Explicit Construction of First Integrals with Quasi-monomial Terms from the Painlev{e} Series

Explicit Construction of First In tegrals with Quasi-monomial T erms from the P ainlev ´ e Series Christos Efth ymiop oulos 1 , T assos Boun tis 2 , and Thanos Manos 2 1 R ese ar ch Center for Astr onomy and Applie d Mathematics, A c ademy of Athens, Sor anou E fessiou 4, 115 27 Athens, Gr e e c e 2 Center for R ese ar ch and Applic ations of Nonline ar Systems, Dep artment of Mathematics, University of Patr as Abstract The Painlev ´ e and weak Painlev ´ e conjectures hav e b een used widely to identify new in- tegrable nonlinear dynamica l systems. F or a system which passes the Painlev ´ e test, the calculation of the integrals relies on a v ariety of metho ds which a re indep enden t fro m Painlev´ e analys is. The present pap er pr opo ses an explicit algo rithm to build first int egrals of a dynamica l system, expressed as ‘quasi- po lynomial’ functions, from the infor mation provided solely by the Painlev´ e - Laurent s eries solutions of a system of ODEs . Res tr ictions on the num b er and form of quasi-mo nomial terms appear - ing in a quasi-p olyno mial int egral are obtained by an applicatio n o f a theor em by Y os hida (1 983). The int egrals ar e obtained by a pro per balancing of the co efficients in a quasi- po lynomial function sele c ted as initial a nsatz for the integral, so that all dep endence on p ow ers of the time τ = t − t 0 is elimina ted. Both r ight and left Painlev ´ e series are useful in the metho d. Alternatively , the metho d can be used to show the non-ex istence of a quasi-p olyno mial first integral. Examples from sp ecific dynamical systems are given. 1 In tro d uction The present pap er deals w ith autonomous dynamical systems describ ed by ordinary differen- tial equatio n s of the form ˙ x i = F i ( x 1 , x 2 , ..., x n ) , i = 1 , 2 , ..., n. (1) where the functions F i are o f the form F i = N X j = 1 a ij x Q ij (2) with x ≡ ( x 1 , x 2 . . . x n ), Q ij ≡ ( q ij 1 , q ij 2 , . . . , q ij n ), x Q ij ≡ Q n k =1 x q ij k k , and the exp onen ts q ij k are assumed to b e rational n um b ers, i.e., the r.h.s. of Eq.(2) is a sum of quasimonomials (Goriely 19 92). A question of particular in terest concerns the existence of first inte gr als of the sys tem of ordinary differen tial equations (1). A first in tegral is a function I ( x i ) satisfying the equalit y dI /dt = ∇ x I · ˙ x ( t ) = 0, where x ( t ) ≡ ( x 1 ( t ) , . . . , x n ( t ) is an y p ossible solution of (1). First in tegrals are imp ortant b ecause they allo w one to constrain the orb its on manifolds of di- mensionalit y low er than n . In particular, a system (1) is called completely integrable if it admits n − 1 indep end en t and single-v alued first inte grals. I ntegrable sys tems exhib it r egular dynamics, wh ile th e lac k of a su fficien t n u mb er of first int egrals very often results in complex, c haotic dynamics. 1 The questio n of the existence of an algorithmic metho d whic h can d etermine a ll the fi rst in tegrals of (1) is an imp ortan t op en problem in the theory of ordinary differen tial equations. Most relev ant to this question are th e metho ds of a) direct searc h or metho d of undetermined co efficien ts (e.g. Hieta r in ta 1983, 1987) b ) normal forms and f ormal in tegrals (see Arnold 1985, Haller 1999 and Gorielly (2001) for a review), and c) Lie group symmetry metho ds (e.g. Lakshmanan and Sen thil V elan (199 2a,b), Marcelli and Nucci (2 003)). Ev en more difficu lt is the question of an alg orithmic metho d probin g t he in tegrabilit y of Eqs.(1). The most relev an t metho d here is si ng u larity analysis . According to the ‘Pai nlev ´ e conjecture’ (Ablo witz, Ramani and Segur 1980 ), a system p ossessing th e Painlev ´ e prop ert y should b e in tegrable. T he P ainlev ´ e p rop ert y means that any global solution of (1) in the complex time plane should b e free of mov able cr itical p oints other than p oles. According to the ‘w eak-P ainlev ´ e conjecture (Ramani et al. 1982, Grammaticos et al. 198 4, Ab enda et al. 2001), ho wev er, certain t yp es o f m o v able branc h points are compatible with integ rabilit y . Algorithms p ro vidin g necessary cond itions for a sys tem to b e Painlev ´ e are a) the classical ARS test (Ablo w itz et al. 1980) b) the p ertur b ativ e Pa inlev ´ e - F uc hs test (F o rdy and Pic kering 1991) wh ic h examines the role of n egativ e resonances, and c) the generalized P ainlev ´ e test of Goriely (1992) w h ic h introdu ces co ord inate transformations clarifying th e nature of the singularit y structure in the complex time domain. On the other hand , there is no algorithm, to the present, determinin g sufficien t conditions for a system to b e P ainlev ´ e. The most imp ortan t ob s tacle is the searc h for essent ial singularities, whic h are not detectable by any of the abov e P ainlev ´ e tests. The presen t pap er e xplores the follo wing question: Is it p ossib le to reco v er the first inte- grals o f a system b y the information pro vided solely b y s ingularit y analysis of its differen tial equations? Our answ er is p artially affirmativ e. W e cannot circum v ent the difficult y con- cerning the c h oice of initial ansatz for the functional form of the integ r al. T he most natural c hoice for quasi-p olynomial equ ations (2) is to consider also a quasi-p olynomial ansatz for th e in tegral, with undetermined quasi-monomial co efficients. This freedom in th e initial ansatz notwithstanding, we sho w in the p resen t pap er that singularit y analysis giv es ind eed the remaining in formation n eeded to reco v er the integ r als. First, as prop osed by Ro ek aerts and Sc h warz (1987), a theorem by Y oshida (1983) on the r elation b et w een Ko walevski exp onen ts and weig h ts of wei gh ted-homogeneous integ rals, can b e used to imp ose restrictions on the degree of the quasi-monomial terms in the in tegral b y analysing the resonances foun d by the Pai nlev ´ e metho d . No w, Y oshida’s theorem for w eighte d-homogeneous in tegrals is app licable on tw o conditions: If b i τ − λ i is a balance of the system ( τ = t − t 0 is the time aroun d the sin gu larity t 0 ), and I is a w eighte d homogeneous in tegral, the theorem holds if a) ∇ I ( b i ) is fin ite, and b ) ∇ I ( b i ) 6 = 0. These conditions shall b e refered to as ‘Y osh ida’s conditions’. The latter imp ose a sev ere restriction in the searc h for first in tegrals, b ecause one cannot sp ecify in adv ance whether these cond itions are satisfied unt il the in tegrals are determined. In conclusion, the theorem of Y oshida has o nly indicat iv e p o wer as regards restrictions on the degree of quasimonomial terms in the in tegral. On the other h and, an analysis of the integrable Hamiltonian systems of t wo d egrees of freedom with a p olynomial p oten tial giv en by Hiterinta (1983) shows that they all satisfy Y oshid a’s condition wh en the balance is tak en equal to the ‘principal’ b alance, in whic h all the b i are differen t from zero. Y et, the exten t of applicabilit y of this result to other t yp es of systems is unknown. In fact, w e w ere able to fi nd also a coun terexample concerning a Hamiltonian prop osed b y Holt (1982). Assuming a qu asi-p olynomial functional form of th e int egral, say Φ( x ; c 1 , c 2 , ..., c M ),as 2 ab o ve, with u ndetermined qu asi-monomial co eficien ts c i , i = 1 , . . . M , the question no w is whether it is p ossible to d etermine the co efficien ts c i b y the series deriv ed via singularity analysis. The answ er to this question is affirm ativ e. Namely , b y the usual P ainlev ´ e tests, the P ainlev´ e-t yp e s eries so lutions around mo v able singularities are fi r st identified x i ( τ ) = 1 τ λ i ∞ X m =0 b m τ m , i = 1 , . . . n (3) Then, the series (3) is substitute d in to the quasi-p olynomial function Φ( x ; c 1 , c 2 , ..., c M ). Th e resulting exp ression is a Puisseux series, i.e., a series in rational pow ers o f τ Φ( x 1 ( τ ) , . . . , x n ( τ ); c 1 , c 2 , ..., c M ) = τ q /p ∞ X m =0 d m ( b i ; c i ) τ m/p (4) with q , p, m in tegers. Th e co efficien ts d m ( b i ; c i ) are nonlinear f unctions of the co efficien ts b i (determined b y (3), i.e., b y sin gu larity analysis), and lin ear functions of the un determined co efficien ts c i . But the function Φ is an int egral of the system if it is constan t along all the solutions of the sy s tem, includin g (3). As a result, the functions d m satisfy the set of linear equations d m ( b i ; c i ) = 0 , i = 0 , 1 , . . . , i 6 = − q (5) This is an infinite num b er of homogeneous linear equations with a finite num b er of u nkno wns (the coefficien ts c i ). If M is the num b er of unknown co efficien ts, Eq.(5) can b e written as A ( b i ) · C = 0 (6) where C = ( c 1 , . . . , c M ) T and A ( b i ) is a matrix with M column s and an in finite n u m b er of lines. Th e en tries of A dep end only on the co efficien ts b i whic h were previously determined b y sin gularit y analysis. In this represen tation, th e first in tegrals are functions with quasi- monomial coefficien ts give n b y the basis v ectors of k er ( A ) (o r linear com b inations of them). In computer algebraic implemen tations of the metho d, w e w ork on a sub -matrix A f defined b y a fi nite num b er of lines, wh ic h is equal or larger than M . A b asis for the su bspace k er ( A f ) is determined b y the sin gular v alue decomp osition algorithm. Then, it is c hec ked w ith direct differen tiation that the resulting expression is an in tegral. Th is completes the determination of al l quasi-p olynomial first integ rals for the giv en system. This metho d was implemente d in a n u mb er of examples presen ted b elow. F ollo wing some preliminary notions exp osed in section 2, the results are p resen ted in section 3, along with v arious details and imp licatio ns in the implemen tation of the algorithm. Section 4 summarizes the main conclusions of the presen t study . 