Invariants at fixed and arbitrary energy. A unified geometric approach

In v arian ts at fixed and arbitrary energy . A unified geometric approac h. Kjell Rosquist ∗ Departmen t of Ph ysics, Sto c kholm Univ ersit y , Albano v a Univ ersity Cen ter, Sto c kholm, Sw eden and Giusepp e Pucacco † Ph ysics Departmen t, Univ ersit y o f Rome ”T o r V ergata”, Via della R icerca Scien tifica, 1, I-00 133 Rome, Italy Published as: J. Phys. A: Math. Gen. 28 , 3235–32 52 (1995) Abstract Inv ariants at arbitrary and fixed energy (strongly and weakly conserv ed q uan tities) for 2-dimensional Hamiltonian systems are treated in a unified w ay . This is ac hieved by u t iliz ing th e Jacobi metric ge- ometrization of th e dynamics. Using Killing tensors w e obtain an in tegrability condition for qu ad ratic inv ariants which inv olves an arbitrary analytic function S ( z ) . F or inv ariants at arbitrary energy the function S ( z ) is a second degree polynomial with real second deriv ative. The in tegrability condition then reduces to Darb oux’s condition for quadratic inv ariants at arbitrary energy . The four types of classical quadratic in v ariants for positive definite 2-dimensional Hamiltonians are shown to correspond to certain conformal transformations. W e deriv e the explicit relation b et ween in vari ants in the ph ysical and Jacobi time ga u ges. In this w ay knowledge about the inv ariant in the physical time gauge enables one to directly write down the components of the corresp onding Killing tensor for th e Jacobi metric. W e als o discuss the p ossibilit y of searc hin g for linear and q uadratic in vari ants at fixed energy and its connection to the problem of t h e th ird integral in galactic dynamics. In our app roac h linear and quadratic inv ariants at fixed energy can b e found by solving a linear ordinary differentia l equ ation of the first or second degree respectively . ∗ Supported b y th e Swe di s h Natural Science Resea r c h Council. E- mail: kr@physto.se. † Supported b y INFN (Istituto Nazionale di Fisica Nucleare). E-mail: pucacco@roma2.infn.it. 1 1 In tro duction The quest for integrable systems is still one of the main are a s of in ter est in class ical dynamics b oth p er se and for the applica tio ns in the related fields of celestial mechanics, ac celerator physics and so on. In galactic dynamics a lo ng-standing a nd to this day unr e solv ed issue co nc e r ns the celebrated third integral. Briefly , the o bserved motion of star s in the ga laxy has a certain regular it y which indicates the existence of a third inv ariant in a ddition to the energy and ang ular momen tum. This picture is confirmed b y numerical simulations. In mo dern terminology , the motion is non-chaotic co n trar y to what one w o uld exp ect for a generic p otential. A particularly str iking numerical exp eriment was made b y H ´ enon and Heiles [1] in 1964 using a mo del p otential no w known as the H´ enon-Heiles po ten tial. They showed that the motion in that po ten tial is reg ular with inv ariant tori up to a certain ener gy v alue E 0 . At this energ y the inv ariant tori abrubtly b egin to break up and the motion becomes c haotic in increasingly large par ts o f the phase space . One conclusion is that ther e can b e no inv ariant which comm utes with the Hamiltonian. On the other hand the very high degree of regula rit y at low energ ies, E < E 0 , seems to indicate the existence of an additiona l inv ariant at those energies . Such an in v ariant, if it exists, must necessarily depend on the ener gy with some kind of singula r b ehaviour at E = E 0 . How ever, no such inv aria n t is known, for any v alue o f the energy . One of the purp oses of this pap er is to discus s the po ssibilit y to find inv ariants at fixed energy v a lues . An ess en tial asp ect of integrability is therefore to check for the existence of integrals of the mo tion in addition to the energy , namely phase-space functions inv ariant with re spect to the phase flow. The search for additional inv ariants ha s therefore been actively pursued for more than a century (Bertrand, 185 2 [2]; Darb oux, 19 01 [3]; for a complete review of the matter see Hietarinta [4]). A less known asp ect of the question is the r elation b et ween the inv aria n ts that most freq uen tly app ear in the a pplications, namely p olynomials in the momenta, a nd Killing tenso r s of the configurational manifold of the dynamical system. This relation is in the form of a straightforw a rd corres p ondence if th e J acobi appr oach to the ge ometrization of the dyna mics of a conser v ative system is a dopted. The framework of the Jacobi geometr ical for m ulation of the dynamics o ffers several conceptual and techn ic a l aids that give the p ossibility of shedding new light o n a n umber of asp ects o f integrabilit y . F rom the techn ic a l p oint o f view, it a llo ws the use of the p ow erful to ols of Riemannian geometry o n the c onfigurational manifold of the corr esponding na tural Jaco bi formulation of the dynamical system. F rom the conceptual po in t of view, it treats inv ar ian ts at fixed energy (the so called configura tional in v a rian ts o f Hall [5]) o n the same fo oting as the in v ar ia n ts at arbitra ry energy , th us rea lizing a unified g eometric approa c h. One aim of the present pap er is to demonstra te the fruitfulness of the geometric approach by providing the generaliza tio n of Darb oux’s conditions to include the configura tio nal inv ariants. In the particular but fundamen ta l case of tw o degrees of freedom, the Killing equatio ns for se c ond rank Killing tenso rs can be so lv ed in full generality , r esulting in a ll the k no wn cases of in teg rabilit y at arbitra ry energ y , b oth the classica l ones already obta ined b y Darb oux [3] and those re cen tly found to ex haust all the p ossibilities in t wo dimensions (Dorizzi et al. [6 ]). The substantially larger family of solutions of the Killing equation in the fixed energy case a llo ws one to find large new classes o f constrained integrable systems, with a backw ar ds pro cedure going fr om the knowledge o f the form of the inv ariants to the a ssessment of the s tructure of the po ten tials admitting constrained integrabilit y . Applying a new prescr iption for tra nsforming Hamiltonian symmetries b et ween different time gauges, the relation b et ween the in v aria n ts in the physical and Jacobi time gauges is explicitly determined. T he relation betw ee n the inv ariants a lso provides a shortcut to ca lc ulate the Killing t ens or from the knowledge of the inv ar ian t in the physical time gauge. 