Configurational invariants of Hamiltonian systems

Configuratio nal in v arian ts of Hamiltonian sy stems Giusepp e Pucacco ∗ Dipartimen to di Fisica – Univ ersit` a di R oma “T or V ergata” INFN – Sezione di Roma I I Kjell Rosquist † Departmen t of Phy sics – Sto ckholm Univ ersit y Abstract In this paper we explore th e general conditions in order t hat a 2-dimensional natural Hamiltonian system p ossess a second inv arian t whic h is a p olynomial in the momen ta and is therefore Liouville integrable. W e examine the possibilit y th at the inv ariant is preserv ed by the Hamiltonian flow on a given energy hypersurface only ( w eak integ rability) and d erive the additional requirement necessary to hav e conserv a- tion at arbitrary energy (strong integrabili ty). Using null complex coordinates, we show that the leading order coefficient of th e p olynomial is an arbitrary h olomorphic function in the case of weak integrabilit y and a p olynomial in the co ordin ates in t h e strongly integrable one. W e rev iew the results obtained so far with strong inv ariants up to degree four and provide some new examples of w eakly integrable sy stems with linear and quadratic in va riants. Published as: Journ al of Math. Phys. , 46 , 05 2902 (20 05) 1 In tro du c tion In 1 983 Hall [1] published a re mark able pap er dev oted to a theo ry o f c onfigur ational invariants of class ical Hamiltonian systems. The main v ir tue of the w ork cons is ted in an eleg ant and pow erful technique to solve the equations for the ex istence of a seco nd po lynomial inv ariant of a rbitrary degree in the momenta fo r 2-dimensional Ha miltonian sys tems. As a result o f this approach, Hall was able to get so me new examples of in tegra ble systems admitting a second in v ariant of degr ee four in the mo men ta. Unfortunately , the pa per was flaw ed by a definitely wrong statement and also by many inaccurate a r- guments and deductions. In fact, Hall purp orted to remedy supp os e d ov ersig hts in previous works o n the search for the second in v aria nt , in particular criticizing the cla ssical a c count by Whittaker [2]. In Hall’s view, Whittaker’s (and a ll o thers’ since then) treatment provides only sufficien t conditions for the existence of a second inv ariant (linear and quadratic in the sp ecific instance), whereas, due to overlooking the link ∗ e-mail: pucacco@roma2.infn.it † e-mail: kr@phy sto.se 1 established on phase-spac e v aria bles by energy conser v a tion, it was not able to find all p ossible solutions. Actually , as it is, this statement is wrong: in lo o king for a str ong second inv ar iant, namely a phas e - space function which comm utes with the Hamiltonian function, Whittaker’s approach is indeed correct and leads to ne c essary and sufficient co nditions for its existence. This point was alr eady stresse d b y Sa r let et al. [3] in their criticism to Hall’s pap er. Moreov er , Hall’s discussio n cont a ined a confusion betw een the conce pt of configur ational (or we ak ) in- v a riant, as a function which exactly commutes w ith the Hamiltonian only on a subset (p ossibly one) o f the energy hypersurfaces (see again [3] and, e.g. [4]), and the notion of what we may c all formal inte gr al as it emerges in the analysis of reg ular po rtions of the phase space of generic non- int eg rable sys tems (see e.g. [5]). In par ticula r, the approximate inv a r iants obtained b y Hall as a result o f his per turbative a pproach have little to do with those that can be obta ined by truncating a normal form expansion. Nonetheless, in spite of all the ab ove shortcomings , the form in which the problem has been se t by Hall and the appro ach followed for its partial solution deserve a tten tion, since they can still b e v ery useful. Already Hietarinta [6], in his acc o unt of the direct metho ds for the sear ch of the second in v aria nt , pr ovides a review of all the known systems a dmitting one o r more c onfigurationa l inv ariants a nd works out again the integrabilit y conditions found by Hall for the existence of weak in v aria nt s up to degree four . How ever, in Hietarinta’s review, the t wo settings of weak a nd strong integrabilit y ar e still k ept w ell sepa rated. More recently , in a series o f works ab out a unified appro a ch to treat b oth kinds of inv aria n ts [7, 8, 9, 14], the same integrabilit y conditions hav e b e en obtained, supplemente d by the additional c onstr aint imp ose d by str ong inte gr ability . This last step is essential for a nea t distinction b e tw een the tw o notions o f weak and strong int eg rability . In these works, this step is a stra ig htf or ward consequence of the geometr ic approa ch in whic h the exis tence of the second in v aria n t is addressed by studying the corresp o nding Killing tensor equations for the Jaco bi metric. This metric dep ends on the mechanical ener g y as a parameter. Therefore, any tensor ob ject on the corres p o nding manifold in general dep ends on the energy . With due car e ab out for ma l rela tions b etw een the t wo approaches, the geometric approach and Hall’s approach are equiv alent. The ess e ntial remark is that, to iden tify the cases of strong integrability , it is sufficient that in the fina l results concer ning the existence of the in v aria nt , a subset can be iso lated whic h is indep endent of the ener g y parameter. In this framework, in [7], quadratic in v ariants at arbitrary and fixed energy for 2 -dimensional Hamiltonian s ystems w ere treated in a unified way , wher eas in [8, 9] the existence of r esp ectively cubic and quartic inv ariants was discussed accompanied with the discovery of s o me new examples not given in ea r lier works (see e.g. [10, 11, 1 2, 13]). In [14] the case o f Hamiltonian systems with vector p otentials is trea ted, extending previous inv estiga tio ns [15, 16, 17, 18, 19, 20]. The a im of the present paper is to discuss the techniques for solving the equa tions for a second inv ar ia nt having a momen tum depe ndence of arbitrary polyno mial deg ree. The analysis is ba s ed o n a com bination of bo th the ab ov e-mentioned approa ches, with particular attention dedicated to the confor ma l transformations used to simplify the equations. The tr eatment is in general effective for b oth class es of in v aria nt s, config- urational (or w