka, 181 -8588 T oky o, Japan, (e-mail: h.yo shida@na o.ac.jp) V ersion of: S eptember 20, 2018 Abstract. W e formulate the necessar y conditions for the maximal super-integrability of a certa in family of c la ssical potentials defined in the constant curv ature tw o- d imensional spaces. W e giv e examples of homogeneous potentials of degree − 2 on E 2 as well a s their equiv alents on S 2 and H 2 for which these necessary conditions ar e also sufficient. W e show explicit forms of the add itional first integrals which alw ays can be chosen polynomial with re spe ct to the momenta and which can be of an arbitrar y high degre e with respect to the momenta. Key words: super-integrable s y stems, integrability , Hamiltonian equations, d if feren t ial Galois integrability obstructions. P ACS num bers: 02.30.Ik; 45.20 . Jj; 02.40.Ky; 45.50.Jf 1 Introduction In this paper we con s ider classical H amiltonian syste ms with n de grees of freedom giv en b y H amiltonian function H ( q , p ) , where q = ( q 1 , . . . , q n ) are canonical coordinates and p = ( p 1 , . . . , p n ) are the canonical mome n t a. W e sa y that such a s ystem is maximally super- integrable if the Hamilton’s equations of motion d d t q i = ∂ H ∂ p i ( q , p ) , d d t p i = − ∂ H ∂ q i ( q , p ) , i = 1, . . . , n , (1.1) admit 2 n − 1 functionally inde pendent first integrals such that among the m n commute. Although such systems are high ly except ional they attract a lot o f attentions . Motiv a- tion for st udy of super -integ rable systems comes fr om the classical as w ell as from quan- tum physics, see, e .g., T empesta et al. [2004]. Classical maximal i nt egrability implies that all bou nded tr ajecto r ies are closed and the motion is periodic. In quantum me chanics maximal super-integrability means the e xistence of 2 n − 1 well defined, algebraically in- depend ent operators (including Hamiltonian) among t hem n pairwise commute see e. g. Gra v el and W inter nitz [2002], Rodrigue z and W internitz [200 2] and re ferences ther e in. F or quantum syst ems the maximal su p er-i ntegrability implies t he degene r acy of energy lev- els. The p roblem of construction of integ rable, super-integrable and maxima lly super- integrable quantum systems from the corre sponding classical ones is v er y complicated, see e.g. Hietarinta [ 1984 ], Gra v el and W inter nitz [2002], and i t will be no t considered in this paper . Recently , some remarkable famili e s of super-integrable syste ms were found. In T rembla y et al. [2009], T rembla y e t al. [2010] the autho rs introduced a family of quantum and classical sys- tems for which the classical Hamiltonian function in polar coordinates ( r , ϕ ) is giv en by H ( 0 ) n = 1 2 p 2 r + p 2 ϕ r 2 ! + V ( 0 ) n ( r , ϕ ) , (1.2) where th e potential V ( 0 ) n ( r , ϕ ) has the for m V ( 0 ) n ( r , ϕ ) : = a r 2 cos 2 ( n ϕ ) + b r 2 sin 2 ( n ϕ ) , (1.3) n is an integer , a , and b are parameters 1 . The syst em is integrable and it has the follo wing first integral G : = 1 2 p 2 ϕ + r 2 V ( 0 ) n ( r , ϕ ) . (1.4) In quantum v ersion momenta are replaced b y appropriate partial deriv ativ e operators. As it wa s s hown in T re mbla y et al. [200 9 ], for small integ er va lue s of n the quantum sy stem is supe r- inte grable, and moreov er the d egree with r e spect to the momenta o f the se cond additional first integral grows with n . On this basis a conjectur e that the s ystem is super- integrable for an arbitrary n w as formulated. L ate r it w as just ified that in fact the quantum syste m is sup er-i nt egrable for any integ er odd n in Quesne [2010] and ev en for rational n in Kalnins et al. [2010b]. For the classical sy stem in T rembla y et al. [2010] it w as shown that all bounded trajectories are closed for all integer and rational v alues of n and in Kalnins e t al. [2010 a] for ms of first integr als w e r e des cribed. The se cond example ne eds a mo r e detailed pres entation. W e consider a po int with the unit mass moving on a sphere S 2 ⊂ R 3 under influence o f Hooke forces. A Hooke centre located at point r ∈ S 2 generates a pote ntial field of forces. The val ue of the pot ential at point γ ∈ S 2 is V = α ( γ · r ) 2 , (1.5) 1 In the cited paper the potential function contains the harmonic oscillator ter m ω 2 r 2 /2. 