2 Geometric represen tation of the dynamics The geometrization o f the dynamics in terms o f the Jacobi geometry has b een known for ab out a century and c a n b e fo und in some text b oo ks [7, 8, 9]. In spite of this it is not widely known or used and therefore we o utline the main ideas in this se c tio n to make the pap er self-contained. Most Hamiltonian systems of ph ys ical interest hav e a geometr ic kinetic energy part, that is to say that the kinetic ener gy is a non- 2 degenerate quadr atic form in the mo men ta p α T = 1 2 h αβ p α p β , (1) where h αβ is a function of the configur ation v aria bles q α . The Ha miltonian itself then has the fo rm H = T + V ( q ) = E . (2) The indep endent v ariable t is often but not always the time. F or simplicit y w e shall r efer to the indep enden t v a riable as the time in this pap er. F or any given energy E of the sys tem w e can use the Hamiltonian H = H − E , (3) to repre sen t the dynamics provided that we imp o se the co nstrain t H = 0 . (4) F or any suc h zero energ y Hamiltonian we can repa r ametrize the system by introducing a new time v ariable t N defined by the r elation dt = N ( p, q ) dt N , (5) together with a redefined Hamiltonian H N = N ( p, q ) H = N T + N ( V − E ) = 0 . (6) The new Hamiltonian will then give the same equations of motion on the constraint surface H N = 0 (see e.g. [10]). W e shall use the term lapse funct io n for N ( p, q ) which defines the independent v ariable gauge. This usage is bo rrow ed fro m applications in general relativity wher e the lapse giv es the rate of ph ys ical time change relative to co ordinate time (see e.g. [11]). The lapse function can be taken as a n y no n-zero function on the phase spa ce. The dynamics of the system c an b e r epresen ted in a purely geometric fo rm ulation by exploiting the repara metr ization freedom. The passage to the geometric representation is a ccomplished by defining a new time v aria ble t J ( Jac obi time ) by the la pse ch o ice N = N J = [2( E − V )] − 1 . (7) Note tha t for a p ositiv e definite kinetic energ y E − V is always nonnega tiv e in the physically allowed region. The cor respo nding Hamiltonian has the for m H N = [2( E − V )] − 1 T − 1 2 . Thus H N has a co nstan t po ten tial ene r gy whic h can be subtracted without a ffecting the equa tio ns o f motion. This leads to the Jac obi Hamiltonian , H J , defined by H J = 1 2 N J h αβ p α p β = 1 2 J αβ p α p β , (8) where J αβ is the Jac obi metric of the system. This defines a geometry whic h contains a ll infor mation a bout the dyna mics . The dynamical metric is given in cov ar ian t form b y J αβ = 2( E − V ) h αβ . (9) The J acobi metric is ther efore conforma lly related to the or iginal kinetic metric h αβ . It follows that the dynamics of the system (2) is eq uiv alent to geo desic motion in the J a cobi geometry 1 ds 2 = J αβ dq α dq β . (10) 1 It may happen that there are other geometries which can also b e used to r epresen t the dynamics. In that case the dynamical geometry defined according to the prescription give n he r e is not unique. F or example an inequiv alent dynamical geometry may sometimes b e obtained b y p erforming a suitable canonical transform ation in some t i me gauge (see e.g. [12]). 3 Note that the Jacobi geo metry dep ends on the e nergy parameter E . This means that in gener al the Ja cobi geometries corres ponding to differe nt energy surfaces a re inequiv alent. The c o mplete geometric re presen ta- tion of the dynamics of the system is therefor e g iv en b y the geo desics o f a 1-par ameter family of geometries. Because of the form of the confor mal fac to r in the Jacobi geometry (9) the geometric representation is only v alid lo cally in co nfig uration space at p oints where V 6 = E . In gener al the equation V = E defines a non-empty ener g y surface in configuration space for a given v alue of E . Ex ceptions o ccur if e.g. V > 0 throughout the configuration space and E ≤ 0. In this pap er we are only concernced with the local existence of inv ar ian ts. Ther efore the failure of the J acobi geometry to represent the g lobal dynamics do es not affect our a na lysis. 3 P ositiv e d efinite t w o-dimensional Hamiltonian systems T o study 2- dimensional systems it is very helpful to use v ar iables which are null (light like) with resp e c t to the dynamical metric. Such v aria bles are naturally adapted to the action of the conformal g roup which pla ys an essential ro le for 2-dimens ional sy stems. The indefinite (Lor en tzian) sig nature ca se was discussed b y Rosquist and Uggla [1 3]. The null v ariables for tha t case are real and the conformal group can b e par ametrized by t wo a rbitrary r eal functions of one v ariable. Our a pproach is to use the method of [1 3] to treat the case o f a p ositive definite dynamical metric. W e consider a general 2-dimensiona l p ositive definite dynamical metric written in the manifestly confor - mally flat form ds 2 = 2 G ( x, y )( dx 2 + dy 2 ) . (11) W e see k a condition on G whic h guarantees the existence o f a s econd rank Killing tensor . T o that end w e follow [13] as closely as p ossible and intro duce null v ariables which in the p ositive definite case are co mplex z = x + i y , ¯ z = x − i y . (12) Here and in the following a bar is use d to denote c o mplex conjugation. The metric then be comes ds 2 = 2 G ( z , ¯ z ) dz d ¯ z . (13) Note that although our notatio n is similar to that in [13] the v ar iables z a nd ¯ z a re complex in the p ositive definite ca s e as opp osed to the Lo ren tzian cas e where they are real. When doing calcula tions it is co n venien t to emplo y either a n orthono rmal frame ω ˆ I using hatted upp e r case La tin indices ( ˆ I , ˆ J , ˆ K , . . . = ˆ 1 , ˆ 