eak) and strong. After that, we imp ose the additional condition needed to isola te the class of strong ly in tegra ble sy stems, o btaining the genera l form of the simplifying family o f transforma tions for each degree of the strong inv ar iant lo o ked for. This in turn implies determining the leading or der terms in the inv ar ia nt itself. An alternative route to this res ult in the g eometric a pproach is based on the inv aria nce, under conformal tr a nsformations, of the conforma l pa r t o f the Killing tensor . Mor eov er, a genera l r eview of the results obtained so far is provided. The plan of the pap er is as follows: in Sectio n 2 w e compa r e the notion of w ea k and strong inv aria nts and, working out in detail the q uadratic case, we correct Hall’s misunderstanding, pr oviding the constraint to b e sa tisfie d in order to g et strong integrabilit y as a restriction of weak int eg rability; in Section 3 w e r e call time repara metrization o f the null Hamiltonian a nd complex (“ nu ll” ) c o ordinates use d to simplify the direct approach; in Section 4 these to ols are exploited to set the general approach to find p oly nomial in v ariants of arbitrar y degree; in Section 5 we recall the main re sults conc e rning inv a r iants of degr ee up to four; in section 6 w e present our conclusions and the prospe c ts for future works. 2 2 Configurational in v arian ts v ersus strong in v arian ts As it is well known (se e, e.g. [2 1]), to gr ant the c omplete integrability of a n N -dimensional Hamiltonia n system, it is necessar y and s ufficient to find N indep endent integrals o f motion or invariants , for short. In the following, we will limit ourselves to the s implest ca s e of a cons e rv ative Hamiltonian system in tw o dimensions ( N = 2). Since the Hamiltonian itself H = 1 2 ( p 2 x + p 2 y ) + V ( x, y ) , (1) is a conserved function, it is enough to find just a second indep endent in v ariant. In his search for po lynomial in v aria nt s, Hall [1 ] exploits wha t is usually c a lled the “direct a pproach” of Darb oux [22] a nd Whittaker [2]. As an “educa ted guess” we ma y start with the working h yp othesis of an inv ar ia nt with a structure analogous to that of the Hamiltonian, namely a seco nd de g ree p olynomia l in the mo men ta. 1 Clearly , o ne could start with a p olyno mial of arbitrar y degree o r even with a more general expression. In the core part of the pap er we will exa mine the g eneral case. Now, as an intro duction a imed at g iving the prop er settings, we work o ut in detail the quadratic case. W e may then look for a phase-space function of the form I 2 ( p x , p y , x, y ) = Ap 2 x + B p x p y + C p 2 y + K, (2) which is prese r ved along the flow given b y eq.(1), namely { I 2 , H } = 0 . (3) In eq.(2), A, B , C and K ar e ea ch functions of x and y . Actually Hall, probably motiv ated b y his interest in accelerato r physics, mana ges to work out the in tegrability of Hamiltonians including also a v ector p otential and henceforth terms which are linear in the momenta. This, in turn, sugg ests the inclusion of analo gous terms in the inv aria n t. Howev er, in the standa rd case of a Hamiltonian inv aria n t under momentum inversion, p → − p , the equations for the co efficients of even and o dd degr ee terms decouple, g r eatly reducing the complexity of the system to solve. Therefore, since the main purp ose o f this pap er is to fully clar ify the issues of weak and s trong integrabilit y , which are una ffected by such generaliza tions, we prefer to limit ourselves to the standa rd case of eq.(1). The treatment o f the vector po ten tial is pres ent ed elsewhe r e [14] where new strongly in teg rable systems with quadratic in v aria nts a r e presented. In the direct appro ach, one inserts the functions (1,2) in to the Poisson brack et (3). The resulting po ly- nomial in the momenta, of third degree in the present ins tance, m ust b e ident ica lly v anishing. In view of the independence and ar bitrariness o f the momen tum co ordinates , e ach co efficient o f the p olyno mial m ust v a nish, determining in turn the follo wing sys tem of PDEs in the co o r dinates: A x = 0 , (4) A y + B x = 0 , (5) B y + C x = 0 , (6) C y = 0 , (7) K x = 2 AV x + B V y , (8) K y = B V x + 2 C V y , (9) where, in order to co mpactify expressions , suffixes denote partial different ia tio n. As men tioned ab ove, Darb oux [22] addressed the problem a nd the equation ensuing from the int eg rability condition for K , that is B ( V y y − V xx ) + 2( A − C ) V xy + 3( B y V y − B x V x ) = 0 , (10) is known a s Darb oux’s e quation . Whittaker [2] also analyzed the problem and, fo r a complete a ccount of the solution of this system, the standard reference is again the review by Hietarinta [6]. 1 Since all the work is made in a Hami ltonian context , contrary to the original treatmen t b y Hall, w e use canonical phase-space coordinates. How ever, we hav e tried to sta y as close as p ossible to his treatmen t as refers to the pro cedure and notations. 3 Equations inv olving only the leading order terms ( A, B and C ) in the inv aria n t are readily solved: A = ay 2 + by + c, (11) B = − 2 axy − bx − dy − e, (12) C = ax 2 + dx + f . (13) Exploiting linear tr a nsformations o f the co or dinates, it can b e sho wn [6, 7] that these functions can b e reduced to four canonical forms that, inserted into Dar bo ux e q uation (10), lea d to the c omplete solution of the problem: there exist four fundamen tal s eparating co ordinate systems (elliptical, po lar, parab olic and Cartesian) in which the p o tential a nd the rema ining unknown of the inv ariant, K , can b e expres sed in terms of c o mbinations of a rbitrary functions of each of the separating co ordinates . Actually , in his refer ence to Whittaker’s result, Hall ment io ns only the elliptical solution, but, as shown in [7] (see also [23]), using conformal tra nsformations, all the s e parable cas e s po ssess the same structure. Moreov er , we remark on the additiona l p ossibility o f complex p otentials, which provides four mor e cases with complex separating co ordinates, which a re listed in [6] a nd can also be obta ine d w ith the techniques in [7]. A t this p o int Hall states that ([1], p.93) “Whittaker’s w or k /.../ w as flawed by his failure to recog niz e that ter ms