2 Preliminary notions F ol lo wing Y oshida (1983) , a system of the form (1) is called scale-in v ariant if the equations remain in v ariant under the scale transformation x i → a λ i x i , t → a − 1 t for some λ i . Th en, the system (1) has exa ct sp ecial solutions of the form x i = b i τ λ i (7) 3 where b i , i = 1 , . . . n is an y of the sets o f r o ots of the system of al gebraic equatio ns F i ( b 1 , b 2 , ..., b n ) + λ i b i = 0 and the time τ = t − t 0 is co nsidered around any mo v able singularit y t 0 in th e complex time plane. Any solution of the form (7) is called a ‘balance’. If th e system (1) is scale-in v ariant and the fun ctions F i are of th e f orm (2), th e exp onen ts λ i asso ciated with any of the balances are rational n um b ers. It should b e stressed that the definition ab o v e d o es not r equire that all the b i b e non-zero. Balances of the form x i ∼ 0 /τ λ i , for some i , are al so considered. T he latter remark is essen tial in ord er to a v oid the confusion whic h is sometimes made b et ween ‘balances’ and ‘dominan t terms’ in the ARS test (see e.g. th e discuss ion b et ween Steeb et al. (1987 ) and Ramani et al. (1988)), and the asso ciated d ifference b et ween ‘Kow alevski exp onent s’ an d ‘resonances’. In th e stand ard ARS algorithm (Ablo witz et al. 1980) the solutions (7) a rise b y the definition of the dominant b eha viors, i.e., whic h is the first step in the implemen tation of the algorithm. The second step in the ARS algorithm is to lo ok for series solutions th at we call ’Pa inlev ´ e series’. These are expansions of the form (3) s tarting w ith dominan t terms of the form (7). They are Laurent (T a ylor) series when the λ i ’s are int egers (p ositiv e int egers), otherwise they can b e series in r ational p o w ers of τ , wh ich are called ‘Puisseux series’. T o build up the series, one first sp ecifies the resonances, i.e. the v alues of r for whic h the co efficien ts of the terms τ r − λ i in the series are arbitrary . In the case when the coefficient s of the balance b i are all non-zero, the resonances are equal to the eigen v alues of the Ko walevski matrix K ij = ( ∂ F i /∂ x j + δ ij λ i ) | x i = b i whic h are called ‘Kow alevski exp onent s’. If, how ev er, some of the b i s are equal to zero, then the r esonances are not equal one to one to the Kow alevski exp onent s, b ut s ome resonances differ from the corresp onding Ko wa levski exp onents by a quan tit y equal to the difference b et ween the exp onen t λ i in th e balance and the exp onent of the first non-zero dominant term in the Laur en t-Puisseux series of x i ( τ ), as sp ecified in the first step of the ARS alg orithm (Ramani et al. 1988 ). A t this p oin t, we are not in terested in whether the system p asses the generalize d P ainlev´ e (or w eak P ainlev ´ e) test. This means that w e d o not require that al l the solutions of the system (1) ca n b e written lo cally (around a m o v able singularit y) in the form (3), or that there is at least one solution of th e form (3) which contai ns n arbitrary constan ts (including t 0 ). On the other hand, w e do c hec k the compatibilit y cond itions to ensure that no loga rithms enter in the series. As regards p ositiv e resonances, compatibilit y is fulfilled automatically for scale- in v ariant systems. In s ummary , w e a re int erested only that the system hav e sp ecial solutions of the form (3 ), but n o other claim on it b eing P ainlev ´ e or not is required. Thus, the results are v alid also for partially in tegrable systems, i.e., systems with a n umb er of first in tegrals smaller than n . Let us no w assume that (1) p ossesses a weig h ted - h omogeneous fi rst in tegral Φ of we igh t M , i.e . an integ ral function Φ which satisfies the relation: Φ( a λ 1 x 1 , a λ 2 x 2 , ..., a λ n x n ) = a M Φ( x 1 , x 2 , ..., x n ) (8) for s ome M . Then we hav e the follo win g Theorem 1 (Y oshida 1983 ): If, for a p articular b alanc e (7) the fol lowing c onditions hold: a) ∇ Φ( b i ) is finite, and b ) ∇ Φ( b i ) 6 = 0 , then M is e qual to one of the Kowalevski exp onents asso ciate d with that b alanc e . 