2) in terms of which the metric is written as ds 2 = ( ω ˆ 1 ) 2 + ( ω ˆ 2 ) 2 , (14) or a complex null fra me Ω I using upp er ca se Latin indices ( I , J, K, . . . = 1 , 2) with the metric written as ds 2 = 2Ω 1 Ω 2 . (15) The fra me compo nen ts a re given b y ω ˆ 1 = (2 G ) 1 / 2 dx , Ω 1 = G 1 / 2 dz , ω ˆ 2 = (2 G ) 1 / 2 dy , Ω 2 = G 1 / 2 d ¯ z , (16) The t wo frames are related b y Ω I = e I ˆ I ω ˆ I or ω ˆ I = e I ˆ I Ω I where e I ˆ I is the transp osed matrix inv erse of e I ˆ I . The tr a nsformation matrix is given b y ( e I ˆ I ) = 1 √ 2 i √ 2 1 √ 2 − i √ 2 ! , ( e I ˆ I ) = 1 √ 2 − i √ 2 1 √ 2 i √ 2 ! . (17) 4 The simplest sy mmetry of the geo desic equations is known as a Killing vector field (see e.g. [14]). A second r a nk Killing tensor is a symmetric tensor K I J = K ( I J ) satisfying the equatio n K ( I J ; K ) = 0 . (18) The existence of such a tensor is e q uiv alent to the existence of a p olynomial inv aria nt of the second degr ee K I J p I p J for the geo desic eq uations. Killing tensor s and second degree in v ar ian ts are the natural g eneral- izations of Killing v ector s K I and the corresp onding fir st deg r ee inv aria n ts K I p I . Any Killing vector gives rise to a Killing tensor K ( I K J ) . Such a K illing tensor is said to b e re ducible . An imp ortant prop ert y of the seco nd rank Killing tensor equations is that they ca n be decomp osed in conforma l (traceless) and tra ce parts a ccording to P ( I J ; K ) − 1 2 h ( I J P L K ); L = 0 , K ; I = − P J I ; J , (19) where the Killing tensor itself is deco mposed in a conformal part P I J and the trac e K = K I I according to K I J = P I J + 1 2 K h I J . (20) Referring to the vector P I = P J I ; J as the confor mal curre n t it follows fro m (19) that for a n y given conformal Killing tenso r the equa tion for the trace c a n be solved if the integrabilit y condition P [ I ; J ] = 0 (21) is s atisfied. The pro cedure to solve the Killing tensor equations is therefore to fir st solve the conformal Killing tensor eq ua tions and then chec k if the integrability condition (21 ) for the trace can b e sa tisfied. In fact it turns out that the c o nformal equations can ea sily be s olv ed leaving the integrabilit y condition as the remaining e q uation to study . Let K ˆ I ˆ J be the comp onen ts of a Killing tensor with resp ect to the orthonormal frame ω ˆ I . Then the null frame comp onents are given by K I J = e I ˆ M e J ˆ N K ˆ M ˆ N which gives K 11 = 1 2 ( K ˆ 1 ˆ 1 − K ˆ 2 ˆ 2 ) − iK ˆ 1 ˆ 2 , K 22 = 1 2 ( K ˆ 1 ˆ 1 − K ˆ 2 ˆ 2 ) + iK ˆ 1 ˆ 2 , K 12 = K / 2 = 1 2 ( K ˆ 1 ˆ 1 + K ˆ 2 ˆ 2 ) , (22) where K = K I I is the tra ce. Since the orthonorma l comp onen ts K ˆ I ˆ J are b y definition r eal it fo llows that K 22 = K 11 . Ther e fore we can r epresen t the null fra me comp onents of the confor mal par t of the Killing tensor by a sing le complex function S ( z , ¯ z ) acc o rding to (cf. [13]) ( K M N ) =  ¯ S G K/ 2 K/ 2 S G  , (23) the trace K b eing a real function. The confo r mal K illing tensor equations ar e for mally iden tical to those obtained in the Lorentzian case [13] C 111 = G 1 / 2 ¯ S ,z = 0 , C 222 = G 1 / 2 S , ¯ z = 0 . (24) It follows that P M N is a conformal Killing tensor precisely if S is a function of z only . One wa y of stating this result is that the equations (24) for the conformal Killing tensor coincide with Cauch y- Riema nn’s equations for cer tain rescaled linea r combinations of the trac e fr ee parts of the Killing tenso r. This result was implicit in [15]. Thus the conformal K illing tensor has the for m ( P M N ) =  ¯ S ( ¯ z ) G 0 0 S ( z ) G  . (25) 5 The tr a ce equations can then b e wr itten in the for m K ,z = − 2 ¯ S ( ¯ z ) G , ¯ z − G ¯ S ′ ( ¯ z ) , K , ¯ z = − 2 S ( z ) G ,z − GS ′ ( z ) , (26) the der iv ative K , ¯ z being given by taking the complex conjugate o f the ab ov e equation. As in the Lorentzian c ase there is an infinite-dimensional family of conformal Killing tensors in this case parametrized by the single complex ana lytic function S ( z ). F ollowing the pro cedure in [13] we no w wr ite down the in teg rabilit y condition for the conforma l curr en t. This is exa c tly what we need to guar an tee the existence of a Killing tensor . In the p ositive definite case the integrability condition b ecomes P [1;2] = − G − 1 G ,z z S ( z ) + G − 1 G , ¯ z ¯ z ¯ S ( ¯ z ) − 3 2 G − 1 G ,z S ′ ( z ) + 3 2 G − 1 G , ¯ z ¯ S ′ ( ¯ z ) − 1 2 S ′′ ( z ) + 1 2 ¯ S ′′ ( ¯ z ) = 0 , (27) or in a mo re compact for m P [1;2] = − i ℑ{ 2 G − 1 G ,z z S ( z ) + 3 G − 1 G ,z S ′ ( z ) + S ′′ ( z ) } = 0 , (28) where ℑ deno tes the imagina ry part. As in [13] we use a conformal transfo r mation to standardize the frame and co ordinate repr esen tation of the conformal Killing tensor . T o that end we introduce the new nu ll frame ˜ Ω 1 = B Ω 1 , ˜ Ω 2 = B − 1 Ω 2 and a new complex co ordinate b y the transforma tion w = H ( z ) with inv erse z = F ( w ). By choosing B = ( ¯ F ′ ( ¯ w ) /F ′ ( w )) 1 / 2 we ensure that the new frame ˜ Ω I has the co ordinate representation ˜ Ω 1 = ˜ G 1 / 2 dw , ˜ Ω 2 = ˜ G 1 / 2 d ¯ w where ˜ G = | F ′ ( w ) | 2 G is the new metric confor mal fa ctor, ds 2 = 2 ˜ Gdwd ¯ w . It follows that the conformal Killing tensor comp onen ts in the new frame a re giv e n by ˜ P 11 = B − 2 P 11 = [ ¯ H ′ ( ¯ z )] 2 ¯ S ( ¯ z ) ˜ G , ˜ P 22 = B 2 P 22 = [ H ′ ( z )] 2 S ( z ) ˜ G . (29) The co ordinates are standar dized by choosing a confor mal transfor mation function H ( z ) which satisfies [ H ′ ( z )] − 2 = S ( z ) implying that ˜ P 11 = ˜ P 22 = ˜ G . Note that det( P I J ) is always p ositiv e . Therefore there is only one type of Killing tensors for p ositive definite Hamiltonians unlik e the situation in th e Lorentzian case (see [13]). F or a s ta ndardized conformal K illing tensor the integrability (2 7) condition s implifies to ˜ G ,w w = ˜ G , ¯ w ¯ w . (3 0) W riting w = X + iY the solution o f this equatio n is ˜ G = Q 1 ( X ) + Q 2 ( Y ) . (31) This is the us ual form of the po ten tial in separ able coo r dinates. The tr a ce equation [see (19)] ca n then b e written K ,X = − 2 ˜ G ,X , K ,Y = 2 ˜ G ,Y , (32) with the solution K = − 2 Q 1 ( X ) + 