in ˙ x 2 and ˙ y 2 are no t indep endent for the purp ose of setting the co efficients of their v arious powers to zero /.../ They are related by co nserv ation of ener gy . Therefore, Whittaker’s c o nstraints were ov er-r estrictive; the solutions he found ar e v alid, but others may also exist.” On these bas es, Hall ela b o rates a genera lized a pproach that, in his view, pr ovides necessar y and sufficient conditions for the ex is tence of the g iven in v a riant. Actually , as we will shortly see, Hall fa ils in recognizing that the p os sible extensio n of the family of s olutions implies a different status for the additional in v aria nt s: t hey are “fixed energ y” or c onfigur ational in v ariants. Configuratio na l (or “conditiona l” or even “weak”) integrals hold for a sp e cifie d p articular value of the ener gy c onstant and were firstly investigated by Birkhoff [24] and cons ide r ed by F omenko [25], K ozlov [26] and others. W eak inv ariants e njoy weak er pro p er ties then their “no bler” cousins, the str ong inv ar iants, which ar e constan t on every energy hype r surface admitted b y the dyna mics of the system. T o clarify this essential p oint, let us a gain work with the quadra tic cas e, concretely introducing Hall’s a rgument. This goes as follows: for every v alue E o f the energy , the function (1) defines the h yp ers ur face 1 2 ( p 2 x + p 2 y ) + V ( x, y ) = E . (14) Let us construct the following linear co mbinations of the squares of the mo menta: Σ = p 2 x + p 2 y , (15) ∆ = p 2 x − p 2 y (16) and observe that, in view of (14), we can imp os e the constraint Σ = 2 G ( x, y ) , (17) where we hav e introduced the “Jac obi” p otential 2 G = E − V . (18) The quadratic terms in the standard form of the in v aria nt o f eq.(2) can no w b e written as Ap 2 x + B p x p y + C p 2 y = 1 2 h ( A + C )Σ + ( A − C )∆ + B p Σ 2 − ∆ 2 i . (19) Exploiting the energy constraint (17) and redefining co efficients, the quadr atic inv ar iant can then b e written as I 2 ( p x , p y , x, y ) = 1 2 D ( p 2 x − p 2 y ) + B p x p y + e K , (20) 2 The origin of this denomination comes from the close relationship b etw een the picture of a s ystem constrained on the fixed energy surface and the geometric pi cture of a geodesic flow ov er a Riemannian manifold endo wed wi th a “Jacobi” metric (see, e.g. [27]). 4 where D = A − C, (21) e K = K + ( A + C ) G. (22) Now, we may pro ce ed along the same lines followed ab ov e. The commutation relatio ns (3) of the weak inv a riant (20) with the Hamiltonian reduce to the system D y + B x = 0 , (23) D x − B y = 0 , (24) e K x = B V y + DV x − GB y , (25) e K y = B V x − DV y − GB x . ( 2 6 ) The integrabilit y condition for e K is now the gener alize d Darb oux e quation B ( V y y − V xx ) + 2 D V xy + 3( B y V y − B x V x ) − 2( E − V ) D xy = 0 , (27) where we have explicitly p ointed out the presenc e of the energy parameter. In practice, since the ener gy of the system is in general an arbitrary real n umber, we can write eq.(27) in the for m f 1 E + f 0 = 0 , (28) where f 1 , f 0 are functions of the co ordinates, b oth explicitly and through V and its deriv atives. If we w ant that the lo oked fo r inv ariant ha s an ident ica lly v anishing Poisson brack et with the Hamilton r e gar d less of the value of the ener gy (in other w o rds that I 2 should be a stro ng inv ariant), eq.(28) must b e satisfied for every v alue of the parameter E so that the tw o equations f 1 = 0 , (29) f 0 = 0 , (30) m ust s e parately be sa tisfied. The first of these equations is simply D xy = 0 . (31) As a co nsequence of this, the second, by a direct comparis on, tur ns out to coinc ide with the standard Darb oux equation (10). This time, the solution of e quation (31), together with (23) and (24) for the leading order terms (no w D and B ), g ives D = a ( y 2 − x 2 ) + by − dx + g , (32) B = − 2 axy − bx − dy − e. (33) Recalling (21) and comparing with (11 – 13), we see that we hav e a rrived at a result completely equiv ale nt to that o f the s tandard Dar b oux-Whittaker approach. In fact, eq.(30) now coincides with eq.(10) and, therefor e , the same po ssible separable po tentials in 2 dimensions can be fo und a lso follo wing this alter native route. In our gener al pres ent a tion in s ection 4, we will see how this s trategy reveals to b e useful a lso in the case of higher degree inv aria nts. While used mostly for its p edag ogical role in the pres e n t instance, what is impo rtant to stress is the key role of the integrability condition in the form (28): in the gene r al case of a po lynomial in v ariant o f degree M , it turns out that eq.(28) is a p olynomial in E of degree M / 2 for M even and ( M + 1) / 2 fo r M o dd. In the cas e of inv ar iance at fix e d energy , the pr oblem admits additional so lutions. No w we hav e that eq.(31) is no more necessary a nd therefor e (23) and (2 4) a re unconstra ined Cauch y-Riema nn equa tions. Instead of (32) and (33), the so lution is now given by a n ar bitr ary a nalytic function: its rea l a nd imaginary 5 parts res p ectively provide the functions D and B . Because of this feature, we will see in the following ho w it is more conv enient to work with complex v aria bles. Summarizing, the ab ov e pro c e dure shows that the strateg y for finding w ea k in v aria nt s is well defined and po int s out where it must b e constra ined to g e t strong in v ariants. This ob jective be ing the most imp orta n t for applica tions, it may app ear that, going this w ay , nothing is ga ined with resp ect to the us ual direct approach. How ever, the proce dur e instead prov es to be very effective in simplifying the system of equa tions resulting from implementing the dir ect appro ach. This simplificatio n is a ctually one of the rea sons why Hall’s w or k is still useful. At the same time, sinc e the approach based on the constrained in v ariant ansatz (20), complemen ted b y a correct use of the gener al Darb o ux equation (2 8), provides the s ame results as the standard appr oach, w e get a simple proo f of the in v alidity of Hall’s criticism tow ar ds Whittak er . How ever, Hall is right in envisaging additional solutions to the pr o blem: in the example we hav e just seen, it is conceiv able to obtain, for p articular v alues o f E , solutio ns of eq.