2 where α is the intensity of the centre . In Borisov et al. [2009] the authors inv est igated the problem o f a point mass moving on a sphere S 2 in the field of odd number n = 2 l + 1 of Hooke cent r e s with equal inte nsities located in a gr e at circle at the v ortexes of t h e regular n -gon . W e can assume that the Hook e cent res are located at p o ints r k ; n = ( sin ϕ k ; n , cos ϕ k ; n , 0 ) , where ϕ k ; n : = 2 π k n for k = 1, . . . , n . (1.6) Thus, the potential has the for m V ( 1 ) n : = n ∑ k = 1 α sin 2 θ sin 2 ( ϕ + ϕ k ; n ) = α n 2 sin 2 θ sin 2 n ϕ . (1.7) In the abo v e w e used t he follo w ing t rigonometric identity n ∑ k = 1 1 sin 2 ( ϕ + ϕ k ; n ) = n 2 sin 2 n ϕ , see e.g. Jakubský e t al. [2005]. Thus denoting a : = α n 2 , w e can write the H amiltonian of the syste m in the following for m H ( 1 ) n = 1 2 p 2 θ + p 2 ϕ sin 2 θ ! + a sin 2 θ sin 2 n ϕ . (1.8) The system is integrable and the follo w ing function F 1 : = 1 2 p 2 ϕ + a sin 2 n ϕ (1.9) is a first inte g ral. In B orisov et al. [2009] it was s hown that this Hamiltonian system is super- integrable as it posses ses t he se cond additional first integral of deg ree either 2 n + 1 or 2 n + 2 with re spect to t he momenta. The proof of this fact is remarkably natural and simply . W e show later that it allo ws t o giv e s ev eral g e neralisations of the abov e tw o examples. The abov e examples ha v e the same, in some sens e nature. In Cartesian coordinates potential (1.3) has the for m V ( 0 ) n = a ( q 2 1 + q 2 2 ) n − 1 [ Re ( q 1 + i q 2 ) n ] 2 + b ( q 2 1 + q 2 2 ) n − 1 [ Im ( q 1 + i q 2 ) n ] 2 . (1.10) Hence it is a rational ho mo g eneous function of de gree − 2. As it is well kno wn a natural Hamiltonian system giv en b y H ( 0 ) : = 1 2 ( p 2 1 + p 2 2 ) + V ( q 1 , q 2 ) , (1.11) with a homogene o us pot ential o f de gree − 2 is integrable because it posse sses the following first integral F 1 : = 1 2 ( q 1 p 2 − q 2 p 1 ) 2 + ( q 2 1 + q 2 2 ) V ( q 1 , q 2 ) . (1.12) 3 For a point on a s phere w e h av e analogous poten t ial. Namely , Hamiltonian sy stem giv en b y H ( 1 ) n = 1 2 p 2 θ + p 2 ϕ sin 2 θ ! + 1 sin 2 θ U ( ϕ ) , (1.13) is integrable with the first integral F 1 : = 1 2 p 2 ϕ + U ( ϕ ) . (1.14) Considering the ab ov e tw o examples, w e can ask whether a potential of the prescribed for m is s uper-integrable. It appears that this quest ion is difficult if w e look for an effectiv e and computable necessary cond itions for the super-i ntegrability . In this pape r we conside r na t u ral Hamil t o nian systems with tw o degre e s of freedom defined on T ⋆ M wh e re M is a t w o dimensional manifold with a cons tant curvature met rics. More s pecifically , M is either sphe re S 2 , Euclidean plane E 2 , or the h y perbolic p lane H 2 . In order to cons ider those thre e cases simultaneously we will proceed as in Herranz et al. [2000], Rañada and Santander [1999] and w e define the following functions C κ ( x ) : =      cos ( √ κ x ) for κ > 0, 1 for κ = 0, cosh ( √ − κ x ) for κ < 0, (1.15) S κ ( x ) : =        1 √ κ sin ( √ κ x ) for κ > 0, x for κ = 0, 1 √ − κ sinh ( √ − κ x ) for κ < 0. (1.16) These functions satisfy the followi ng identities C 2 κ ( x ) + κ S 2 κ ( x ) = 1, S ′ κ ( x ) = C κ ( x ) , C ′ κ ( x ) = − κ S κ ( x ) . (1.17) W e consider natural syste ms Hamiltonian sys tems with pot ential V ( r , ϕ ) defined b y H ( κ ) = 1 2 p 2 r + p 2 ϕ S 2 κ ( r ) ! + V ( r , ϕ ) . (1.18) The form of the kinetic energy corresponds to the metric on M with const ant curv ature κ . Our aim is to distinguish a special class of super-integrable potent ials. I nspired