2 Q 2 ( Y ) . (33) The in tegra bilit y conditio n (27) can b e in terpreted as fo llows. Giv en any a nalytic function S ( z ) there is a family of po tentials (the solutions of (27)) which is integrable at z ero energy . F or future reference we note that the orig inal potential is given in terms o f the standar dized po tential by the relation G = | S ( z ) | − 1 ˜ G . (34) 6 W e wish to express the in v ar ia n t I J = K M N p M p N in terms of the co ordinate mo men tum compo ne nts. The subscr ipt J is used to distinguish the inv ar ia n t in the Jacobi time gaug e from the inv aria n t I in the ph y sical time g auge (see section 6). By (16 ) the null fra me comp onents ar e related to the complex co ordinate comp onen ts b y ( p 1 , p 2 ) = G − 1 / 2 ( p z , p ¯ z ). The inv ar ian t ca n then b e written in complex co ordina tes as I J = S ( z ) p z 2 + ¯ S ( ¯ z ) p ¯ z 2 + G − 1 K p z p ¯ z . (35) Using the relation p z = 1 2 ( p x − ip y ) , p ¯ z = 1 2 ( p x + ip y ) , (36) implied by (12) we obtain the inv ar ian t in real c o or dina tes as I J = 1 2 ℜ ( S )( p x 2 − p y 2 ) + ℑ ( S ) p x p y + 1 4 G − 1 K ( p x 2 + p y 2 ) . (37) Note that the la s t term is equal to K H J . 4 Arbitrary energy in v arian ts A common situation is that one is interested in in v ariants whic h are v alid for a rbitrary v alues of the energ y . Inv ariants of that t yp e arise if the in teg rabilit y condition itself is indep enden t of the energy . This happe ns precisely if ℑ{ S ′′ ( z ) } = 0. Since S ′′ ( z ) is an ana lytic function it follows that S ′′ ( z ) is a re al constant a nd therefore S ( z ) m us t be a seco nd degree p olynomial S ( z ) = W ( z ) := az 2 + β z + γ , (38) where a is a real constant and β and γ ar e complex constants. The integrability co ndition (27) then simplifies to ℑ{ 2 G ,z z S ( z ) + 3 G ,z S ′ ( z ) } = 0 . (39) This is nothing but Darb oux’s condition for quadra tic co nstan ts of the motion [3, 4] written in complex v a riables. The general integrability condition (27) therefore g eneralizes Darb oux’s condition to include als o quadratic inv aria n ts a t fixed e ne r gy . F or a given analytic function S ( z ) the po ten tial may b e written a s G ( X , Y ) = | F ′ ( w ) | − 2 ( Q 1 ( X ) + Q 2 ( Y )). In the arbitrar y energ y c a se w e hav e S ( z ) = W ( z ) = [ H ′ ( z )] − 2 . The equation H ′ ( z ) = [ W ( z )] − 1 / 2 can then be integrated resulting in (without loss of generality we may put a = 1 if a 6 = 0 since (39) is in v ariant under a rea l scaling of S ) w = H ( z ) =    log( √ W + z + β / 2) + const , ( a = 1 ) , 2 β − 1 √ β z + γ + const , ( a = 0 , β 6 = 0) , γ − 1 / 2 z + const , ( a = 0 , β = 0) . (40) Inv erting these relations we must distinguish betw e e n the four ca ses (i) a = 1 , ∆ := p β 2 / 4 − aγ 6 = 0 , (ii) a = 1 , ∆ = 0 , (iii) a = 0 , β 6 = 0 , (iv) a = 0 , β = 0 , (41) giving r ise to the following explicit formulas for F ( w ) z = F ( w ) =        ∆ cosh( w − w 0 ) − β / 2 , (i) e w − w 0 − β 2 , (ii) 1 4 β ( w − w 0 ) 2 − γ /β , (iii) γ 1 / 2 w − w 0 . (iv) (42) 7 where w 0 is an arbitrar y consta n t. W e ar e primar ily interested in prop er co nformal transfor mation, i.e. S ( z ) 6 = 1. Therefore w e are free to p erform tr a nslations and rotations in z and w to simplify for m ulas. In particular w e may put w 0 equal to zero b y a translation of the or igin in the w -pla ne. W e can also rotate the z -plane to transform ∆ to a po sitiv e real num b er in case (i). Likewise β may b e assumed r eal in case (iii) and γ can b e taken as real a nd p ositive in case (iv ). The ar gumen t we used to set a = 1 ca n then be used to set β = 4 and γ = 1 for conv e nie nce in c a ses (iii) and (iv) resp ectively . W e also translate the z -origin obtaining finally F ( w ) =      ∆ cosh w , (i) e w , (ii) w 2 , (iii) w , (iv) (43) where ∆ is now to b e understo o d as a po sitiv e real num b er. Differentiation of the expres sions (43) yields F ′ ( w ) =      ∆ sinh w , (i) e w , (ii) 2 w , (iii) 1 , (iv) (44) implying tha t the conforma l transformation facto r s tak e the for ms | S ( z ) | = | F ′ ( w ) | 2 =        ∆ 2 (sinh 2 X + sin 2 Y ) = p ( r 2 + ∆ 2 ) 2 − 4∆ 2 x 2 , (i) e 2 X = r 2 , (ii) 4( X 2 + Y 2 ) = 4 r , (iii) 1 , (iv) (45) where r := p x 2 + y 2 . In ca se (i) the conformal co ordinate transforma tion is given b y x = ∆ co sh X co s Y , y = ∆ sinh X sin Y . (46) This is the transfor mation used to define the St¨ ac kel p otential [16] which was also discussed b y Darbo ux [3]. Case (i) therefore gives St¨ ack el’s classic a l integrable p otential. F ro m (45) a nd (34) we find that the potential is given by the well-known form ula [4] G ( x, y ) = Q 1 ( X ( x, y )) + Q 2 ( Y ( x, y )) ∆ 2 [sinh 2 X ( x, y ) + sin 2 Y ( x, y )] = A ( ξ ( x, y )) + B ( η ( x, y )) ξ ( x, y ) + η ( x, y ) , (47) where ξ ( x, y ) := 2∆ 2 sinh 2 X = r 2 − ∆ 2 + p ( r 2 + ∆ 2 ) 2 − 4∆ 2 x 2 , η ( x, y ) := 2∆ 2 sin 2 Y = − r 2 + ∆ 2 + p ( r 2 + ∆ 2 ) 2 − 4∆ 2 x 2 . (48) and the functions A a nd B are a rbitrary functions of their ar gumen ts. The ca se (ii) conformal tra nsformation is given by x = e X cos Y , y = e X sin Y . (49) In par ticular it follows tha t Y = a rctan( y/ x ) =: φ is a p olar ang le. Using (45) it then follows that the po ten tial ca n b e written a s G ( x, y ) = A ( r ) + r − 2 B ( φ ) , (50) where A and B are arbitra ry functions. This is again a cla ssical in tegra ble case [4] known as the Eddington po ten tial. Cas e (iii) is characterize d b y the conformal transfor ma tion x = X 2 − Y 2 , y = 2 X Y . (51) 8 case conformal trans formation w = H ( z ) conformal Killing tensor comp onen t S ( z ) = [ H ′ ( z )] − 2 Separating co ordinates (i) ln z ± √ z 2 − ∆ 2 z 2 − ∆ 2 Elliptical (ii) ln z z 2 Spherical (iii) √ z 4 z Parabo lic T able 1 : Conformal transformation functions giving rise to systems with a second inv ariant whic h is linear or quadratic in the mo menta. Using (45) this g ives immediately the likewise w ell-known classic a l in tegra ble potential [4] G ( x, y ) = [ A ( r + x ) + B ( r − x )] /r , (52) where a gain A and B a r e arbitrary functions. Finally in case (iv) the conformal tr ansformation factor is unit y and so the p oten tial is s imply given b y the explicitly s eparated form G ( x, y ) = A ( x ) + B ( y ) , (5 3 ) in terms of the arbitrar y functions A a nd B . This completes the lis t of the four c lassical cases . The three nontrivial c onformal transfor mations giving rise to systems with a se c ond linear o r quadratic in v aria nt are tabulated in table 1. 