4 There has b een a n um b er of theorems in the literature linking K ow alevski exp on ents with the weigh ts of homogeneous or quasi-homogeneous in tegrals of a sys tem. Some charac teris- tic pap ers on th is sub ject are Llibre an d Zh ang (20 02), Tsygvinsev (2001), Goriely (1996) and F urta (1996 ). How ever, the application of Y oshida’s original theorem in the searc h for w eighte d-homogeneous integ rals seems to b e the most p ractical. F urthermore, Y oshida’s th e- orem can b e reformulat ed in an int eresting w a y: consider the Pa inlen ´ e series starting with one of the balance s (7). It follo w s that the series can b e w r itten a s: x i = x iE + R i = b i τ λ i + ... + A i τ r − λ i + R i (9) where r is the m axim um Ko wale vski exp onent asso ciated with this balance an d A i is the corresp onding arbitrary co efficien t en tering in the P ainlev´ e series (9) f or the v ariable x i . The sum x iE = b i τ λ i + ... + A i τ r − λ i will b e called essential p art of the series and the remaining part R i r emainder of the series. The remainder R i starts with terms of degree r − λ i + 1 /p where p is th e denominator of λ i written as a ratio nal λ i = q /p with q , p copr im e in tegers. A quasi-p olynomial in tegral Φ has the form Φ = X c k 1 ,k 2 , ...,k n n Y i =1 x k i i (10) where the exp onents k i are rational num b ers . If the in tegral Φ is wei gh ted-homogeneous of w eight M , w e ha ve n X i =1 k i λ i = M (11) due to (8). T aking in to accoun t the fact that the r emainder R i starts with terms of degree O ( τ r − λ i +1 /p ), it follo ws that the con tribu tion of R i in x k i i is in terms of degree O ( τ − λ i k i + r +1 /p ) or higher. This means that if the series (9) is substituted in the integ ral Φ( x i ), the con tribu- tion of R i in Φ( x i ) is of degree O ( τ − P λ i k i + r +1 /p ) = O ( τ − M + r +1 /p ) or higher. But r ≥ M . Th us, the remainder R i con tribu tes only to terms of p ositive d egree in τ . On the other hand , since Φ is an in tegral, the time τ as a denominator must b e eliminated in Φ( x i ). But sin ce there are no negativ e pow ers of τ generated in Φ by R i , it f ollo ws that all the n egativ e p o w ers of τ are already eli minated by sub stituting the exp ression x iE alone in to Φ. Hence, we ha v e the follo wing Prop osition 2: If the c onditions of Y oshida’s the or em hold for a weighte d-homo gene ous inte gr al of (1) and a p articular b alanc e (7), then the expr ession Φ( x iE ) , wher e x iE ar e the essential p arts of Painlev´ e series x i ( τ ) initiate d with the same b alanc e , do es not c ontain si n- gular terms in τ . Consider next the case when the f unctions F i in (1) are n ot homogeneous. By the restrictions imp osed b y (2 ), it follo ws that the fun ctions F i can be decomp osed in sums of the form F i = F ( m i 0 ) i + F ( m i 0 +1 /p ) i + ... + F ( m i 0 + q /p ) i (12) where the functions F ( j ) i are homogeneous of degree j , w ith j, m i 0 rational, p, q in teger, and p is the denominator in the s implest fraction giving m i 0 . In this case, if the system (1) has 5 P ainlev´ e t yp e solutions, the dominan t b eh aviors and associated resonances of t hese solutions are determined by the homogeneous term of the highest d egree F ( m i 0 + q /p ) i . On the other hand, the functions F ( j ) i , j < m i 0 + q /p m u st hav e a sp ecial form to ensure that compatibilit y conditions are fu lfilled and the ser ies solution is of the Pai nlev ´ e typ e. Finally , as regards p oten tial fi r st in tegrals, th e assumption that they co nsist of a sum of quasi-monomial terms implies that they can also b e wr itten as sum s of the form (12). The selection of terms in the quasi-p olynomial inte gral can b e determined by Y oshida’s theorem (or prop osition 2) implemen ted in the scale-in v ariant systems ˙ x i = F ( m i 0 ) i ( x i ) (13) and ˙ x i = F ( m i 0 + q /p ) i ( x i ) (14) resp ectiv ely (Nak aga wa 2002 ). 3 Explicit c on stru c tion of in tegrals with quasi-monomial terms 3.1 An elemen tary example Consider the t wo-dimensional nonlinear system ˙ x 1 = x 2 , ˙ x 2 = − x 1 − 3 x 2 1 (15) The o nly first int egral of this system Φ = x 2 1 + x 2 2 + 2 x 3 1 (16) can b e r eco v ered b y elemen tary means. How ever, we will use this example to illustrate the steps used b y the present metho d. Th e corresp onding homogeneous system con taining the terms of maxim u m d egree ˙ x 1 = x 2 , ˙ x 2 = − 3 x 2 1 (17) has the unique balance x 1 = − 2 /τ 2 , x 2 = 4 /τ 3 , with Ko w alevski exp onen ts (equal to reso- nances) − 1 and 6. Com b atibilit y conditions are fulfilled for the system (15), which admits the Lauren t series solution x 1 ( τ ) = − 2 τ 2 − 1 6 − 1 120 τ 2 + a 4 τ 4 + O ( τ 6 ) x 2 ( τ ) = 4 τ 3 − 1 60 τ + 4 a 4 τ 3 + O ( τ 5 ) (18) where a 4 is an arbitrary p arameter. F ol lo wing Y o shida’s theorem, w e sh all lo ok for an inte gral of th e system (15) b y requesting that this in tegral b e a sum of w eight ed- homogeneous fun ctions of w eight not higher than M = 6. Sin ce the equations (15) are p olynomial, the int egral will also b e assumed p olynomial. According to the defin ition of th e weigh ted-homogeneous functions (8), th e undetermined in tegral con tains terms of th e form x q 1 1 x q 2 2 the exp onents of whic h are r estricted by the relation 2 q 1 + 3 q 2 ≤ 6. Th is lea v es only s ix p ossibilities, namely ( q 1 = 1 , q 2 = 0), ( q 1 = 0 , q 2 = 1), 6 ( q 1 = 2 , q 2 = 0), ( q 1 = 1 , q 2 = 1), ( q 1 = 0 , q 2 = 2), ( q 1 = 3 , q 2 = 0). Th us the integ ral is assumed to ha v e the form Φ = c 10 x 1 + c 01 x 2 + c 20 x 2 1 + c 11 x 1 x 2 + c 02 x 2 2 + c 30 x 3 1 (19) Up to no w , the steps are exactly as prop osed b y Ro ek aerts and Sch wa rz (1987) . A t this p oint, ho wev er, we do n ot pro ceed by the ‘direct metho d’; instead, the series (18) is substituted into (19). Then , terms of equal p o w er in τ are separated and their resp ectiv e co efficien ts are set equal to zero. W e must determine at least as many equations as the n u m b er of u nkno wn co efficien ts c ij , i.e ., six equations. These are: O r der O (1 /τ 6 ) : 16 c 02 − 8 c 30 = 0 O r der O (1 /τ 5 ) : − 8 c 11 = 0 O r der O (1 /τ 4 ) : 4 c 20 − 2 c 30 = 0 O r der O (1 /τ 3 ) : 4 c 01 − 2 3 c 11 = 0 O r der O (1 /τ 2 ) : − 2 c 10 + 2 3 c 20 − 2 15 c 02 − 4 15 c 30 = 0 O r der O (1 /τ ) : 0 = 0 These equ ations can b e written in matrix form:          0 0 0 0 16 − 8 0 0 0 − 8 0 0 0 0 4 0 0 − 2 0 4 0 − 2 / 3 0 0 − 2 0 2 / 3 0 − 2 / 15 − 4 / 15 0 0 0 0 0 0                   c 10 c 01 c 20 c 11 c 02 c 30          =          0 0 0 0 0 0          (20) or simp ly A f · C = 0 (21) where C is a six-dimensional v ector and A f is a 6 × 6 matrix with constan t en tries. The singular v alue deco mp osition of A f yields a one-dimensional n u ll space: k er ( A f ) = λ (0 , 0 , 1 , 0 , 1 , 2 , 0) (22) The basis v ector (0 , 0 , 1 , 0 , 1 , 2) corresp onds to the fi rst int egral Φ = x 2 1 + x 2 2 + 2 x 3 1 . A few remarks are here in order: a) the last line of A f has only zero en tries, since it corresp onds to the iden tity 0 = 0 for the O (1 /τ ) terms . This is n ot a problem, b ecause lines w ith zero elemen ts are allo w ed b y the singular v alue deco mp osition alg orithm whic h determines the subspace ker ( A f ). b) T he entries of A f are constan t num b ers whic h dep end only on the co efficients of the Lauren t series (18). T h is is the crucial remark; it imp lies that the information on the fi rst in tegral is contai ned in the Pa inlev ´ e series b uilt by sin gularit y analysis. c) The arbitrary p arameter a 4 in (18) do es not ap p ear in A f . T his phenomenon is not generic. In general, all the arbitrary parameters of the P ainlev ´ e ser ies app ear in A f . In the computer implement ation of the algorithm, w e pr o ceed by giving fix ed v alues to the arbitrary parameters. Although the c h oice of v alues affects the con v ergence of the P ainlev ´ e series, it do es not influ en ce the presen t algorithm whic h is based only in the formal prop erties of the series. 7 3.2 F urther examples Of particular interest in nonlinear dynamics are autonomous Hamiltonian systems of t w o degrees o f f reedom of the form H ≡ 1 2 ( p 2 x + p 2 y ) + V ( x, y ) (23) where V ( x, y ) is of the form (2). The easiest examples are systems with a p olynomial p oten tial (e.g. Hietarin ta 1983, 1 987). F or example: H ≡ 1 2 ( p 2 x + p 2 y + x 2 + y 2 ) − x 2 y − 2 y 3 (24) This system passes the P ainlev ´ e test and it is integ rable (Boun tis et al., 1982). Keeping only the highest order terms of the p oten tial ( − x 2 y − 2 y 3 ) yields the prin cipal balance x = 6 i/τ 2 , y = 3 /τ 2 with resonances r = − 3 , − 1 , 6 , 8, w h ic h are equal to the corresp ond- ing Ko walevski exp on ents. The Painlev ´ e series generated by the Hamilto nian (24) and the ab o ve principal balance satisfies the compatibilit y conditions of the ARS test. Assumin g n o w a p olynomial fi rst in tegral Φ of (24), only the monomial terms of w eight less or equal to 8 will b e included to it. Since the leading term s of the momen ta are p x ∼ p y ∼ O (1 /τ 3 ), the selected monomial terms are: x 4 , x 3 y , x 2 y 2 , xy 3 , y 4 , p 2 x x, p x p y x, p 2 y x, p 2 x y , p x p y y , p 2 y y (w eight 8) p x x 2 , p x xy , p x y 2 , p y x 2 , p y xy , p y y 2 , (weigh t 7) x 3 , x 2 y , xy 2 , y 3 , p 2 x , p x p y , p 2 y , x 3 , x 2 y , xy 2 , y 3 (w eight 6) p x x, p x y , p y x, p y y , (we igh t 5) x 2 , xy , y 2 , p x , p y , x, y (w eights 4,3,2) It s h ould b e stressed that there are several other r estrictions th at redu ce the num b er of eligible terms. F or examp le, the integral Φ is either eve n or o d d in the momen ta (Nak aga wa and Y oshida 2001). F u rthermore, the linear terms can b e omitted by app ropriate transformations. Ho we v er, in the p ractical implemen tation of the algorithm th ese restrictions only in tro duce a complication, b ecause, if an integ ral Φ exists, th e singular v alue decomp osition algorithm selects th e b asis for the corresp ondin g n ull space of the matrix A f without needin g any extra information on restrictions whic h are sp ecific for the system under study , e.g. hamiltonian or other. F ol lo wing the selection of m onomial terms, the algorithm pro ceeds in bu ilding the ho- mogeneous system (6) as well as the finite r estriction A f of the matrix A . In this case, th e