b y the examples d iscussed abov e we consider po tentials o f the form V ( κ ) ( r , ϕ ) : = 1 S 2 κ ( r ) U ( ϕ ) . (1.19) These potentials are separable. In fact G : = 1 2 p 2 ϕ + U ( ϕ ) , (1.20) 4 is a first integral of the sys t em and w e hav e also H = 1 2 p 2 r + 1 S 2 κ ( r ) G . (1.21) In order to for mulate our main resu lt let us ass ume that the r e exist s ϕ 0 ∈ C such that U ′ ( ϕ 0 ) = 0 and U ( ϕ 0 ) 6 = 0. Und er t his assumption w e define the following quantity λ : = 1 − 1 2 U ′′ ( ϕ 0 ) U ( ϕ 0 ) . (1.22) The most import ant result of t his paper is for mulated in the following th e orem which giv es necessar y cond itions for the s u per-integrabili ty of sy stems (1.21 ) with pote n t ial (1.19). Theorem 1.1. Assume that potenti al V ( κ ) given by (1.19) satisfies the following assumption: ther e exists ϕ 0 ∈ C such that U ′ ( ϕ 0 ) = 0 and U ( ϕ 0 ) 6 = 0 . If V ( κ ) is super-integrable, then λ : = 1 − 1 2 U ′′ ( ϕ 0 ) U ( ϕ 0 ) = 1 − s 2 , for a certain n on-zer o rational number s. The neces sary conditions for t he s uper-integrabili ty giv en b y Theorem 1.1 are deduced from an analysis of the dif fere n t ial Galois group of the v ariational equations along the de- scribed particular so lution. He re w e refe r to our paper Maci e jewski et al. [2008] where t he reader will find a description of applic ations of the dif ferent ial Ga lois the ory to a study of the integrability and the super-integrabil ity as w e ll as an analysis of the case κ = 0. Indeed the stateme n t of Theorem 1.1 for t he case κ = 0 is just a rep hrase o f the previous result written in polar coordinates, as it will be s een in the next section. 2 Relation with kno w n necessar y conditions f or super-integrability For a systems on E 2 giv en b y natural Hamiltonian (1.11) with a homog eneous potential V ( q 1 , q 2 ) of deg r e e k equations of motion ha v e the for m d d t q i = p i , d d t p i = − ∂ V ∂ q i , i = 1, 2 . (2.1) One can look for their particular solution of the for m q ( t ) = ϕ ( t ) c , p ( t ) = ˙ ϕ ( t ) c , where c ∈ C 2 is a non-zero v e cto r , and ϕ ( t ) is a scalar function. As it is easy to se e such a solution e xists provided that c is a non -zero solution o f g rad V ( c ) = c , and ϕ ( t ) satisfies ¨ ϕ + ϕ k − 1 = 0 . V ector c is called the Darboux point of potential V . Then the necessary conditions for the integ rabili t y and the sup er-i n t egrability which come from an analysis of the dif ferent ial Galois group o f the v ariational equations along the described particular solution are expresse d by means o f one eige nv alue of the Hes sian matrix V ′′ ( c ) , see Mo rales [1999] fo r t he inte grability obs t ructions, and Maciejewski et al. [2008] for super-integrability conditions. For matrix V ′′ ( c ) v e ctor c is an eigenv ector with the corres ponding eigenv alue ( k − 1 ) . Thu s the o t her eigenv alue is giv en by λ = T r V ′′ ( c ) − ( k − 1 ) = ∇ 2 V ( c ) − ( k − 1 ) . (2.2) 5 The ment ione d abov e ne cessary conditions for the integrability h av e the for m of arithmetic restrictions impos ed on λ . For example, in our previous paper Maciejewski et al. [2008], w e pro v e d, among o ther things the follo wing. Theorem 2.1. If a Hamiltonian system given by (1.11 ) , with a homogeneous pote n tial V ( q 1 , q 2 ) of degr ee k, | k | ≤ 2 is super-integ rable, then for each Darboux point c the corr esponding eigenvalue λ satisfies the following conditions: • if k = 2 , then λ = s 2 , where s is a non-zer o ratio n al number; • if k = 1 , then λ = 0 ; • if k = − 1 , then λ = 1 ; • if k = − 2 , then λ = 1 − s 2 , where s is a non-zer o ratio n al number . In the polar coordinates homogeneous p otential ha v e the for m V ( q ) = V ( r cos ϕ , r sin ϕ ) = r k U ( ϕ ) , (2.3) and a Darboux point is giv en b y ( c 1 , c 2 ) = c ( cos ϕ 0 , sin ϕ 0 ) , (2.4) where ϕ 0 is a solution of U ′ ( ϕ ) = 0 such that U ( ϕ 0 ) 6 = 0. The Laplacia n ∇ 2 V o f function V ( q ) takes