5 The Killing v ector sub case Although Killing vectors c o rresp ond to reducible se c o nd ra nk Killing tensors it is nevertheless w orthwhile to g iv e a separ ate treatment of that sub case. As will be shown b elo w the integrabilit y condition for Killing vectors is a first o rder differential equa tion. In so me applications it can a dv antageous to inv estig ate this simpler case b efore tac kling the se c ond rank Killing tensors. A Killing vector can b e described b y a function Z ( z , ¯ z ) acco rding to (cf. [13]) K 1 = G 1 / 2 ¯ Z , K 2 = G 1 / 2 Z . (54) The K illing vector equations then bec o me K (1;1) = K (2;2) = Z , ¯ z = 0 , (55) K (1;2) = 1 2  Z ,z + ¯ Z , ¯ z + G − 1 G ,z Z + G − 1 G , ¯ z ¯ Z  = 0 . (56) This shows that Z dep ends only on z a nd is ther efore a complex analytic function. It follows that the remaining Killing vector equation (56) r educes to a form ana logous to the integrabilit y condition fo r the second r a nk Killing tensor s (27). It can also b e written in the for m K (1;2) = G − 1 ℜ{ GZ ′ ( z ) + G ,z Z ( z ) } = G − 1 ℜ{ ( GZ ) ,z } = 0 . (5 7 ) An y Killing vector gives rise to a second rank Killing tensor g iv en by K I J = K I K J + ch I J , (58) where c is a n arbitrary (re a l) constant. The co nformal Killing tensor comp onents then b ecome P 11 = K 11 = ( K 1 ) 2 = G ¯ Z 2 , P 22 = K 22 = ( K 2 ) 2 = GZ 2 , (59) 9 Referring to (23) it follows that the confo r mal Killing tensor is determined by the analytic function S = Z 2 . (60) As in the se cond rank case, in v aria nce at arbitrary energy in volves a f ur ther restriction coming from the requirement that (56) should b e inv ar ian t with r espect to energy r e definitions. This means tha t w e must impo se the conditio n ℜ [ Z ′ ( z )] = 0 , (61) leading to the simplified Killing vector equation G ,z Z + G , ¯ z ¯ Z = 0 . (62) where Z ( z ) = U ( z ) := ibz + δ , (63) and b a nd δ are real and complex constants resp ectiv ely . It follows tha t the cor respo nding second rank conformal Killing tensor is deter mined b y S ( z ) = [ U ( z )] 2 = − b 2 z 2 + 2 i bδ z + δ 2 . (64) Thu s S ( z ) is an even square for a r e ducible Killing tensor so the condition ∆ 2 = β 2 / 4 − aγ = 0 is alwa ys satisfied. 6 T ransforming Ha miltonian symmetries b et w een differen t time gauges In this section we a ddress the pro blem of how an inv ariant is affected when we transform from one time gauge to a nother. Supp ose we ar e given a Hamiltonia n H with a second inv aria n t I so that { I , H } = 0. In another time gauge the Hamiltonia n is given by H N = N H = N ( H − E ) where N = dt/d ˜ t . O f particular int er est for the purp oses of this pap er is the trans formation b et ween the physical a nd Jac obi time gauges . In that cas e, g oing from the ph ysical time ga uge to the J a cobi time gauge, dt = N dt J , w e hav e N = | 2 V | − 1 . Since the Poisson brack et { I , H N } = H { I , N } , (65) is in g eneral non- v anishing off t he zero energy surface H = 0, the original inv ariant do es not have a v anishing Poisson brack et with the Hamiltonian in the new time gaug e. Borrowing usag e fro m Dirac’s theory o f constrained Hamiltonians we may say that the inv aria n t is only weakly conserved in the new time g auge. F or a linear in v aria n t, e.g. the Killing vector c ase, one ca n a lways choos e v aria bles such tha t I = p ˜ x and ˜ x is a cyclic v aria ble in the Ha miltonian. Then for the Jacobi time ga ug e, N dep ends only o n the remaining v a riable and hence { I , H N } = 0 implying that I is s trongly conserved in the new time gauge. Supp ose now that w e hav e a 2 -dimensional sys tem with a second inv ariant given in the ph y sical time gauge . T he n, the system is integrable a nd we ca n express the lapse explicitly as a function N = f ( t ) of the physical time t . Integrating the rela tio n d ˜ t = dt/f ( t ) then gives the new time as an explicit function of the old time at least up to a qua drature. In principle, the so lutio ns can consequently always be transfor med to a nother time gauge. One therefore exp ects that the integrability prop erties o f a dynamica l system ar e indep enden t of the time gauge. In or der to exploit fully t he geometrica l formulation of mec hanics it is desirable to find a corres p onding inv ariant whic h is strongly conser v ed in the Jacobi time gauge. As we shall see this is actua lly po ssible at least in the c a ses considered in this pap er. T o find the in v ar ian t I N in the ne w time gauge it is conv enient to use a n ansatz of the form I N = I + R H N . (66) 10 W e look for a condition on R whic h guar an tees that I N is a constant of the motion for H N . T o that end w e form the Poisson brack et { I N , H N } = N H N h { R, H } − { I , N − 1 } − H N { R, N − 1 } i . (67) Requiring this expre s sion to v a nish off the zero energy sur fa ce yields the condition { R, H } = { I , N − 1 } + H N { R, N − 1 } . (68) This is a necessary and sufficient condition for I N to b e a co ns tan t of the mo tion fo r H N . In general a solution to this partial differen tia l equatio n would b e difficult to find. F or our purpos es, howev er, it turns out that we can actually find a so lutio n b y a simple pro cedure which w e now o utline. Supp ose now that I is a quadra tic in v ariant. Then if N is a function on the configura tion space, the brack e t { I , N − 1 } is a linear function of the momenta. Supp ose we find a function R on the configuratio n space which satisfies { R, H } = { I , N − 1 } . Then since H N { R, N − 1 } = 0 we ha ve a solution of (68). In this wa y the pro cedure to find the inv ar ia n t in the new time g a uge is r educed to calc ula ting the Poisson brack et { I , N − 1 } and finding a function whose tim e deriv ative coincides with that brack et. Although w e do no t know under which conditions this pro cedure w o r ks it does work in the cas e s considered in this pap er. T o summar ize we first calculate the function { I , N − 1 } and chec k whether it can b e expr e ssed a s a total time der iv ative of some function R . If in addition { R , N − 1 } = 0 then R is the required function which satisfies (68). 