(28) not included in those of 29 and 29. In [7], w e hav e pro duced several cla sses of solutions corresp o nding to E = 0. The inv estigatio n of such we akly inte gr able systems (WIS) offer s sev er a l additional is sues to study . W e men tion some of them: – pos sible solutions defined for a con tinuous but finite range of energy v alues E 1 < E < E 2 ; – WIS w hich are a ctually SIS ( stro ngly inte gr able syst ems ) with a more general form for the second inv a riant (Examples: the Kepler pro blem, the Sarlet et al. case [3]); – the main prop erty of the integrable dynamics on the E s urface may help in understanding s o me of the main features in the non-int eg rable regime (see e.g. [28]). 3 The n ull Hamiltonia n and time reparametrization As alr e ady put forw a r d by Hietarinta (see [6], sect.7.2), canonical p oint transfor mations generated b y ana- lytical functions preserve the for m of a nul l Hamiltonian. This inv aria nce allo ws a straig ht for ward w ay to reduce the set of equations to b e solved in the search fo r a po lynomial inv a r iant. In the present section we show how, in the case of 2- dimensional systems, the a b ove transfor mations ar e actually co nformal trans- formations and ar e r e lated with the time repa r ametrization o f the dynamics. These to ols, exploited in the remaining part of the paper , w ere implicit in Hall’s work. In general the Hamiltonian itself has the form H = T + V ( q ) = E , (34) where T is a quadratic form in the mo ment a . The indep endent v ar iable, let us say t , is often but not always the time. F o r an y given ener gy E of the s ystem, to repres ent the dynamics, we can use the nul l Hamiltonian H 0 = H − E , (35) provided that we impose the constr a int H 0 = 0 . (36) F or a n y such zero energy Hamiltonian we ca n reparametrize the system by in tro ducing a new time v ar iable ¯ t defined by the rela tio n dt = N ( p, q ) d ¯ t , (37) together with a redefined Hamiltonian ¯ H 0 = N ( p, q ) H 0 = N T + N ( V − E ) = 0 . (38) The new Hamiltonian will then give the same equations o f motion on the co nstraint s ur face ¯ H 0 = 0. W e shall use the term lapse fu n ction for N ( p, q ) which defines the independent v ariable gauge. This usage is bo rrow ed fro m cosmological a pplica tions where the la pse gives the rate of physical time change relative to co ordinate time. The lapse function can b e taken as a ny non-zer o function on the pha se space. 6 It is simpler to work with complex , or “null”, coo rdinates, z = x + iy , p z = p = 1 2 ( p x − ip y ) , (39) ¯ z = x − iy , p ¯ z = ¯ p = 1 2 ( p x + ip y ) . (40) The null Hamiltonian ca n then b e written in the form H 0 = 2 p ¯ p − G ( z , ¯ z ) = 0 , (41) where G is the function int r o duced in (18). T he equations of mo tion given by (41) are dz dt = 2 ¯ p, (42) dp dt = G z = − V z , (43) and corr e sp o nding complex conjugates. G and V a re always assumed to b e real functions. In the following we will spar e to men tion explicitly to the complex conjugates. W e use a conforma l transformation to standa rdize the fr ame and co or dinate representation of the inv a ri- ant. T o that end w e intro duce a new complex coo rdinate w by means of the transformation z = F ( w ) . (44) The conformal transformatio n z → w = X + iY , (45) given by the holomorphic function (4 4) determines the canonical po int tra ns formation w = F − 1 ( z ) , P = F ′ p, (46) so that (1) transforms int o the new null Ha miltonian e H 0 = 2 P ¯ P − e G ( w, ¯ w ) | F ′ | 2 = 0 . (47) Let us in tro duce the “standa rd” n ull Hamiltonian H S = 2 P ¯ P − e G ( w , ¯ w ) = 0 , (48) that implies the use of the new “time” s , suc h that d ds = | F ′ | 2 d dt . (49) Equations of motion (42 – 43) beco mes dw ds = 2 ¯ P , (50) dP ds = e G w . (51) A t the same time, a co ns erved quantit y stays co nserved if transfor med b etw een the tw o gaug es (4 7) and (48). In terms of real v ariables, (48) is given by H S = 1 2 ( P 2 X + P 2 Y ) − e G ( X, Y ) = 0 . (52) 7 4 In v arian ts of arbitrary degree With the choice of p olynomial inv ariants, the time reflectio n symmetry of Hamiltonian (1) a llows a further simplification in the pro cedure. In fact, it can b een proved (see Hietarinta [6], sect.2.3 and [29]) that the algebra of commutin g functions with the Hamiltonian, which is an even function with resp ect to time reflection, has “g o o d” time reflection parit y . Therefor e, a p olynomial second inv aria nt can be either even or o dd p o lynomial in the momen ta. W e ca n therefore assume a phase-space function of the follo wing form: I M = [ M / 2] X k =0 M − 2 k X j =0 p j x p M − 2 k − j y A ( j,M − 2 k ) ( x, y ) , (53) where the functions A ( j,M − 2 k ) ( x, y ), no t necessarily p olyno mials, ha ve to b e determined. In the first sum- mation, with [ M / 2] w e deno te the greatest integer less than M / 2 , so that if, e.g ., M = 1, k takes only the v a lue zero. 4.1 The direct approac h in the Cartesian frame The simples t procedur e (also called the dir e ct ap pr o ach ) is no w to compute the Poisso n bra ck ets of I M with H , collect terms with v a r ious power of moment a end let v anish their resp ective co efficients. Using at first the form (53) in the us ual Ca rtesian frame, we then get a system of partial differential equations of the form: ( j + 1) A ( j +1 ,k +1) ∂ x V + ( k + 1 − j ) A ( j,k +1) ∂ y V = ∂ x A ( j − 1 ,k − 1) + ∂ y A ( j,k − 1) , (54) with j = 0 , ..., k and k = M + 1 , M − 1 , M − 3 , ..., 0 o r 1 and it is implicit that A ( s,t ) = 0 if s < 0 or s > t and t < 0 or t > M . In order to simplify form ulas , in the present section we reintroduce the standard notation ∂ to denote partia l differentiation with resp ect to the v a r iable in the subscript. The set (54) is a co mplica ted set of PDEs, in general ov erdetermined: if M is even w e hav e ( M + 2)( M + 4) / 4 equations for ( M + 2 ) 2 / 4 unknowns; if M is o dd we have ( M + 3) 2 / 4 equa tions for ( M + 1)( M + 3) / 4 unknowns. Actua lly , in the general case, to these figures the po tent ia l V en ters as an additional unknown. 