the for m ∇ 2 V = ∂ 2 V ∂ q 2 1 + ∂ 2 V ∂ q 2 2 = 1 r  ∂ ∂ r  r ∂ V ∂ r   + 1 r 2 ∂ 2 V ∂ ϕ 2 in p olar coordinates, and for V = r k U ( ϕ ) , ∇ 2 V = k 2 r k − 2 U ( ϕ ) + r k − 2 U ′′ ( ϕ ) . Thus, as computed b y S ansaturio et al. [19 97 ], w e hav e λ = k 2 c k − 2 U ( ϕ 0 ) + c k − 2 U ′′ ( ϕ 0 ) − ( k − 1 ) = 1 + c k − 2 U ′′ ( ϕ 0 ) = 1 + U ′′ ( ϕ 0 ) k U ( ϕ 0 ) , (2.5) and subst itution k = − 2 reproduces λ in (1.22 ). T herefore the s t atement of Theorem 2.1 with k = − 2 giv es Theore m 1.1 immediately . S o the nov elty of The orem 1.1 is to confir m that the same statement holds, independe nt of the v alue of the cur v ature κ . 3 Proof of Theorem 1.1 According to Theore m 1.2 in Maciejewski et al. [2008], if the considered system is maximally super-integrable, t hen the identity compo n e nt the differential Galois group of t h e nor mal v ariational e q u ations along a particular s olution is just the identity . The assumptions in Theorem 1.1 guarantee that t 7 − → ( ϕ 0 , r ( t ) , 0, ˙ r ( t ) ) (3.1) 6 is a particular solution of the syst em provi d ed that r ( t ) satisfie s ¨ r = 2 U ( ϕ 0 ) C κ ( r ) S κ ( r ) 3 . (3.2) W e consider a particular solution with the energy e , i.e., w e fix e = 1 2 ˙ r 2 + U ( ϕ 0 ) S κ ( r ) 2 . (3.3) V ariational e quations along particular solution (3.1) ha v e the form     ˙ R ˙ Φ ˙ P R ˙ P Φ     =     0 0 1 0 0 0 0 S − 2 κ ( r ) 2 S − 4 κ ( r ) [ 2 κ S 2 κ ( r ) − 3 ] U ( ϕ 0 ) 0 0 0 0 − S − 2 κ ( r ) U ′′ ( ϕ 0 ) 0 0         R Φ P R P Φ     . (3.4) Since the motion takes place in the plane ( r , p r ) t he no r mal part of va riational equations is ˙ Φ = S − 2 κ ( r ) P Φ , ˙ P Φ = − S − 2 κ ( r ) U ′′ ( ϕ 0 ) Φ . W e rewrite this sys tem as one equation o f the second order ¨ Φ + a ( r , p r ) ˙ Φ + b ( r , p r ) Φ = 0, where a ( r , p r ) = 2 C κ ( r ) S κ ( r ) p r , b ( r , p r ) = S − 4 κ ( r ) U ′′ ( ϕ 0 ) . Making t h e follo wing change of the indepe n d ent v ariable t 7→ s = S κ ( r ) , w e transfor m this equation into a linear equation with rational coe f ficient s Φ ′′ + p ( s ) Φ ′ + q ( s ) Φ = 0, (3.5) where p ( s ) = ¨ s + ˙ s a ˙ s 2 = 1 s + es es 2 − B + κ s κ s 2 − 1 , q ( s ) = b ˙ s 2 = B ( λ − 1 ) s 2 ( es 2 − B ) ( κ s 2 − 1 ) , B : = U ( ϕ 0 ) . (3.6) Finally , w e perfor m one more transfor mation o f the inde pendent v ariable putting z = es 2 − B ( e − κ B ) s 2 . (3.7) After this transfor mation equation (3.5) reads d 2 Φ d z 2 + P d Φ d z + Q Φ = 0, (3.8) 7 where P = 2 z − 1 2 z ( z − 1 ) , Q : = λ − 1 4 z ( z − 1 ) . (3.9) This is the Gauss hy pergeometric equation for which the diff e rences of exponents at z = 0, z = 1 and z = ∞ are ρ = 1 2 , σ = 1 2 , τ = √ 1 − λ , (3.10) respectiv ely . Putting w = Φ exp  1 2 Z P ( ζ ) d ζ  , (3.11) w e transfor m t his equation to its reduced for m d 2 w d z 2 = r ( z ) w , (3.12) where r ( z ) = − 4 λ z ( z − 1 ) + 3 16 z 2 ( z − 1 ) 2 . (3.13) This equation coincides with equation (A.9) in Macieje wski et al. [2008] in which w e subst i- tute k = − 2. Thus, w e can apply Proposition A.3 and Proposition A.4 from Maciejewski et al. [2008] t o equation (3.12), and this exactly giv es t he t hesis of our theorem. 4 T r embla y-T urbiner-W inter nitz (TTW) system: A family of super- integrable systems in Euclidean flat space E 2 4.1 Finding the potential by use of the necessar y conditions for super-integrability In general, v alues of λ in (3.12) can depend on the angle ϕ 0 , or , o n t he Darboux point, and in order to be supe r- inte grable, the only ne cessary cond ition is that λ = 1 − s 2 , for a certain non-zero rational number s . How ev er , the r e is a special cla s s of s uper-integrable potentials, for which λ takes the sam e admissible va lue for all choic e s of the angle ϕ 0 . Indeed, the known su per-integrable pot ential (1.3) is strongly characterised by posses sing t his property as it is shown below . First, condition λ = 1 − s 2 with express ion (1.22) giv es − 1 2 U ′′ ( ϕ 0 ) U ( ϕ 0 ) = − s 2 . (4.1) Let us change the depend ent v ariable U ( ϕ ) , or the angular part of the potent ial, by U ( ϕ ) = 1 [ f ( ϕ ) ] 2 . Then at a point where U ′ ( ϕ 0 ) = 0, w e ha v e f ′ ( ϕ 0 ) = 0, and furthe rmo r e , − 1 2 U ′′ ( ϕ 0 ) U ( ϕ 0 ) = f ′′ ( ϕ 0 ) f ( ϕ 0 ) . (4.2) 8 Next w e force the additional requ ire me nt that λ takes t h e same v alue λ = 1 − s 2 , f o r all Darboux p oints U ′ ( ϕ 0 ) = 0. T hat is, w e ha v e t he re lation f ′′ ( ϕ 0 ) = − s 2 f ( ϕ 0 ) , (4.3) for all ϕ 0 such that f ′ ( ϕ 0 ) v anishes . Here s is a non-zero rational number , wh ich is indepen - dent of ϕ 0 . This requirement doe s not det er mine function f uniquely . How ev e r , w e can find examples of f satisfying this cond ition. The simplest and nai v e example is the function f which satisfies the dif fere ntial equation f ′′ ( ϕ ) = − s 2 f ( ϕ ) . (4.4) This is equiv alent t o assume that f satisfies ( 4.3 ) for all ϕ 0 ∈ ( 0, 2 π ) . The n w e find tw o independe nt solutions for f f 1 ( ϕ ) = cos ( s ϕ ) , f 2 ( ϕ ) = sin ( s ϕ ) and therefore U 1 ( ϕ ) = 1 cos 2 ( s ϕ ) , U 2 ( ϕ ) = 1 sin 2 ( s ϕ ) . (4.5) are the desired angular part of the supe r- inte grable potent ial. No t e that any linear combina- tion of U 1 ( ϕ ) and U 2 ( ϕ ) , namely U = aU 1 ( ϕ ) + b U 2 ( ϕ ) = a cos 2 ( s ϕ ) + b sin 2 ( s ϕ ) (4.6) has the s ame property , but unless ab = 0 the v alue of λ is λ = 1 − ( 2 s ) 2 , instead of λ = 1 − s 2 . Indeed , for the potent ial (1.3) w e hav e t h e follo wing Lemma 4.1. If ϕ 0 ∈ C is a solution of equation U ′ ( ϕ ) = 0 such that U ( ϕ 0 ) 6 = 0 , then λ = 1 − 1 2 U ′′ ( ϕ 0 ) U ( ϕ 0 ) = ( 1 − n 2 , if a b = 0, 1 − ( 2 n ) 2 , otherwise . (4.7) Pro of. For the potent ial (1.3) w e ha v e U ( ϕ ) = a cos 2 ( n ϕ ) + b sin 2 ( n ϕ ) , (4.8) so U ′ ( ϕ ) = 2 n a sin ( n ϕ ) cos 3 ( n ϕ ) − 2 n b cos ( n ϕ ) sin 3 ( n ϕ ) , (4.9) and U ′′ ( ϕ ) = 2 n 2 a 1 + 2 sin 2 ( n ϕ ) cos 4 ( n ϕ ) + 2 n 2 b 1 + 2 cos 2 ( n ϕ ) sin 4 ( n ϕ ) . (4.10) Let u s assume that a b 6 = 0. In a case when a 6 = 0 and b = 0, from (4.9) w e hav e sin ( n ϕ 0 ) = 0. Thus, U ( ϕ 0 ) = a , and, b y (4.10), U ′′ ( ϕ 0 ) = 2 n 2 a , so we ha v e relation (4.7) . In a s imilar w a y we show that th is for mula is val id in the case a = 0 and b 6 = 0. 9 If a b 6 = 0, then from (4.9) w e find that b = a tan 4 ( n ϕ 0 ) . Using this relation w e obtain U ( ϕ 0 ) = a cos 4 ( n ϕ 0 ) , and U ′′ ( ϕ 0 ) = 8 n 2 a cos 4 ( n ϕ 0 ) , and this finishes the proof. 4.2 Checking the super -integrability by separ a tion of variables As w e hav e shown that if n is a non-zero rational number , then p otential (1.3) satisfies the ne cessary condition for the s uper-integrabili t y , ne xt w e are g o ing t o p ro v e that t his potential is i nd eed super-integrable. W e demonstrate this g iving an explicit form of the second additional first integral F which is functionally ind e penden t tog ether with H ( 0 ) n and G giv e n b y (1.2) and (1.4) , respectiv ely . Ho w ev er , in order to d e monstrate how to d eriv e this first integ ral w e cons ider at first simplified case when b = 0 in po tential (1.3). Then Hamiltonian takes the for m H = 1 2 p 2 r + p 2 ϕ r 2 ! + a r 2 cos 2 ( n ϕ ) = p 2 r 2 + 1 r 2 G , (4.11 ) where G is the first integral g iv en by G = p 2 ϕ 2 + a cos 2 ( n ϕ ) . (4.12) In order to perfor m the explicit integration w e introduce as in B orisov et al. [2009] a new independe nt v ariable τ such that d τ /d t = 1/ r 2 . Then w e find that p r = r ′ r 2 , p ϕ = ϕ ′ , where pr ime denotes the d if fere ntiation with respect to τ . I n e f fect w e ha v e H = r ′ 2 2 r 4 + 1 r 2 G , and G = ϕ ′ 2 2 + a cos 2 ( n ϕ ) , i.e., w e effectiv ely separated v ariables Z d r r p 2 ( H r 2 − G ) = τ + C 1 , Z cos ( n ϕ ) d ϕ p 2 ( G cos 2 ( n ϕ ) − a ) = τ + C 2 . (4.13) The explicit forms o f t hese elementary inte grals are following 1 √ 2 G arctan s H r 2 − G G = τ + C 1 , (4.14) 10 and 1 n √ 2 G arcsin " r G G − a sin ( n ϕ ) # = τ + C 2 . (4.15) From (4.14) and (4.15) w e d educe that I = n √ 2 G ( C 2 − C 1 ) = arcsin " r G G − a sin ( n ϕ ) # − n arctan s H r 2 − G G , (4.16) is a firs t integral of the syste m. Using it we find an algebraic first integral. T o this end w e perfor m a sequence of t ransfor mations applying the follo wing for mulae arcsin z = − i ln  i z + p 1 − z 2  , arccos z = − i ln  z + p z 2 − 1  , arctan z = i 2 ln  1 − i z 1 + i z  . Using them and making some simplifications w e obtain I = − i ln ( p ϕ cos ( n ϕ ) p 2 ( G − a ) + i r G G − a sin ( n ϕ ) ! ( √ 2 G − i r p r ) n ( 2 H ) n / 2 r n ) . From t he abov e for mula we ded uce that I 1 = ( 2 H ) n / 2 q 2 ( G − a ) exp ( i I ) = 1 r n  p ϕ cos ( n ϕ ) + i √ 2 G sin ( n ϕ )  ( √ 2 G − i r p r ) n , (4.17) is a first integ ral o f the sys tem. For rational n this inte gral is an algebraic function of Cartesian vari ables ( q 1 , q 2 , p 1 , p 2 ) . If the considered system is real, then one w ould like to p osses s real fir s t integ rals. T aking the real and ima g inary parts of I 1 (assuming that all v ariables are real) we obtain real first integrals. Let us assume for simplicity that n is a positiv e integer . The n I 1 = r − n  p ϕ cos ( n ϕ ) + i √ 2 G sin ( n ϕ )  n ∑ k = 0  n k  ( 2 G ) ( n − k ) /2 ( − i ) k r k p k r , and from this w e obtain F 1 = Re I 1 = [ n / 2 ] ∑ k = 0 ( − 1 ) k  n 2 k  ( 2 G ) n − 2 k 2 p 2 k r r n − 2 k  p ϕ cos ( n ϕ ) + n − 2 k 2 k + 1 r p r sin ( n ϕ )  , F 2 = Im I 1 = [ n / 2 ] ∑ k = 0 ( − 1 ) k  n 2 k  ( 2 G ) n − 2 k − 1 2 p 2 k r r n − 2 k  2 G sin ( n ϕ ) − n − 2 k 2 k + 1 r p r p ϕ cos ( n ϕ )  . (4.18) Here [ x ] de n o tes the integer part o f x . W e note that alw a ys one of these first inte grals is a polynomial in moment a ( p r , p ϕ ) . For n ev en ex p ression ( 2 G ) ( n − 2 k ) /2 is a p olynomial and as result F 1 is a polyno mial in the mo me nta. S imilarly one can ded uce t h at for n odd F 2 is a polynomial in the momenta. L et us note that if we put neg ativ e n in (4.11), then the p o tential does not change, thus w e can assume t hat alw a ys n > 0 . The same is true also for more general fo r m of p otential (1.3). 11 For positiv e rational n = n 1 / n 2 from (4.17) also a polyno mial in the momenta first integral can be constructed. Namely w e consider the new first integral I 2 : = I n 2 1 = r − n 1  p ϕ cos ( n ϕ ) + i √ 2 G sin ( n ϕ )  n 2 ( √ 2 G − i r p r ) n 1 . S eparating real and imaginary p arts of this first integral w e find t hat for n 1 ev en and n 2 odd integral F 1 : = R e I 2 is polynomial in mo me nta. Moreov er , for odd n 1 integral F 2 : = I m I 2 is polynomial in momenta independ ently of the parity of n 2 . The described direct approach w orks per fe ctly in the same wa y for the general for m of the potential (1.3) and it giv es the following for m of the first integ ral I 1 = r − 2 n ( √ 2 G − i r p r ) 2 n h √ 2 G sin ( 2 n ϕ ) p ϕ − 2i ( G cos ( 2 n ϕ ) + b − a ) i . (4.19) Assuming that n is a positiv e integer , then I 1 = r − 2 n h √ 2 G sin ( 2 n ϕ ) p ϕ − 2i ( G cos ( 2 n ϕ ) + b − a ) i 2 n ∑ k = 0  2 n k  ( 2 G ) ( 2 n − k ) /2 ( − i ) k r k p k r , (4.20) and the real and imaginary parts o f this complex function giv e add itional first integrals F 1 = n ∑ k = 0 ( − 1 ) k  2 n 2 k  ( 2 G ) n − k p 2 k r r 2 ( n − k )  G sin ( 2 n ϕ ) p ϕ − 2 ( n − k ) 2 k + 1 ( G cos ( 2 n ϕ ) + b − a ) r p r  , F 2 = n ∑ k = 0 ( − 1 ) k  2 n 2 k  ( 2 G ) n − k p 2 k r r 2 ( n − k )  2 ( n − k ) 2 k + 1 sin ( 2 n ϕ ) r p r p ϕ + 2 ( G cos ( 2 n ϕ ) + b − a )  . In the abo v e formulae F 1 = Re ( I 1 ) /2 √ 2 G and F 2 = Im ( I 1 ) . Proceeding in the w a y similar to the previous case w e can also cons truct polynomial in the momenta first integrals for positiv e rational n . Obtained results can be rewritten immediately for Hamiltonian s ystems with indefinite flat for m of kinet ic energy , which in polar coordinates are giv en b y the following Hamilton function