7 The relat ion b et w een the quadratic in v arian ts in the ph ysical and Jacobi time gauges W e w is h to see how the in v aria n t (37 ) app ears in the physical time gauge in ter ms o f the conformal f unctio n S ( z ) and the tra ce K . T o that end we use the pr ocedure o utlined in s ection 6 going “ bac kwards” fro m the Jac o bi time g a uge to the physical time gaug e. Referring to sectio n 2 the sta rting p oint is now the Jacobi Hamiltonian H = H J = 1 4 G − 1 ( p x 2 + p y 2 ) co ns idered at the ener gy v alue 1 / 2 s o H = H J − 1 / 2. The transfor mation from the J acobi time to the ph ys ical time is then given by dt J = N dt wher e N = 2 G . According to (66) we write the physical time inv aria n t as I = I J + R N H . Using the presc r iption in section 6, R should satisfy the equatio n { I J , N − 1 } = { R, H J } . Ca lculating the left hand side { I J , N − 1 } with I J given b y the expr ession (37) gives { I J , N − 1 } = G − 3 ( S GG ,z + 1 2 K G , ¯ z ) p z + G − 3 ( ¯ S GG , ¯ z + 1 2 K G ,z ) p ¯ z . (69) Assuming that R is a function o n the configuration spa ce the rig h t hand side is given by { R, H J } = G − 1 ( R , ¯ z p z + R ,z p ¯ z ) . (70) Comparing equa tions (69) and (70) then leads to R ,z = ¯ S G − 1 G , ¯ z + 1 2 K G − 2 G ,z , R , ¯ z = S G − 1 G ,z + 1 2 K G − 2 G , ¯ z . (71) A t this p oint it is convenien t to introduce a function Q = R + (1 / 2) K G − 1 . The equations (7 1) then reduce to Q ,z = − 1 2 ¯ S , ¯ z , Q , ¯ z = − 1 2 S ,z (72) 11 where w e hav e used the trace equation (2 6 ). The integrabilit y condition for this equation coincides with the condition for integrability at arbitrary e nergy , ℑ S ′′ ( z ) = 0. Using (38) w e then find that the solution of (72) is given by Q = − az ¯ z − 1 2 ¯ β z − 1 2 β ¯ z + Q 0 , (73) where Q 0 is a (real) integration cons tan t. Collecting o ur results we find that the expr e ssion for the inv ariant in the physical time ga uge is I = I conf + 1 2 Q ( p x 2 + p y 2 ) + K 2 − GQ , (74) where I conf = P M N p M p N = 1 2 ℜ ( S )( p x 2 − p y 2 ) + ℑ ( S ) p x p y , (75) is the conformal part o f the in v ariant. It follows that this part is the sa me in both time gauges. It is also seen that Q can be identified with the trace of the ph ysica l inv ar ian t with resp ect to the ph ysical metric δ ab where a and b are co ordinate indices taking the v alues x and y . The relation b et ween the in v ariants pr o vides a sho r tcut to calcula te the Killing tens or from knowledge of the ph y s ical inv ariant. W e shall o utline this pro cedure and then illustr ate with an ex a mple. Let the physical inv ariant be g iv en by an expr ession of the form I = Q ab ( x, y ) p a p b + f ( x, y ) , (76) The physical Hamiltonian is given by H = 1 2 δ ab p a p b + V ( x, y ) , (77) W e can r ead off the function Q by Q = δ ab Q ab . T aking the conformal par t and identifying with (75) we obtain S = 2 ( P xx + iP xy ) , (78) where P ab = Q ab − 1 2 Qδ ab and P y y = − P xx . Finally we o bta in the Killing tensor tra c e as K = 2( f + GQ ) putting G = E − V . Let us now illustra te the above r esults by an e xample. W e take the Kepler p otent ia l V = − µ/ r wher e r = p x 2 + y 2 with its well-kno wn non-trivia l qua dratic inv ariant, the Laplace-Rung e-Lenz vector (see e.g . [17]) with comp onent s L 1 := e x = 1 µ ( xp y 2 − y p x p y ) − x r , L 2 := e y = 1 µ ( y p x 2 − xp x p y ) − y r . (79) The ho mogeneous and inhomog eneous parts of the inv aria n ts L 1 and L 2 are conseque ntly given by Q ab (1) =  0 − y / (2 µ ) − y / (2 µ ) x/µ  , f (1) = − x/r , (80) and Q ab (2) =  y /µ − x/ (2 µ ) − x/ (2 µ ) 0  , f (2) = − y / r . (81) It follows that the confor mal Killing tensor comp onents are given b y P xx (1) = − P y y (1) = − x/ (2 µ ) , P xy (1) = − y / (2 µ ) , P xx (2) = − P y y (2) = y / (2 µ ) , P xy (2) = − x/ (2 µ ) , (82) while the traces a re given b y K (1) = 2( E /µ ) x , K (2) = 2( E /µ ) y . (83) 12 8 Applications to in tegrabilit y at fixed energy An imp ortant lesson to b e le a rned from the pr e sen t work is that integrabilit y a t fixed energy and arbitrar y energy (weak and strong conser v ation laws) are just t wo a s pects of the sa me pheno meno n. In particular a fixed energ y inv ar ian t in the physical time gaug e corresp onds to an arbitra ry energy inv ariant if the system is geo metrized b y going to the Ja cobi time gaug e. T o illustra te how this works in practice we g iv e a few examples in sectio n 8 .1 of fixed energy inv ariants beginning with the Kepler p otential. Surprisingly , we find t wo apparently unkno wn linear inv aria n ts at zero ener gy for the Kepler problem. I t is r emark able that it is still p ossible to dis co ver new prop erties of such a simple and well-known sy s tem. This is in fact a sign of the power of the geometric formulation of dynamica l sys tems . In section 8.2 w e discuss how conformal transformatio ns can be used to g enerate systems which are integrable a t fixed energ y . 