4.2 The direct approac h in the n ull frame Int r o ducing the complex n ull frame (39,40), we ha ve the following generic expression of the in v ariant: I M = 2Re    M / 2 X k =0 C 2 k p 2 k    , M ev en (55) I M = 2 Re    ( M − 1) / 2 X k =0 C 2 k +1 p 2 k +1    , M odd , (56) where the complex functions C 2 k , k = 0 , ...M / 2 or C 2 k +1 , k = 0 , ... ( M − 1 ) / 2 depend on z , ¯ z and, with the exclusion of the leading or der co efficients C M , implicitly on E . Their explicit expressio n in terms of the co efficients of the inv ar ia nt in Cartesian form are C ( M ) 2 k = M 2 − k X a =0  G 2  a ×    2( a + k ) X ℓ =0 i 2( a + k ) − ℓ " a X r =0 ( − 1) r  ℓ ℓ − a + r  2( a + k ) − ℓ 2( a + k ) − ℓ − r  # A ( ℓ, 2( a + k ))    , (57) 8 when M is even a nd C ( M ) 2 k +1 = M − 1 2 − k X a =0  G 2  a ×    2( a + k )+1 X ℓ =0 i 2( a + k )+1 − ℓ " a X r =0 ( − 1) r  ℓ ℓ − a + r  2( a + k ) + 1 − ℓ 2( a + k ) + 1 − ℓ − r  # A ( ℓ, 2( a + k )+1)    , (58) when M is o dd. In these express ions a sup ers cript ( M ) has bee n introduced in or der to let them apply in g eneral: namely , C ( M ) a denotes the a − th co efficient of the inv ar iant of deg ree M . In the following this sup e rscript will not be use d unless it is strictly needed to preven t confusion. As usual, in (57) and (58)  n k  = n ! ( n − k )! k ! (59) denotes the binomial co e fficient . The forms (55) and (56) of the in v aria nt do not co nt a in cross terms, na mely terms with p ow er s of p ¯ p , since they hav e been elimina ted exploiting the constraint 2 p ¯ p = G ( z , ¯ z ) (60) dictated by (41). A simple chec k of this statement can b e p erformed with the quadratic inv aria n t with which we are familiar from Section 2. E q.(57) with M = 2 give C (2) 2 = A 22 − A 02 + iA 12 , (6 1) C (2) 0 = A 00 + ( A 02 + A 22 ) G. (62) Posing A 22 = A, A 12 = B , A 02 = C , A 00 = K and C (2) 2 = D + iB , C (2) 0 = e K w e again find relations (21,22) and see that the expression (20) of the in v ariant is equiv alent to I 2 ( p, ¯ p, z , ¯ z ) = C (2) 2 p 2 + ¯ C (2) 2 ¯ p 2 + e K . (63) The use of the energy cons tr aint leads to the max ima l reductio n in the num b er of eq ua tions ens uing fro m the direct approach. How ever, in the case of in tegrability at arbitr ary ener g y , to this r e duce d n umber one m ust add the int eg rability co nditions which results in a set o f equations g eneralizing (29). The constra int (60) can also b e used to “homogeneize” the p olynomia l inv a riant. In this form, used in the geometric framew o rk in c onnection with the geodes ic flow o ver a Riemannian manifold, the coefficients of the inv ar iant give r ise to a symmetric M-rank tensor, known as a Kil ling tensor . The commutation r e la tion of the functions (55) or (56) with the null Hamilto nian (41) gives the sy stem of equations ∂ ¯ z C k − 1 + 1 2 ( ∂ z C k +1 ) G + 1 2 ( k + 1) C k +1 ∂ z G = 0 , k = 0 , 1 , . . . , M , (64) where it is implicitly assumed that C j = 0 for j < 0 and for j > M . The set o f equation (64) must be supplement ed by the closure equations ∂ ¯ z C M = 0 (65) and ℜ{ ∂ z ( C 1 G ) } = 0 . (66) Clearly , this last condition m ust be stated o nly in the c a se of o dd degree , since for M even, a c lo sure condition of the form ℑ{ C 0 } = 0 (67) is implicit since C 0 is real by definition. 9 In b oth cases, using the n ull complex co ordinates, we have M + 2 rea l equations fo r M + 1 real unknowns: the substantial r eduction of the num b er of indep endent equations with resp ect to that in the Car tesian frame was already remarked b y Hieta r inta (see [6], se c t.7 .5). How ever, he thought that reintroducing the explicit depe ndence o n ener g y would r eestablish the orig inal num b e r, wherea s, as we will see, the gain in saving equation still remains when going to the Cartesian frame. 4.3 Solving t he equations for the Mth-degree inv arian t A t present a general solution of system (64) is still lacking. In [7, 8, 9] we found the general solutions for 1 ≤ M ≤ 4. In the pres ent subsection we illustrate the asp ect o f the pr o cedure which are common to all Mth-degree inv ariant. The first steps are essentially the s ame as in the original paper by Hall. Lo oking at the system ab ov e, w e see that eq.(65) is rea dily so lved: C M = C M ( z ) , (68) that is C M is an arbitrary holo morphic function. The first imp orta n t result we g et is ther efore that the leading or der coefficie nt of a polyno mial inv ariant is given b y an arbitrar y holomorphic function. This fa c t is already known in the case of homogeneous p olynomial inv ariants (Kolokoltso v [30], Kozlov [26]) and was obtained by Bir khoff ([24], c ha p. 2) for M ≤ 2. W e rema rk that, in a greement with no tations in (57) and (58 ), here we are referring to C ( M ) M , that is the lea ding or der co efficient of the p o ly nomial inv ar iant of degree M with v arious v alues of M . Therefor e, in order to av oid co nfusion with low er o rder coefficients in a p olynomial with a given M and also to conform with the notation in previous works, in the following we will denote the leading order coe fficient with S M . In order to atta ck the remaining equations in (64), we may fur ther s implify them by p erforming a co ordinate transforma tion. This is a c o nformal transformatio n of the form (44), where the genera ting function F ( w ) is chosen s uch that F ′ ( w ( z )) = S M 1 / M . (69) In this case, w e see that the first equation of the chain (64) c a n be re wr itten as S M 2 M − 1 ∂ ¯ w C M − 2 + M 2 ∂ w e G = 0 , (70) where, in agreement with (47), the new “confor mal” potential e G ( w, ¯ w ) = | F ′ ( w ) | 2 G = ( S M ¯ S M ) 1 / M G, (71) has been intro duced. Defining the function e C M − 2 ( w, ¯ w ) = S 2 − M M M C M − 2 , (72) eq.(70) be c o mes: ∂ ¯ w e C M − 2 + M 2 ∂ w e G = 0 (73) and, defining in an analogous w ay , e C 2 k ( w, ¯ w ) = S − 2 k M M C 2 k , k = 0 , 1 , . . . , [ N / 2] , (74) the rest of the system beco mes ∂ ¯ w e C j − 2 + 1 2 e G j − 1 ∂ w ( e C j e G j ) = 0 , (75) with the ranges j = 4 , 5 , . . . , N − 2 if N is even a nd j = 3 , 4 , . . . , N − 2 if N is odd. 10 4.4 The K¨ ahler p ot ential Eq.