H = 1 2 p 2 r − p 2 ϕ r 2 ! + V , (4.21) with potential V = a r 2 cosh 2 ( n ϕ ) + b r 2 sinh 2 ( n ϕ ) . (4.22) Coordinates ( r , ϕ ) are related to the Cartesian coordinates b y th e formulae q 1 = r cosh ( ϕ ) , q 2 = r sinh ( ϕ ) . (4.23) This syste m is separable in ( r , ϕ ) coordinates with first integral G = 1 2 p 2 ϕ − a cosh 2 ( n ϕ ) − b sinh 2 ( n ϕ ) . (4.24) 12 One more add itional first integral has the for m I 1 = r − 2 n ( √ 2 G − r p r ) 2 n h √ 2 G sinh ( 2 n ϕ ) p ϕ + 2 ( G cosh ( 2 n ϕ ) + a + b ) i . (4.25) It can be obtained either b y a direct integ ration, or from integ ral (4.19 ) by substitutions ϕ → i ϕ , p ϕ → − i p ϕ G → − G , √ G → i √ G , b → − b . (4.26) One can construct also another first integral I 2 = r − 2 n ( √ 2 G + r p r ) 2 n h √ 2 G sinh ( 2 n ϕ ) p ϕ − 2 ( G cosh ( 2 n ϕ ) + a + b ) i (4.27) from integral (4.19) choosing the other square root of − G , i.e. making th e substitut ion √ G → − i √ G in (4 . 19 ). Then, for n ∈ N , e ither I 1 + I 2 , or I 1 − I 2 , is po lyn o mial in momenta first integral. In general case for po sitiv e rational n = n 1 / n 2 , eithe r F 1 : = I n 2 1 + I n 2 2 , or F 1 : = I n 2 1 + I n 2 2 is a first integral which is polyn o mial in momenta. 4.3 Other form of the additional integral, polynomial in momenta In the p revious se ction w e sho wed that the additional firs t inte gral is polynomial in po lar momenta p ϕ and p r . Here w e s how an approach which allo ws to demonstrate that t h is integral is e x p ressible in ter ms of po lynomials close ly related with the Cheb yshev polyno- mials. The obtained for m of the first integral shows that is rational in Cartesian v ariables ( q 1 , q 2 , p 1 , p 2 ) and p o lynomial in moment a ( p 1 , p 2 ) . Let us introduce d ouble spherical coordinates q 1 = r cos ϕ , q 2 = r sin ϕ , p 1 = p cos ψ , p 2 = p sin ψ . (4.28) Let us consider for example natur al Hamil t onian with p otential (1.10) for a = 1 and b = 0. In polar coordinates (4.28) Hamiltonian t akes the for m H = 1 2 p 2 + 1 r 2 cos 2 ( n ϕ ) , and Hamiltonian equations transfor m into ˙ r = p cos ( ϕ − ψ ) , ˙ ϕ = − p r sin ( ϕ − ψ ) , ˙ p = − 2 r 3 cos 3 ( n ϕ )  n − 1 2 cos ( ( n + 1 ) ϕ − ψ ) − n + 1 2 cos ( ( n − 1 ) ϕ + ψ )  , ˙ ψ = − 2 pr 3 cos 3 ( n ϕ )  n − 1 2 sin ( ( n + 1 ) ϕ − ψ ) + n + 1 2 sin ( ( n − 1 ) ϕ + ψ )  . (4.29) Let us note that tr ans fo r mation (4.28) is not canonical. In these coordinates Jacobi first integral takes the for m I 0 = r 2 p 2 sin 2 ( ϕ − ψ ) + 2 cos 2 ( n ϕ ) . 13 Let us look for an additional first integral of the for m I = p n sin ( n ψ ) + [ n / 2 ] ∑ i = 1 ( − 2 ) i p n − 2 i r 2 i cos 2 i ( n ϕ ) n − 1 − i ∑ m = i − 1 a i , m sin [ 2 ( n − m − 1 ) ϕ − ( n − 2 m − 2 ) ψ ] , where a im are unknown constant coef ficients . Substitution o f these for mulae into condition ˙ I = 0 y ields the following r e currence equation on a im [ i ( n − 1 ) + m + 1 ] a i + 1, m + 1 − [ i ( n + 1 ) + m + 2 ] a i + 1, m = ( n + 1 ) ( n − m − i − 1 ) a i , m − ( n − 1 ) ( m − i + 2 ) a i , m + 1 , i = 1, . . . ,  n − 2 2  , m = i , . . . , n − i − 2. (4.30) It has the follo wing so lution a i , m = ( m + 2 − i ) i ( n − m − i ) i − 1 ( 1 ) i − 1 ( 2 ) i − 1 , (4.31) where ( a ) n = a ( a + 1 ) · · · ( a + n − 1 ) is the Pochhammer symbol. W e d efine homogeneou s polynomials f n and g n and F n , G n in the following wa y ( q 1 + i q 2 ) n = f n + i g n , ( p 1 + i p 2 ) n = F n + i G n . From this definition th e con n e ction of polynomials f n , g n as w e ll as F n , G n with Chebyshev polynomials is ob vious, s ee e.g. S ection 2 in Freudenburg and Fr e udenburg [2009]. Then the first integral I in the Cartesian coordinates has the for m I = G n + [ n / 2 ] ∑ i = 1 ( − 2 ) i f 2 i n n − 1 − i ∑ m = i − 1 a i , m r 2 [ ( i − 1 ) ( n − 1 ) + m ] p 2 ( m − i + 2 ) S i , m , where S i , m : = Im h ( p 1 − i p 2 ) n − 2 m − 2 ( q 1 + i q 2 ) 2 ( n − m − 1 ) i . Let us notice that Im h ( p 1 − i p 2 ) α ( q 1 + i q 2 ) β i = F α g β − G α f β . Hence the abov e firs t integral is rational in ( q 1 , q 2 , p 1 , p 2 ) and p o lynomial in ( p 1 , p 2 ) . In the similar wa y one can treat the gen e ral potent ial (1.10), how ev er in this case the for m of the first integral is much more complicated. 