8.1 Degeneracy of t he Laplace-Runge-Lenz v ector at zero energy Consider now the Kepler p otential, V = − µ/r , using p olar co ordinates defined by x = r cos φ , y = r sin φ . In this ca se we hav e one linear inv ar ian t at arbitra r y energy , the angula r momentum p φ . W e a re interested in finding out whether there e xists a nother linear in v aria n t at some fixed energy v alue. If this is the case the Jacobi metric (9) has tw o Killing v ectors . Howev er, a 2-dimensio nal space with t wo Killing v ector s mu s t necessarily a lso hav e a third Killing vector ([18], Theor em 8.15) and the geometry is then a space of c onstan t curv ature. This can easily b e deter mined by computing the scalar curv ature (2) R = 2 G − 2 G ,z ¯ z − 2 G − 3 G ,z G , ¯ z , (84) of the metric (13) and checking if it is constant. F or the K epler po ten tial we hav e G = E + µ ( z ¯ z ) − 1 / 2 and the sca lar curv a ture be c o mes (2) R = − E µ 2[ µ + E ( z ¯ z ) 1 / 2 ] 3 . (85) This s ho ws that the J a cobi geometry has constant curv ature only if E = 0 and then the ge ometry is ac tually flat. Of co urse the fla tness of the Jacobi geometry in this case also follows directly from the form o f the metric since G is then a pro duct of functions of z and ¯ z . The t wo extr a Killing vector fie lds can immediately be written down if we in tr o duce Cartesia n coor dina tes, ( X, Y ), for the Jacobi geometry by the transfor mation w = X + iY = √ 2 z leading to the manifestly Euclidea n form ds J 2 = 4 µ ( dX 2 + d Y 2 ) . (86) The r elation to the origina l coor dinates is given b y the par abolic transfor mation x = 1 2 ( X 2 − Y 2 ) , y = X Y . (87 ) This tra nsformation do es not hav e a unique in verse. How ever, in the r egion X = ℜ ( w ) > 0 we may se lect the inv er se transformation to b e X = √ r + x , Y = (sgn y ) √ r − x . (88) It follows that the Killing vector fields are K (1) = ∂ / ∂ X , K (2) = ∂ /∂ Y and K (3) = − Y ∂ / ∂ X + X ∂ /∂ Y = 2 ∂ / ∂ φ . The tw o extra Killing vector fields are thus the transla tion sy mmetries K (1) and K (2) . The corre- sp onding inv ariants are I 1 := p X = ( r + x ) 1 / 2 p x + (sgn y )( r − x ) 1 / 2 p y = (2 r ) 1 / 2 cos( φ/ 2) p r − (2 / r ) 1 / 2 sin( φ/ 2) p φ , I 2 := p Y = − (sg n y )( r − x ) 1 / 2 p x + ( r + x ) 1 / 2 p y = (2 r ) 1 / 2 sin( φ/ 2) p r + (2 / r ) 1 / 2 cos( φ/ 2) p φ . (89) These tw o in v ar ian ts ar e related b y the q uadratic formula I 1 2 + I 2 2 = 8 µH 0 J where H 0 J = − (2 V ) − 1 T = 1 2 is the Jacobi Hamiltonian for the zero energy system. In terms of the ph ysical Hamiltonian, the cor respo nding 13 relation is I 1 2 + I 2 2 = 4 µ (1 − H/ V ) = 4 µ . In fact here we hav e the k ey to the physical interpretation of I 1 and I 2 . T o see this let us introduce a n in v aria n t φ 0 at ze ro energy by the r elations I 1 = − 2 µ 1 / 2 sin( φ 0 / 2) , I 2 = 2 µ 1 / 2 cos( φ 0 / 2) . (90) Now using (89) to s olv e for the radial momentum yields p r = p 2 µ/r sin[( φ − φ 0 ) / 2]. Inserting this v alue int o the Hamiltonian constraint H = 0 and solv ing for r giv es the familia r relation r = p φ 2 /µ 1 + cos( φ − φ 0 ) . (91) This shows that φ 0 is nothing but the angular integration constant. It follows that changing the v alue o f I 1 (or I 2 ) only affects the parametr iz ation of the or bit while leaving the o rbit itself inv ariant. F rom this p oint of view these inv ar ia n ts a re gauge symmetr ie s of the K epler system a t zero energ y . The in v ar ia n ts I 1 and I 2 are in fact closely related to the comp onen ts of the Laplace-Runge-Lenz v ector . Expressing tho se comp onen ts as e x = e cos φ 0 , e y = e s in φ 0 , (92) where e is the eccentricity ( e = 1 at zero ener gy) and compa ring with (90) it is evident that L 1 = − (2 µ ) − 1 I 1 2 + 1 , L 2 = − (2 µ ) − 1 I 1 I 2 . (93) This can also b e s een directly b y calculating for exa mple I 1 2 from the expressio n giv en in (89) with r esult I 1 2 = 2( r + x ) H + 2 µ (1 − L 1 ) , (94 ) where we hav e use d (79). Now solving for L 1 at zero energy g iv es again the first of the relations (93). The rela tions (93 ) imply that the seco nd rank Killing tensors corresp onding to the comp onents o f the Laplace-Runge- Lenz v ector ar e reducible a t zer o energy . The inv ariants in the Jaco bi time g auge can be found fro m the relatio ns (82), (83), (78) and (37). This gives J 1 = − (2 µ ) − 1 x ( p x 2 − p y 2 ) − µ − 1 y p x p y , J 2 = (2 µ ) − 1 y ( p x 2 − p y 2 ) − µ − 1 xp x p y , (95) where J 1 and J 2 are the Jacobi inv aria n ts corr esponding to L 1 and L 2 resp ectiv ely . F rom (89) we then find the re lations J 1 = − (2 µ ) − 1 I 1 2 + 2 H J , J 2 = − (2 µ ) − 1 I 1 I 2 . (96) The reducibilit y of the Killing tensors K M N (1) and K M N (2) corres p onding to J 1 and J 2 is therefore expres sed by the for m ulas K M N (1) = − (2 µ ) − 1 K M (1) K N (1) + J M N , K M N (2) = − (2 µ ) − 1 K ( M (1) K N ) (2) . (97) W e also wish to under stand the co mm utation relatio ns for the fixed e ner gy inv ar ian ts in the physical time gauge. T o facilitate the calculatio ns the ph y s ical Hamiltonian is first express ed in terms of the Jacobi Hamiltonian by H = T + V − E = ( V − E )(1 − 2 H J ) , (98) where we hav e us e d T = 2 ( E − V ) H J . W e can no w exploit the fa ct that the in v ariant commutes with the Jacobi Hamiltonian to o btain { I , H } = { I , H} = (1 − 2 H J ) { I , V } = H ( V − E ) − 1 { I , V } . (99) This rela tion shows that we only need to compute the Poisson bra c ket with the p oten tial. It als o follows that the bracket { I , H } in g eneral depends linear ly on H . Using (99) to calculate the brack e ts for the Kepler inv ar ian ts we find { I 1 , H } = r − 1 ( r + x ) 1 / 2 H = √ 2 r − 1 / 2 cos( φ/ 2) H , { I 2 , H } = − (sgn y ) r − 1 ( r − x ) 1 / 2 H = − √ 2 r − 1 / 2 sin( φ/ 2) H . (100) 14 8.2 Some other examples of in tegrability at fixed energy As discussed for example by Hietarinta [4], conformal tr ansformations provide links b etw een physically different sy stems which are integrable a t some fixed energy . In pa rticular if the original p oten tia l ˜ V is separable, ˜ V = Q 1 ( X ) + Q 2 ( Y ), then the tra nsformed system has the p otential V = | H ′ ( z ) | 2 [ Q 1 ( ℜ ( H ( z )) + Q 2 ( ℑ ( H ( z )) − E ] , (101) where z = x + iy and H ( z ) = X + iY . One o f the results of the present work is that w e hav e iden tified those conformal transfor mations fo r which the new p otential in this situatio n is actually integrable at arbitra ry energy (see table 1). Conv ersely , if the conformal tr ansformation is not contained in table 1 then the resulting po tential do es not ha ve a linear or q uadratic in v ariant a t arbitrary energies. Hietarinta considered conformal transformatio ns of the fo rms H ( z ) = z m (with m = − 1 , − 2 , 1 2 , 2), e z and ln z . Note in particular that of these H ( z ) = z 1 / 2 and H ( z ) = ln z pro duce systems which a re int eg rable a t ar bitrary energ y if the original potential is separable. In this subs ection we give some further exa mples of s imple systems