(70), or better its trans formed form (7 3), plays a sp ecia l role. In view of the fact that the new po ten tial e G is still a real function, (73) can be so lved int r o ducing a real function K ( w, ¯ w ) s uch that e G = ∂ 2 w ¯ w K , (76 ) e C M − 2 = − M 2 ∂ 2 ww K . (77) The firs t e q uation says that the Jacobi p otential is the Laplacia n of the function K ( w , ¯ w ). Since, in the geometric picture, G is the confor mal factor o f the metric elemen t, using K it gives rise to a Hermitian form and therefor e it is referr e d to as the K¨ ahler p otential . Without los s o f gene r ality , in ([9], sect. IV) it has bee n shown that the K¨ ahler po ten tial can b e a ssumed to be of the form K = E [ F ( w ) ¯ F ( ¯ w ) + 2 ℜ{ Λ( z ( w )) } ] − Ψ , (78) where Λ is an arbitrary holomorphic function independent of E and the real pr ep otential Ψ is suc h that ∂ 2 w ¯ w Ψ = F ′ ¯ F ′ ∂ 2 z ¯ z Ψ = | F ′ | 2 V . (79) In the c ase of in v ariants up to the four th degr ee, M ≤ 4, we see that the closur e equations (6 6) and (67) are themselves e xpressed o nly in terms of the K¨ ahler p otential. In these c a ses, to get a complete solution of the problem, it remains only to solve an integrabilit y condition for K ( w , ¯ w ). In the ca ses o f linear and quadratic in v aria nts, the integrability condition is linear a nd we have the genera l solution dep ending respectively on one and t wo arbitra ry r eal functions (see [7]). In the hig her degr ee cases, the integrability conditions are nonlinear and a general solutio n is lacking: moreover, in g eneral, they giv e rise to an overdetermined system for the unknowns Λ and Ψ . This is consistent with the fa ct tha t only isolated cases of integrable Hamilto nia n sys tems with inv aria nt of degree hig her than tw o are known. F a iling to w or k in full genera lit y , a useful a pproach is to mak e an ansatz for o ne of the unkno wn (e.g. Λ ) and s olve for Ψ. In this w ay sev er al new integrable and supe r integrable systems hav e b e e n found w ith a cubic ([8]) and a quartic ([9]) second in v ariant. 4.5 The leading order t erm of strong in v ariants Lo oking for a strong inv a riant, so that the integrability condition must be solv ed fo r arbitrary v alues of E , it tur ns out that the function C M can no longer b e arbitrar y . In our previous works we hav e s hown its po lynomial structure with deg ree equal to M for M ≤ 4 a nd guessed that this is the generic b ehavior for any v alue of M . In this sectio n we prov e that this conjecture is tr ue b y explicitly computing the function with the aid of the already known solution in the Cartesia n ca se. In [7] for the linear and quadratic cases we hav e found resp ectively the co nditions ℜ{ S ′ 1 ( z ) } = 0 , (80) ℑ{ S ′′ 2 ( z ) } = 0 . (81) In [8], for the cubic case, the condition ℜ  d 3 dz 3 S 3  = 0 (82) was found, whereas in [9], for the quartic case, the condition ℑ  d 4 dz 4 S 4  = 0 (83) was found. Therefore w e can guess the genera l co nditions ℜ  d M dz M S M  = 0 , M o dd , (84) ℑ  d M dz M S M  = 0 , M even . (85) 11 Conditions fro m (80) to (83) have be en obta ined working o n the int eg rability condition for the K¨ ahler po tent ia l K . Since it is not po ssible to write an explicit expressio n for the integrabilit y condition at arbitra ry M , we prove (84) and (85) by explicitly calculating them. The easiest way to pro ceed is to exploit the solution in Cartesia n co ordina tes and after that p erforming the transformation to complex co o rdinates: the reason for not working directly in the complex frame is that, in (55) and (56), part of information on the structure o f S M is hidden in the low er or der co efficients through the ener gy constra int : this information c o uld be recov ered only having the solution of the complete system of equations for C k up to the integrability for C 0 . Comparing (55) and (56) with (53) we get the re la tion S M ( x + iy ) ≡ C M = M X m =0 i M − m A ( m,M ) ( x, y ) . (86) As it is natural, the leading order complex co efficient depends only o n leading order Cartesia n coefficients. Considering (54) with k = M + 1, we get the equation for the leading order Cartesian co efficients, namely ∂ x A ( j − 1 ,M ) + ∂ y A ( j,M ) = 0 , j = 0 , 1 , ..., M + 1 , (87) whose solution can be immediately found (see Hietar inta [6], se c t.3.0): A ( m,M ) = m X k =0 M − m X j =0 ( − 1) k ( j + k )! j ! k ! a ( j + k,m − k ,M ) x j y k (88) where a ( j + k,m − k ,M ) are r e al integration constan ts. F r o m the relation ∂ z = 1 2 ( ∂ x − i∂ y ) (89) we hav e the express io n for the r − th deriv ative ∂ r z = 1 2 r r X s =0 r ! ( r − s )! s ! ( − i ) s ∂ r − s x ∂ s y . (90 ) In order to compute the oper ator (90) w e need the following in termediate result ∂ r − s x ∂ s y x j y k = j ! ( j − r + s )! k ! ( k − s )! x j − r + s y k − s , for k ≥ s and j ≥ r − s (91) whereas the res ult is zer o if k < s or j < r − s . Using it, the action of the op erato r (90) o n the monomial x j y k is ∂ r z x j y k = 1 2 r r X s =0 ( − i ) s r ! ( r − s )! s ! j ! k ! ( j − r + s )!( k − s )! x j − r + s y k − s . (9 2) Actually , we ar e int er ested only in the in the a ction of ∂ M z : in this c a se, since eac h deriv ative co rresp onds to decreasing the degree of the monomial by one, only the highest degree ter ms, with j + k = M , survive the action of ∂ M z . Applying (92) we g e t: ∂ M z x M − k y k = 1 2 M M X s =0 ( − i ) s M ! ( M − s )! s ! k !( M − k )! ( s − k )!( k − s )! x s − k y k − s . (93) Analogously to those in (91), we have that b oth conditions s − k ≥ 0 a nd k − s ≥ 0 must b e satisfied. They implies just s = k so that (93) turns out to be simply ∂ M z x M − k y k = 1 2 M ( − i ) k M ! ( M − k )! k ! k !( M − k )! = ( − i ) k 1 2 M M ! . (94) 12 The same result holds exchanging k with M − k : ∂ M z x k y M − k = ( − i ) k 1 2 M M ! . (95) Let us no w denote with ˆ A ( m,M ) the highest degree part of A ( m,M ) : ˆ A ( m,M ) = ( − 1 ) m M ! ( M − m )! m ! a ( M , 0 ,M ) x M − m y m . (96) Using (94) we then hav e ∂ M z A ( m,M ) = ∂ M z ˆ A ( m,M ) = i m 2 M M ! 2 ( M − m )! m ! a ( M , 0 ,M ) . (97) Remem b ering (86) we finally get ∂ M z S M = M X m =0 i M − m ∂ M z A ( m,M ) (98) = M X m =0 i M − m i m 2 M M ! 2 ( M − m )! m ! a ( M , 0 ,M ) (99) = i m 2 M M ! a ( M , 0 ,M ) M X m =0 M ! ( M − m )! m ! (100) = i m M ! a ( M , 0 ,M ) (101) which is just what we wanted to prov e since it is equiv ale n t to (84) and (85). 5 Examples W e illustra te the applica tions of the gener al appr oach describ ed so far with a selection of results, many of which are new. In particular, we provide some weakly in tegr able sy stems w ith linea r and quadr atic inv aria nt s and a recip e for highe r order exa mples. In order to compa ctify formulas, fr o m hereinafter with the subsc r ipt we denote the partial deriv ative with resp ect to the corresp onding v ariable (except when used to denote a comp onent of momentum). 