dimension of A f should b e M × 38, with M ≥ 38, sin ce th ere are 38 unkn own coefficient s of the a b o ve monomial terms. A t this p oint , it d o es not matter which balance and Lauren t- generated series are u s ed to b uild the matrix A f . In the ab o v e example, th e p rincipal balance leads to a ’sp ecial solution’, since there are only t wo p ositiv e resonances ( r = 6 , 8) mean- ing that th ere are three arbitrary parameters in total en tering in the series. On the other hand, the general La urent series solution with four arbitrary constan ts is giv en by a d ifferen t 8 balance, namely x = 0 /τ 2 , y = 1 /τ 2 so that x starts with dominan t terms of O (1 /τ ), i.e. x ( τ ) = A τ + a 1 τ + B τ 2 + a 3 τ 3 + a 4 τ 4 a 5 τ 5 + O ( τ 6 ) y ( τ ) = 1 τ 2 + b 0 + b 2 τ 2 + b 3 τ 3 + C τ 4 + O ( τ 5 ) (25) p x ( τ ) = − A τ 2 + a 1 + 2 B τ + 3 a 3 τ 2 + 4 a 4 τ 3 5 a 5 τ 4 + O ( τ 5 ) y ( τ ) = − 2 τ 3 + 2 b 2 τ + 3 b 3 τ 2 + 4 C τ 3 + O ( τ 4 ) where A, B , C together with t 0 = t − τ are arbitrary , and b 0 = (1 − A 2 ) / 12, a 1 = A (1 − 2 b 0 ) / 2, b 2 = ( b 0 − 6 b 2 0 − 2 Aa 1 ) / 10, a 3 = (2 b 0 a 1 − a 1 + 2 Ab 2 ) / 4, b 3 = − AB / 3, a 4 = (2 Ab 3 − B − 2 B b 0 ) / 10, a 5 = (2 AC + 2 a 1 b 2 + 2 b 0 a 3 − a 3 ) / 18. In this solution, the resonances r = − 1 , 0 , 3 , 6 d o not coincide one by one to th e Ko w alevski exp onen ts ( − 1 , 1 , 4 , 6) d educed b y the Ko wale vski matrix associated with th e ab o ve bala nce. Nev erth eless, the g eneral solution (25), as w ell as an y other solution wo r k equally we ll in determining th e matrix A f . In our case, b y p erforming the singular v alue decomp osition of A f , the sub space k er ( A f ) yields tw o indep enden t in tegrals in inv olution, with coefficien ts (up to the computer precision) Φ 1 = 0 . 21194 49884 55( y 2 + p 2 y − 4 y 3 ) + 0 . 0496127 30813 ( x 2 + p 2 x ) +0 . 2164 43010 189( xp x p y − p 2 x y − x 2 y 2 − 1 4 x 4 ) (26) − 0 . 2074 46966 722 x 2 y Φ 2 = 0 . 03596 01214 14( y 2 + p 2 y − 4 y 3 ) + 0 . 3424164 78352 ( x 2 + p 2 x ) − 0 . 4086 08475 918( xp x p y − p 2 x y − x 2 y 2 − 1 4 x 4 ) (27) − 0 . 4805 28718 746 x 2 y in term s of whic h w e can express the Hamiltonian H = 2 . 16456459 36295Φ 1 + 1 . 14658628 9822Φ 2 (28) and find a s econd int egral orthogo n al to the hamiltonia n I 2 = − 2 . 164 56459 36295 Φ 2 + 1 . 14658 62898 22Φ 1 (29) The integ r al I 2 is a linear com bination of the hamiltonian and of th e in tegral I b giv en b y Boun tis et a l. (1982) I 2 = 0 . 16517521 23636227( 2 H − 12 7 I b ) (30) Let us note that an alternativ e wa y to obtain the matrix A f is b y considering only the essen tial parts of the Laurent series (25), for as m an y differen t sets of v alues of the arb itrary parameters as requested in order to hav e a complete determination of the M × 38 elemen ts of A f , with M ≥ 38. This approac h is preferable in computer implemen tations of the algorithm, b ecause one do es n ot need to calculate the terms of th e Laurent series (25) b eyond the highest p ositiv e resonance. F u rthermore, Prop osition 2, in s tead of Y oshida’s theorem, can b e used to select the quasiminomial basis set. 9 A fin al remark concerns the applicabilit y of Y oshida’s condition ∇ Φ( b i ) 6 = 0. W e chec k ed whether th is condition is satisfied in Hient arin ta’s table (1983) of int egrable Hamiltonian systems of t wo degrees of freedom with a p olynomial homogeneous p otenti al. T here are six non-trivial ca s es: (a) V ( x, y ) = 2 x 3 + xy 2 with in tegral Φ = y p x p y − xp 2 y + x 2 x 2 + 1 4 y 4 (b) V ( x, y ) = 16 3 x 3 + xy 2 with in tegral Φ = p 4 y + 4 xy 2 p 2 y − 4 3 y 3 p x p y − 4 3 x 2 y 4 − 2 9 y 6 (c) V ( x, y ) = 2 x 3 + xy 2 + i √ 3 9 y 3 with inte gral Φ = p 4 y + 2 √ 3 ip x p 3 y + 2 √ 3 iy 3 p 2 x − (2 y 3 + 2 i √ 3 xy 2 ) p x p y + (4 i √ 3 x 2 y + 2 i √ 3 y 3 + 4 xy 2 ) p 2 y + 4 √ 3 ix 3 y 3 + 2 √ 3 ixy 5 − x 2 y 4 − 5 9 y 6 (d) V ( x, y ) = 4 3 x 4 + x 2 y 2 + 1 12 y 4 with in tegral Φ = y p x p y − xp 2 y + ( 1 3 2 x 3 + xy 2 ) y 2 (e) V ( x, y ) = 4 3 x 4 + x 2 y 2 + 1 6 y 4 with in tegral Φ = p 4 y + 2 3 y 4 p 2 x − 8 3 xy 3 p x p y + (4 x 2 y 2 + 2 3 y 4 ) p 2 y + 1 9 ( y 8 + 4 x 2 y 6 + 4 x 4 y 4 ) and (f ) V ( x, y ) = x 5 + x 3 y 2 + 3 16 xy 4 with in tegral Φ = y p x p y − xp 2 y + 1 2 x 4 y 2 + 3 8 x 2 y 4 + 1 32 y 6 . In all these cases, the p rincipal balance, with all b i differen t from ze ro, satisfies Y oshida’s condition. Note that case (f ) the pr incipal balance leads to ‘weak- P ainlev´ e’ solutions, but the asso ciated integ ral Φ is easily reco verable b y the present algorithm. I n fact, the role of Y oshida’s condition is to exclude fr om Theorem 1 inte grals w hic h are comp osite fun ctions of a simpler int egral. F or example, consider the in tegral I = Φ 2 , where Φ is the p olynomial integral in an y of the ab o v e six Hamiltonian systems. In all of them Φ is weigh ted-homogeneous for some