whic h are int eg rable at a fixed energy . Consider first the p olynomial function S ( z ) = iz 2 . This is the simplest p olynomial which gives a p oten tial which is not a utomatically in tegra ble at ar bitrary e ne r gy . The co rresp onding confor mal tra nsformation is given b y w = H ( z ) = 2 − 1 / 2 (1 − i ) ln z or in terms of the rea l v ariables X = 1 √ 2 ( θ + ln r ) , Y = 1 √ 2 ( θ − ln r ) . (102) F rom the relation (3 4) it then follows that the p oten tial given b y G = r − 2 [ A ( re θ ) + B ( r e − θ )] , (103) is integrable at zer o ener gy for a r bitrary functions A a nd B . F or functions c o n taining linear and quadratic terms the p oten tial takes the form G = r − 1 ( a 1 e θ + a 2 e − θ ) + a 3 e 2 θ + a 4 e − 2 θ , ( 1 04) where the a i ( i = 1 , . . . , 4) are a r bitrary constants. As ano ther example we ta ke a function o f the form S ( z ) = z k where k 6 = 0 , 1 , 2 is a real co nstan t. This leads to w = H ( z ) = m − 1 z m where m = − k / 2 + 1 6 = 0 , 1 2 , 1. The conforma l trans fo rmation can then b e written X = m − 1 r m cos( mθ ) , Y = m − 1 r m sin( mθ ) . (105) The co rresp onding potential is G = r − k [ A ( X ) + B ( Y )] . (106) Cho osing for exa mple A ( X ) = a 1 m s X s and B ( Y ) = a 2 m s Y s we ha ve G = r − 2+ m ( s +2) [ a 1 cos s ( mθ ) + a 2 sin s ( mθ )] . (107) Spec ia lizing to the case m = 2 while k e e ping s arbitrary and using cos(2 θ ) = ( x 2 − y 2 ) r − 2 , sin(2 θ ) = 2 xy r − 2 yields finally the p oten tial G = r 2 [ a 1 ( x 2 − y 2 ) s + a 2 2 s x s y s ] , (108) which is therefore integrable at zero energy . 15 9 Concluding remarks It was shown in section 3 tha t co nfo r mal transformations g iv en by analytic functions H ( z ) for which the condition ℑ{ S ′′ ( z ) } = 0 with S ( z ) = [ H ′ ( z )] − 2 is sa tisfied give r ise to the class ical p otent ia ls which admit quadratic (or linea r) s econd inv a r ian ts at ar bitrary energ ies. Confor mal tr ansformations which do not sa tisfy that condition give p otentials which admit quadratic second inv ariants only at a fixed energy . Our approach unifies the descr iption of quadratic inv ariants at arbitrary and fixed ener gies. In particula r the in tegr abilit y condition (27) is v alid for b oth types o f in v ariants. It reduces to Darb oux’s class ic al condition for arbitrar y energy inv aria n ts when ℑ{ S ′′ ( z ) } = 0 . Whether a unifica tion can also be achiev ed for t hir d degree inv ariants or hig her remains a n op en problem. An in trig uing asp ect o f the integrability condition (27) is the p ossibility that for a given p oten tial function G = E − V there could exis t a family of solutions S ( z , E ) with a con tinuous dep endence on the energy . This would lead to new fa milies of integrable p oten tia ls wit h energy dependent quadr atic inv aria nts. At this p oint we cannot ex clude the existence of such s olutions of the integrability condition. A related res ult w as given by Hietarinta [4] who showed that the po ten tial x/y is in fact integrable by energy depe nden t inv aria n ts which are certain transcendental functions of the mo menta. F or a given p otential V the integrability conditio n (27) with G = E − V ca n b e used to deter mine energy v alues for which there exists a n in v ar ian t of at most second degr ee. This inv olves s olving a linear differential equation of the sec o nd order . F or linear inv ar ian ts it is sufficient to so lv e the linear equation (56). W e c o nsider this p ossibility to test for linea r and qua dratic integrability a t fix ed energy to b e an impo rtan t applica tio n of the geo metric appro ac h to Hamiltonian dynamics. Another approach is to lo ok for conditio ns inv olving cur v ature inv aria nts s uc h a s the technique used in section 8.1 to find ca ses with additional inv aria n ts. In fact, conditions inv olving curv ature inv a r ian ts fo r (1 +1)-dimensional mo dels hav e recently been found (using tw o differe nt approa c hes) for po ten tials which do not req uire the manifest linear inv ariant presen t in the Kepler pr oblem [19, 20]. There ar e indications tha t at least the a pproach of [1 9] can b e genera lized to inco rpor ate q uadratic in v ariants as w ell. It is of cons iderable in terest to develop these techn iq ues and use them to lo ok for fixed energy inv aria n ts o f physically interesting mo dels such as the H ´ enon-Heiles p otential and other s. References [1] M. H´ eno n and C. Heiles, AJ 69 , 73 (19 64). [2] J. Bertrand, J. Math. XVI I , 121 (1852). [3] G. Dar boux, Archiv es N´ eer landaises (ii) VI , 3 7 1 (1901). [4] J. Hietarinta, Phys. Rep. 147 , 87 (1987 ). [5] L. S. Hall, Physica 8D , 90 (1983). [6] B. Dorizzi, B. Gramma tico s, and A. Ramani, J. Math. P h ys. 24 , 228 2 (1983). [7] R. Abra ham and J. E. Ma rsden, F oun datio ns of me chanics (Be njamin Cummings, New Y ork, USA, 1978). [8] V. I. Arnold, Mathematic al metho ds of classic al me chanics (Springe r -V erlag, New Y ork, USA, 1978 ). [9] C. L a nczos, The variational princip les of me chanics (Dov er, New Y ork, USA, 198 6 ). [10] C. Ugg la, K. Rosquis t, and R. T. Jantzen, Phys. Rev. D 42 , 404 (199 0). [11] C. W. Misner, K . S. Thorne, a nd J. A. Wheeler , Gr avitation (F reeman, San F rancisco, USA, 1973). [12] A. Ash tek ar , R. T ate, and C. Uggla, Int . J. Mo d. Phys. D 2 , 15 (1993 ). 16 [13] K. Ro squist and C. Uggla, J. Math. Phys. 32 , 34 12 (1991). [14] B. F. Sch utz, Ge ometric al metho ds of m athema t ic al physics (Cam br idg e University Press, Cambridge, U.K., 198 0). [15] V. N. Ko lok ol’ts ov, Math. USSR Izvestiya 21 , 291 (19 83). [16] J. Binney a nd S. T remaine, Galactic d ynamics (Princeton Univ er sit y Pres s, Princeton, New Jersey , 1987). [17] H. Goldstein, Classic al me chanics , 2 ed. (Addison-W esley , Rea ding, Massach usetts, 19 80). [18] D. Kramer, H. Stephani, M. A. H . MacCallum, and E. Herlt, Exact S olutions of the Einstein Equations (VEB Deutscher V erlag der Wissenschaften, Berlin, GDR, 19 80). [19] M. Golia th and K. Ros q uist, USIP Rep ort 95-01 , Departmen t o f Physics, Sto c kholm University (un- published). [20] C. Ugg la, M. Br adley , and M. Marklund, pr e pr in t (unpublished). 17