5.1 W eakly in tegr able systems with linear in v arian t s The ansatz is I 1 = S p + ¯ S ¯ p , (102) where for simplicit y we have suppressed the subscript in the S function. The system of equations ensuing from the conserv ation condition is the follo wing : S ¯ z = 0 , (103) ℜ{ ( GS ) z } = 0 . (104) Equation (103) agrees with (68), confirming that S can b e an arbitra ry analytic function, S = S ( z ) . (105) According to (69), the conformal transforma tion is given b y dz dw = F ′ ( w ) ≡ S ( z ( w )) (106) 13 or equiv alent ly w = X + iY = Z dz S ( z ) . (107) Int r o ducing the “conformal” potential e G = | F ′ | 2 G = | S | 2 G = S ¯ S G, (1 0 8) eq.(104) reduces to ℜ{ e G w } = 0 , (109) which is rea dily solved in e G = g ( Y ) , (110) where g is an arbitra r y rea l function a nd, according to the definition of the co ordinate transformation, Y is the imaginary part of w . In this co ordina tes, the inv ar iant assumes the ‘normal form’ I 1 = P + ¯ P (111) or simply I 1 = P X . (112) Equation (104), in view of (105) and recalling the definition of G , can be rewr itten as ℜ{ S ′ ( E − V ) − S V z } = 0 . (113) W e r ecognize the structure of eq.(28), where now f 1 = ℜ{ S ′ ( z ) } = 0 (114) and f 0 = ℜ{ ( S V ) z } = 0 (115) and w e see that (11 4) coincides with (80) and solution (110) now a pplies to V , s o that we ha ve V = g ( Y ( x, y )) | S | 2 , (116) with g arbitrary . The general form of the second in v aria nt in the or iginal real coo rdinates is I 1 = ℜ{ S } p x + ℑ{ S } p y . (117) W e may provide tw o int er esting classes of weakly integrable sys tems a dmitting linear inv ariants. The firs t is obta ined by the simple obs erv ation that, if w e cho ose the level surface E = 0, it is no longer nece ssary that condition (114) b e satisfied. Any analytic function S = S ( z ) provides a s o lution through the corr esp onding conformal transformation. If Y , as ab ov e, denotes the new coo rdinate Y = Im  Z dz S ( z )  , (118) then the solution is given by the pair (116 – 11 7), pr ovided we consider only motions at the energ y level E = 0. The second class of weakly integrable systems is obtained with the following tric k. L e t us co nsider the analytic function f ( z ) and consider then the conformal transfor ma tion (106) with S ( z ) given by S = 1 a + f ′ ( z ) , (119) 14 with a co nstant. Reca lling definition (10 8), let us consider the “flat” conforma l p otential e G = 1. In this case, relation (108), using (119) gives G = E − V = 1 | S | 2 = a 2 + a ( f ′ + ¯ f ′ ) + | f ′ | 2 . (120) W e ca n there fo re in terpr et a 2 as the fixed v alue of the energy constant and get as a consequence the family of potentials V ( z , ¯ z ; E ) = − √ E ( f ′ + ¯ f ′ ) − | f ′ | 2 . (121) T o co mplete the solution, we hav e to write explicitly the co o r dinate transforma tion genera ted by (1 19), that is w = Z dz S ( z ) = az + f ( z ) . (122) The inv ariant is still of the form (117). 5.2 W eakly in tegr able systems with quadratic in v arian t s The ansatz is I 2 = S p 2 + ¯ S ¯ p 2 + e K . (123) The system of equations ensuing from the conserv ation condition is the following: S ¯ z = 0 , (124) e K ¯ z + S G z + 1 2 S ′ G = 0 . (125) Equation (124) is already familiar. Since e K is real, equation (125) has the following integrability condition ℑ{ S ′′ G + 3 S ′ G z + 2 S G z z } = 0 . (126) How ever, as ab ov e, we can directly simplify eq.(12 5). Accor ding to (69), the conforma l transforma tion is now given b y dz dw = F ′ ( w ) ≡ p S ( z ( w )) (127) or equiv alent ly w = Z dz p S ( z ) . (128) With the conformal p o tent ia l e G = | F ′ | 2 G = | S | G = p S ¯ S G, (129) eq.(125) reduces to e K ¯ w + e G w = 0 , (130 ) that is the first example of the set (73). Its int eg rability co ndition is ℑ{ e G ww } = 0 (131) which is rea dily solved b y e G = e A ( w + ¯ w ) + e B ( w − ¯ w ) , (132) where e A and e B are arbitr ary r eal functions. Putting this so lution in (125) g ives e K = e B − e A. (133) 15 Comparing with (129) and using real “separating” co o rdinates, X = ℜ{ w } , Y = ℑ{ w } , (1 34) the solution for the original function G is then G = e A ( X ; E ) + e B ( Y ; E ) | S | . (135) The inv ariant takes the ‘normal for m’ I 2 = P 2 + ¯ P 2 + e B − e A = 1 2 ( P 2 X − P 2 Y ) + e B − e A (136) or, in the original real coo rdinates, I 2 = 1 2 Re( S )( p x 2 − p y 2 ) + Im( S ) p x p y + e B ( Y ( x, y )) − e A ( X ( x, y )) . (137) Equation (131) can be re w r itten as ℑ{ S ′′ ( E − V ) − 3 S ′ V z − 2 S V z z } = 0 . (138) W e a gain recognize the structure of eq.(28), where no w f 1 = ℑ{ S ′′ ( z ) } = 0 (139) and f 0 = ℑ{ S ′′ V + 3 S ′ V z + 2 S V z z } = 0 (140) and we see that (139) co inc ides with (8 1 ). It can be prov en (see [28], section 2 ) that this condition, in addition to warrant strong int eg rability , is also necessary and sufficient in order that the confor mal factor | S | can b e written as a sum of t wo functions of X and Y , sa y | S | = S X ( X ) + S Y ( Y ) . (1 41) Since from (140) we hav e that solution (135) now applies to V , w e g et V = A ( X ) + B ( Y ) | S | . (142 ) with suitable A and B . In this case, X and Y a re pro pe r ly referred to as sep ar able co ordinates. F r om (135) and (141), w e deduce the rela tions e A ( X ; E ) = E S X − A ( X ) , (143) e B ( Y ; E ) = E S Y − B ( Y ) , (144) so that, eliminating the energy parameter through the Hamiltonian E = H = 1 2 ( P 2 X + P 2 Y ) + A + B S X + S Y , (145) the general form of the strong quadratic in v aria nt is I 2 = S Y ( P 2 X + 2 A ) − S X ( P 2 Y + 2 B ) S X + S Y . (146) 16 W e mention here s ome in tere s ting examples of weak integrability with qua dratic inv a riants. Consider first the poly nomial function S ( z ) = i z 2 . This is the simplest polynomia l whic h gives a p otential which is not automatically int eg rable at arbitrary energy . The corres po nding confor mal transformation is given by w = 2 − 1 / 2 (1 − i ) ln z (147 ) or in terms of the real v aria bles using pola r coor dinates X = 1 √ 2 ( θ + ln r ) (148) Y = 1 √ 2 ( θ − ln r ) . (149) F r o m the relation (135) it then follows that the p o tent ia l given by V = r − 2 [ A ( re θ ) + B ( re − θ )] , (150) is in tegr a ble at zero energy for arbitrary functions A and B . In [28], the cla s s o f systems with A ( X ) = 1 2 ( C − sin √ 2 X ) , (15 1) B ( Y ) = 1 2 ( C − sin √ 2 Y ) , (152) where C is a real constant, has b een inv estigated. Using p olar co ordina tes, the explicit form of the p otential is V ( r , θ ) = C − sin θ cos(ln r ) r 2 (153) and that of the second in v aria n t is I 2 ( p r , p θ , r, θ ) = r p r p θ − cos θ sin(ln r ) (154) where rp r = xp x + y p y (155) p θ = xp y − y p x . (156) A natural question one can ask is if the system is still integrable at ar bitrary energy v a lues. As it is well known, it is po ssible to answer this question by perfor ming careful numerical in vestigations, but, on pur e ly analytical grounds, the pr oblem is in general quite difficult. I n [28], this p otential has b een pr ov en to be non-integrable at v alues of the energy − ǫ 1 < E < ǫ 2 , with small enoug h ǫ 1 , 2 , for every v alue of C in the int er v a l (0 , 1]. The pro of has been obtained by applying the P o inc a r´ e non-existence theorem of additional inv a riants (see, e.g. [31, 32]), working in the transfor med coor dinates X , Y , ta king as unperturb ed part of the Hamiltonian the integrable system at E = 0 and as a small p er turbation the term asso cia ted to the conformal factor. A second example is that given by a g enerating function of the form S ( z ) = z − 2 . (157) The conformal transformatio n ca n then b e written in terms of the rea l v aria bles as X = 1 2 r 2 cos 2 θ = 1 2 ( x 2 − y 2 ) , (15 8) Y = 1 2 r 2 sin 2 θ = xy . (159) The corres po nding p otential is V ( X, Y ) = r 2 [ A ( X ) + B ( Y )] . (160) 17 since the conformal factor is | S ( X , Y ) | = 1 2 √ X 2 + Y 2 = 1 r 2 . (161) Cho osing for example A ( X ) = 4 X 2 , (162) B ( Y ) = (2 + a ) Y 2 + b, (163) where a is a constant such that 0 ≤ a ≤ 2 and b > 0 , w e get the family of p otentials V ( x, y ) = r 2 ( x 4 + y 4 + ax 2 y 2 + b ) . (164) This po tent ia l has a re la tive maximum at the origin wher e V (0 , 0) = 0, absolute minima placed symmetrica lly in the four q uadrants aro und the origin and gr ows as r 6 for r → ∞ . It allows b ound motion for all the admissible v alues of the energ y . F or a = 2 it is rotationally symmetric and therefor e it is “sup erintegrable” at zero energy . The second in v aria n t is I 2 ( p x , p y , x, y ) = x 2 − y 2 2 r 4 ( p 2 x − p 2 y ) − 2 xy r 4 p x p y + x 4 + y 4 − (4 + a ) x 2 y 2 . (165 ) What is rema r k a ble in this cas e is that already a t energ ies slig htly b elow or ab ov e the integrable level E = 0 , the dynamics is strong ly chaotic . This shows tha t identifying weakly integrable sys tems do es no t ne c essarily provide a ne arly integrable s y stem. 5.3 W eakly in tegr able systems with higher-order in v arian t s Exploiting the results obtained in [8, 9], several families of weakly integrable systems a dmitting cubic and quartic inv ar iants can b e o btained. Both cases ca n b e des crib ed w ith a similar approach. The inv ar iants hav e the form I 3 = 2 Re { S 3 p 3 + R 3 p } , (166) I 4 = 2 Re { S 4 p 4 + R 4 p 2 } + C 0 . (16 7) The conformal transformatio ns (6 9) res pe c tively give the nor mal forms I 3 = 2Re { P 3 + e R 3 P } , (168) I 4 = 2Re { P 4 + e R 4 P 2 } + C 0 , (169) with e R M = S 2 − M M M R M , M = 3 , 4 , (170) in agreement with definition (72). Introducing the K¨ ahler potential, eq.(77) provides the solutions for e R M : e R M = − M 2 K ww . (171) In the cubic case, the closure equation (66) expressed in terms of the K¨ ahler p otential gives ℜ{ ( K ww K w ¯ w ) w } = 0 . (172) In the quartic case, the closure equa tion is that deter mined b y eq.(75) that, with j = 2 and the nota tio ns adopted here, gives ( C 0 ) ¯ w = − 1 2 e G ( e R 4 e G 2 ) w . (173 ) 18 The reality of C 0 requires the in teg r ability condition ℑ{ ( K ww w K w ¯ w + 2 K ww ¯ w K ww ) w } = 0 . (174) Equations (172,17 4) illustrate the imp ortant difference betw een the previo us linear and quadra tic cases ( M = 1 , 2) and the higher order ( M > 2) ca ses. In the linear (see eq.(109)) and the quadra tic ca se (s e e eq.(131)), the integrability conditions are line ar differ e ntial equations : their solution is in terms of one or t wo arbitra r y functions resp ectively . Now, equations (172) and (174) are b oth nonline ar partial differential equations: they a ppea r very complicated and in pr a ctice imp ossible to solve in full generality . Only isola ted systems can b e identified. Using the ge ne r al expr ession (78), we get in b oth M = 3 and M = 4 cases a quadr atic equation of the for m f 2 E 2 + f 1 E + f 0 = 0 , (175) where the f k , k = 0 , 1 , 2 depe nd on S , Λ, Ψ a nd their deriv atives. Lo ok ing for str ong inv ariants, S is determined as ab ov e (cfr. eq.(82) and eq.(83)) and several so lutions can b e obtained by suitable assumptions on Λ and Ψ [8, 9]. Lo oking for weak inv aria n ts, a large clas s of solutions can be obtained in the simplest event uality E = 0. In fact, in this ca se the only condition we hav e to satisfy is f 0 = 0 and it ca n b e pr ov en that this equation is a nonlinear PDE in the v aria ble Ψ only . In fact, we get resp e ctively ℜ{ (Ψ ww Ψ w ¯ w ) w } = 0 , M = 3 (176) and ℑ{ (Ψ ww w Ψ w ¯ w + 2Ψ ww ¯ w Ψ ww ) w } = 0 , M = 4 . (177) Now, any solution of these equa tio ns in whatever co or dinate system c a n b e used to genera te a new weakly int eg rable system (a t E = 0) using an arbitra ry conformal transfor mation not included in the families (82) and (83). 6 Conclusions Much of the structure and ideas pr esented in this w or k has come from the g eometric formulation us ing the Jacobi metric as described in [7] a nd further de velop ed in [8] and [9]. One of the important results of those earlier works was the unification o f the no tions of weak and strong integrability in a co mmon framework. Another was the identification of the c r ucial ro le of the co nformal trans formations for analy zing integrabilit y in tw o dimensions. How ever, while the geometric picture has been very useful in these and other respects, it is clear from the present work that an approach which is mo re closely related to the mo r e traditiona l Hamiltonian picture can also b e very effective. 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