w eigh t M . Th us I is weig hted-homogeneous of w eight 2 M . Sub stituting the sp ecia l solution (7) in the inte gral Φ yields Φ( b i /τ λ i ) = τ − M Φ( b i ). Sin ce Φ is an in tegral, it sh ould b e time-indep enden t, thus Φ( b i ) = 0. S im ilarily , I ( b i ) = 0. Ho w ever, while ∇ Φ( b i ) 6 = 0, we ha ve ∇ I = 2Φ ∇ Φ, th us ∇ I ( b i ) = 0. Th u s, while Φ satisfies the Y oshida’s condition, I do es not. Th is en s ures that w hile M is necessarily a Ko wale vski exp onen t, the m ultiples of it, e.g. 2 M are not necessarily Ko w alevski exp onents. View ed und er this con text, it app ears that th e condition ∇ Φ( b i ) 6 = 0 mak es a ‘n atural’ choice of the simp lest in tegral among an infinity of p ossible in tegrals whic h are comp osite functions of Φ. Ho we v er, there are in teresting coun terexamples which c hallenge this p oin t of v iew. One example is the homoge neous limit of the Holt (1982 ) Hamiltonian H = 1 2 ( p 2 x + p 2 y ) − 3 4 x 4 / 3 − y 2 x − 2 / 3 (31) 10 In this case, the p oten tial is a homogeneous su m of quasi-monomials, of degree 4 / 3. The second integral reads: Φ = 2 ˙ y 3 + 3 ˙ y ˙ x 2 − 6 ˙ y y 2 x − 2 / 3 + 9 ˙ y x 4 / 3 − 18 ˙ xy x 1 / 3 (32) and it is w eigh ted-homogeneous of degree M = − 6. The principal balance is x =  1 3  3 / 2 τ 3 , y =  i 3 √ 2  τ 3 (33) with resonances (=Ko w alevski exp onents) r = − 1 , − 2 , − 3 , − 4. Th is case is remark able b e- cause a) all the r esonances are negativ e, and b) they are not equal to the weig h t of the integral M = − 6. Thus, Y oshida’s theorem do es not apply in the case of this balance. S ubstituting the balance b = ((1 / 3) 3 / 2 , 1 / 3 √ 2 , 3(1 / 3) 3 / 2 , 1 / √ 2), i.e., Eq.(33) in the gradient of the integ ral (32), ∇ Φ( x, y , ˙ x, ˙ y ) = (4 ˙ y y 2 x − 5 / 3 + 12 ˙ y x 1 / 3 − 6 ˙ x y x − 2 / 3 , (34) − 12 ˙ y y x − 2 / 3 − 18 ˙ xx 1 / 3 , 6 ˙ x ˙ y − 18 y x 1 / 3 , 6 ˙ y 2 + 3 ˙ x 2 − 6 y 2 x − 2 / 3 + 9 x 4 / 3 ) yields ∇ Φ ( b ) = 0. Thus none o f the resonances has to b e equal to the w eigh t of the in tegral. Ho we v er, the f orm of Φ or ∇ Φ d o es not su ggest that these f u nctions are comp osite f unctions of some simpler in tegral. On the other h an d , there is a second balance of the Hamilto nian (31), n amely b = ((1 / 6) 3 / 2 , 0 , 3(1 / 6) 3 / 2 , 0), whic h co r resp onds to the dominan t b eha vior x =  1 6  3 / 2 τ 3 , y = Aτ 4 (35) with A arbitrary . The resonances h er e are r = 0 , − 1 , − 4 and − 7, b ut th e Ko wale vski ex- p onents are r K = 1 , − 1 , − 4 , − 6 (t wo of them differ by one from the resp ectiv e resonances). No w, the Ko walevski exp onent r k = − 6 is equal to the w eigh t of the integral . If we look at ∇ Φ, Eq.(34), w e see that the comp onen ts ∂ Φ /∂ y and ∂ Φ /∂ ˙ y con tain terms indep endent of y and ˙ y . Th u s, for this p articular balance ∇ Φ( b ) 6 = 0, i.e. , Y oshida’s condition is satisfied. Th us, in the ab s ence of a counterexa mple, w e form ulate the follo wing Conjecture 3: In any sc ale-invariant system of the form (1,2 ), which p ossesses a weighte d- homo gene ous first inte gr al Φ , at le ast one of the b alanc es b satisfies the c ondition ∇ Φ( b ) 6 = 0 . Returning to the Holt Hamiltonian, the next step is the s electi on of quasi-monomial terms in the initial ansatz for a quasi-polynomial in tegral. Guided by the form of the Hamiltonian, natural exp onen ts are adopted for the p ow ers to which the momen ta p x , p y , and of the v ari- able y are raised, wh ile the v ariable x is considered as raised to p o w ers m/ 3, where m is in teger (p ositiv e or negativ e). E ven u nder th ese r estrictions, there is an infinit y of p ossible quasi-monomial terms of weig ht -6. F or example, the terms p i x m/ 3 y (2 − m ) / 3 , where p i is either p x or p y , are of w eigh t − 6 for all m ∈ Z . Thus an arbitrary low er limit h as to b e set to m . This is c h osen as the lo west b ound of m in the Hamilto nian, namely m = − 2. Neve rtheless, failure to find an integral w ith these r estrictions on the quasi-monomial terms do es not im- ply that an in tegral do es not exist, b ecause of the arbitrariness with r esp ect to the lo we st b ound of negativ e exp onen ts considered. This problem do es not exist when there are only 11 p ositiv e exp onen ts pr esent in the qu asi-monomial terms of the equations of motion (or of the Hamiltonian). With the ab ov e restrictions, the quasi-monomial terms c onsidered are: p 3 x , p 2 x p y , p x p 2 y , p 3 y , p 2 x x 2 / 3 , p 2 x x − 1 / 3 y p x p y x 2 / 3 , p x p y x − 1 / 3 y , p 2 y x 2 / 3 , p 2 y x − 1 / 3 y p x x 4 / 3 , p x x 1 / 3 y , p x x − 2 / 3 y 2 , p y x 4 / 3 p y x 1 / 3 y , p y x − 2 / 3 y 2 , x 2 , x y , y 2 The final step is to b uild the matrix A f as in the pr evious examples, i.e. by replacing th e P ainlev´ e series in the initial ansatz for th e in tegral. An in teresting p oin t is that in the case of the Holt Hamiltonian (31) w e ha v e to consider left Painlev´ e series (Pick erin g 1996), i.e., series in descendin g p o wers of τ . This is b ecause the bala nces ∼ τ 3 do not imply singular b eha vior as τ → 0. In this case, the limit | τ | → ∞ repr esen ts a s ingularit y , but the left Painlev ´ e series are co nv ergent for all τ with | τ | > ǫ for some real p ositive ǫ . The series are constructed as: x i ( τ ) = τ λ ∞ X k =0 b k τ − k (36) where λ > 0. Resonances and compatibilities are chec ked in the same wa y as in the usual P ainlev´ e test. By this metho d, we w er e able to obtain the integ r al (32) b y a prop er balancing of the quasi-monomial co efficien ts c i so as to eliminate the co efficien ts of the terms of successiv e descending pow ers o f τ in the integral expr ession. As a fi nal remark, it should b e stressed that the selection of a quasi-p olynomial ansatz for the in tegrals of a system of the form (1), with the fun ctions (2) b eing quasi-p olynomial, is not exhaustiv e. T his can be easily exemplified in a case with p olynomial functions. The Bogo y- a vlensky - V olterra B-t yp e s y s tems are giv en in n orm alized coord inates u i b y th e follo w ing set o f autonomous nonlinear ODEs: ˙ u 1 = u 2 1 + u 1 u 2 ˙ u i = u i u i +1 − u i u i − 1 , i = 2 , ..., n − 1 (37) ˙ u n = − u n u n − 1 the r.h.s. of E qs.(37) are homogeneous functions of second d egree in the v ariables u i . The system (37 )) admits balances of the form of the form: u i = a i τ (38) where τ = t − t 0 is the time n ear a sin gularit y t 0 in the complex t-plane. In the case of the principal bala nce, the a i are non-zero solutions o f the set of algebraic equations: − a 1 = a 2 1 + a 1 a 2 − a i = a i a i +1 − a i a i − 1 , i = 2 , ..., n − 1 (39) − a n = − a n a n − 1 12 and they are gi v en by the recursion formulas a k +2 = a k − 1 , a 1 = ( − 1) n [ n + 1 2 ] , a 2 = − 1 − a 1 (40) for k = 1 , ..., n . The resonances of the prin cipal b alance (=Ko wa levski exp onents) are giv en by the char- acteristic equation, i.e., s etting the determinant of the Kow alevski m atrix equal to zero. Th e determinan t has a tridiago nal form, i.e., det               a 1 − r a 1 0 0 0 0 ... 0 − a 2 − r a 2 0 0 0 0 0 − a 3 − r a 3 0 0 0 . . . . 0 . . . . . . . . . . . 0 − a n − 1 − r a n − 1 0 . . . 0 − a n − r               = 0 (41) whic h can b e solv ed easily yielding the resonances r k = ( − 1) k k , k = 1 , ..., n (42) Assuming that the conditions for Y oshida’s theorem h old, the we igh t of a w eigh ted- homogeneous in tegral of (37) should b e one of th e resonances (42 ). Indeed, we find an in tegral b y singularity analysis for any of the p ositiv e resonances given by equation (42). The in tegrals are giv en by the recurren t relatio ns: I ( m ) n = I ( m ) n − 1 + u 2 n I ( m − 2) n − 2 + 2 m/ 2 − 1 X k =0 [ I (2 k ) n − m − 1+2 k n Y j = n − m +1+2 k u j ] (43) where the conv en tion I (0) n = 1 and I ( m ) n = 0 for all m, n with n = 2 , 3 , . . . and m > n is adopted. F or n = 3, the resonances are r = − 3 , − 1 , 2 and a p olynomial in tegral is I (2) 3 = c 2 = u 2 2 + u 2 3 + 2 u 1 u 2 + 2 u 2 u 3 (44) Ho we v er, it is simp le to see that this is not th e only fi rst int egral of the system (37). Defining u = u 2 + u 3 , and us in g an y constan t v alue c of the integral (44), the equations of motion tak e the form ˙ u = 1 2 ( u 2 − c 2 ) (45) In tegratio n of (45) yields u = − c coth( c ( t − t 0 ) 2 ) (46) Using u instead of u 2 as a new in dep endent v ariable yields the equation: ˙ u 3 + uu 3 = u 2 3 (47) 13 Whic h can b e solv ed for u 3 yielding u 3 = − c cosh( c ( t − t 0 )) − 1 sinh( c ( t − t 0 )) − c ( t − t 0 ) + 2 cγ (48) where γ is an integ ration constan t. By eliminating the time b et we en th e solutions (46) and (48), a new first in tegral of the original equations is found: I tr = − 2 γ = 1 c ln( u 2 + u 3 − c u 2 + u 3 + c ) + 2 u 1 + u 2 + u 3 u 1 u 3 (49) whic h is a tran s cenden tal fun ction of the v ariables u j . Th is int egral could n ot ha v e b een found b y th e in itial polynomial ansatz . 4 Conclusions In this pap er we ha ve explored the question of whether it is p ossible to reco ver the fir st in tegrals of systems of first-order nonlinear ordinary d ifferen t equations in volving quasi- p olynomial f unctions of the indep end en t v ariables based on the information provided by singularit y analysis. The mai n conclusions are: a) Th e theorem of Y oshida (1983 ) constrains the choi ce of an initial ansatz for an integ ral with undetermined parameters, lea v in g, ho w ev er, an infinit y of p ossible choic es. b) Th e condition of Y osh ida’s th eorem ( ∇ Φ( b i ) 6 = 0 and fi nite) h olds for all the inte grable Hamiltonian systems of tw o degrees of freedom included in Hietarin ta’s (1983) table, if b i is set equal to the principal balance b i 6 = 0 for all i = 1 , . . . 4. 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