4.4 Generalisation of TTW system to a system on S 2 and H 2 In this section w e consider syste ms g iv en by the H amiltonian function (1.18) with pote n t ial V ( κ ) n ( r , ϕ ) : = 1 S 2 κ ( r ) U ( ϕ ) , (4.32) 14 where U ( ϕ ) = a cos 2 ( n ϕ ) + b sin 2 ( n ϕ ) . This is a natural generalisation of the s ystem considered i n the previous subsection o n t o the s p aces with a constant no n-zero cur v ature 2 . It is not difficult to show that for this potential Le mma 4.1 applies. That is, if th e potential is super-integrable, the n n is non-zero rational number . W e s how that in fact those po t entials are supe r- inte grable, i .e ., that t he necessar y conditions of Theo rem 1.1 are also sufficient. T o this end it is enough t o perfor m the explicit inte gration s imila r to that done in the previous s ubsections. It giv es the follo wing for m o f the first integral I =  √ 2 G C κ ( r ) S κ ( r ) + i p r  2 n  √ G p ϕ sin ( 2 n ϕ ) + i √ 2 ( G cos ( 2 n ϕ ) + b − a )  , (4.33) and the first integral G t ake s t he for m G = 1 2 p 2 ϕ + U ( ϕ ) . Assuming that n ∈ N ⋆ : = N \ { 0 } , a , b ∈ R , and taking real and imaginary parts of (4.33) w e obtain t he follo w ing explicit for ms o f fir s t integrals I 1 = √ G Re I = n ∑ j = 0 ( − 1 ) j  2 n 2 j  ( 2 G ) n − j  C κ ( r ) S κ ( r )  2 n − 2 j − 1 p 2 j r  G p ϕ sin ( 2 n ϕ ) C κ ( r ) S κ ( r ) − 2 ( n − j ) 2 j + 1 p r ( G cos ( 2 n ϕ ) + b − a )  , I 2 = Im I √ 2 = n ∑ j = 0 ( − 1 ) j  2 n 2 j  ( 2 G ) n − j  C κ ( r ) S κ ( r )  2 n − 2 j − 1 p 2 j r  C κ ( r ) S κ ( r ) ( G cos ( 2 n ϕ ) + b − a ) + n − j 2 j + 1 p r p ϕ sin ( 2 n ϕ )  . Analogous calcula t ions can be repe ated for pot ential U ( ϕ ) = a cosh 2 ( n ϕ ) + b sinh 2 ( n ϕ ) . The Jacobi first integ ral for it takes the for m (4.24) and t h is one obtained from separation of v ariables in po lar coordinates is I =  √ 2 G C κ ( r ) S κ ( r ) + p r  2 n  − √ G p ϕ sinh ( 2 n ϕ ) + √ 2 ( G cosh ( 2 n ϕ ) + a + b )  =  − √ G p ϕ sinh ( 2 n ϕ ) + √ 2 ( G cosh ( 2 n ϕ ) + a + b )  2 n ∑ j = 0  2 n j  ( 2 G ) ( 2 n − j ) /2  C κ ( r ) S κ ( r )  2 n − j p j r . 2 During the final stage of preparation the w ork of Kalnins et al. [2010c] appeared where the reader will find other examples of super-integr able systems o n constant curvature s paces. 15 It can be also obt ained directly from (4.33) using the substitu t ions (4.26). One can also construct first integral u sing dif fe rent root of − G i.e. making the substitution √ G → − i √ G in (4.33 ) and w e obtain the following for m I =  √ G p ϕ sinh ( 2 n ϕ ) + √ 2 ( G cosh ( 2 n ϕ ) + a + b )  2 n ∑ j = 0 ( − 1 ) j  2 n j  ( 2 G ) ( 2 n − j ) /2  C κ ( r ) S κ ( r )  2 n − j p j r . Proceeding in the w a y similar to the previous cases we can also construct polynomial in the momenta first integrals for positiv e rational n . Ackno wl e dgements The authors are v er y th ank ful t o anonymous refere es for the ir re marks, comment s and sugges tions that allo w e d to impro v e cons iderably this paper . The first two authors are v ery grate ful t o Alexey V . Borisov , Iv an S. Mamaev and Alexan- der A. Kilin for v aluable discuss ions d uring their visit in Izhevsk. This r e search has been partially supporte d b y grant No. N N202 2126 33 of Ministr y of S cience and H igher Ed u cation of Poland and by EU funding for the Marie-Curie Research T raining N etw ork AstroNe t. The third author ’s w ork was partial ly s upported b y t he Grant-in-Aid for S cientific Re- search of Japan S ociety for t h e Promotion of S ciences (JSPS), No .18540 226. References A. V . Boriso v , A. A. Kilin, and I. S. Ma maev . 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