aa r X i v : . [ m a t h - ph ] D ec Generalised Eisenhart lift of the Toda chain
Marco Cariglia ∗ DEFIS, Universidade Federal de Ouro Preto, Campus Morro do Cruzeiro, 35400-000 Ouro Preto, MG - Brasil
Gary Gibbons † DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK (Dated: November 7, 2018)The Toda chain of nearest neighbour interacting particles on a line can be described both in termsof geodesic motion on a manifold with one extra dimension, the Eisenhart lift, or in terms of geodesicmotion in a symmetric space with several extra dimensions. We examine the relationship betweenthese two realisations and discover that the symmetric space is a generalised, multi-particle Eisenhartlift of the original problem, that reduces to the standard Eisenhart lift. Such generalised Eisenhartlift acts as an inverse Kaluza-Klein reduction, promoting coupling constants to momenta in higherdimension. In particular, isometries of the generalised lift metric correspond to energy preservingtransformations that mix coordinates and coupling constants. A by-product of the analysis is thatthe lift of the Toda Lax pair can be used to construct higher rank Killing tensors for both thestandard and generalised lift metrics.
PACS numbers: 02.30.Ik, 45, 45.10.Na
I. INTRODUCTION
Integrable systems occur rarely in real world applica-tions and yet are extremely valuable theoretically. Apartfrom the advantage of offering the possibility to studya given system analytically, one of the reasons why in-tegrable systems are important is that they lie at theintersection of a number of research areas, among whichdynamical systems, geometrical methods and fundamen-tal theories.In the study of dynamical systems one of the funda-mental questions is to understand what exactly differ-entiates integrable systems from chaotic ones, especiallyin the thermodynamical limit. For example it is stillan open question to understand what happens to Fermi-Pasta-Ulam systems [1] for very long time scales: inte-grable systems like the Toda lattice [2] and the KdVequation [3] among others have been used to approxi-mate them. On the other hand, geometry and geometri-cal methods play an important role, as for example thereare several integrable systems in one dimension that canbe described in terms of geodesic motion on a symmet-ric space. Some authors have also conjectured that thismight be the case for all one-dimensional integrable sys-tems, for example Perelemov and Olshanetsky [4] andMarmo and collaborators [5]. Lastly, integrable systems,even the simplest one dimensional ones, crop up in funda-mental theories of physics: notable examples are amongothers the (super) Calogero model [6] that has been ar-gued to provide a microscopic description of the near-horizon limit of the extreme Reissner-Nordstr¨om black-hole, and Toda and Toda-like systems, which appear in ∗ [email protected] † [email protected] the study of gauge field theories [7, 8], supergravity the-ories [9], and Kaluza-Klein theories [10, 11].In this work we focus our attention on one specific in-tegrable system, the Toda chain. It describes a chainof particles on a 1-dimensional line, interacting via anexponential, nearest neighbour potential. It has beenintroduced by Toda in [2], who showed that it admitssolitonic ”travelling waves” solutions. Constants of mo-tion where displayed by Henon [12] and Flaschka [13].In particular they showed that the Toda chain is a fi-nite dimensional analog of the KdV equation. The Todachain with n particles is one of the special classical Hamil-tonian systems described by Olshanetsky and Perelo-mov in [4] such that their dynamics can be describedby geodesic motion on a symmetric space with dimen-sion bigger than n . For the Toda chain the symmetricspace is X − = SO ( n ) \ SL ( n, R ). At the same time, itis a known result by Eisenhart [14] that every classicalHamiltonian system with n degrees of freedom, kineticenergy of the form P ni,j =1 h ij ˙ q i ˙ q j , in the absence ofmagnetic field interaction, and with a potential V inde-pendent of time, is equivalent to geodesic motion on aspace with dimension n + 1 and metric g = n X i,j =1 h ij dq i dq j + dy V . (1.1)In particular for positive potentials, like the one for theToda chain we are considering in this work, the lift metricis Riemannian. The original work by Eisenhart displayedthe metric (1.1) as a special case of a more general con-struction. When V depends on time, or there is interac-tion with an external magnetic field, it is possible to con-sider a Lorentzian lift metric in n +2 dimensions, and theequations of motion of the original system are obtainedby considering null trajectories. After the original workby Eisenhart it took a number of years after the sameidea was independently re-discovered in [15], from thereprompting further work, among which [16, 17]. In recentyears the technique has been used on several occasions: anon-exhaustive list of examples is given by a geometricalrevisiting of integrable motion and its relation to chaos[18, 19], building of new examples of non-trivial higherorder Killing tensors in curved space [20, 21], the geo-metrical lift of certain types of Dirac equation with fluxinto a free Dirac equation [22].The relation, if any, between these two types of geo-metrical lifts of the system was unknown prior to thiswork. We investigated the problem and found that theOlshanetsky Perelomov lift (OP) in fact corresponds toa generalised Eisenhart lift, where a new dynamical vari-able ω a is introduced for each coupling constant in thesystem, and X − is endowed with an SL ( n, R ) right in-variant metric. The original Eisenhart lift can be recov-ered by setting a linear combination of the ω variablesequal to the y variable in (3.1), and performing a di-mensional reduction to n + 1 dimensions. A consequenceof the construction is that studying the isometries of themetric on X − corresponds to finding transformations be-tween the coordinates of the original Toda system andthe coupling constants such that the energy is left un-changed. Also, as a by-product of the techniques usedwe have also found new examples of higher rank Killingtensors.The rest of the work is organised as follows. In sec.II we present the Toda chain. In sec. III we discuss theEisenhart lift of the Toda chain, in particular we obtainnew higher rank Killing tensors. In sec. IV we present theOP symmetric space lift of the Toda chain, analyse therelevant dynamical variables and find an SL ( n, R ) rightinvariant metric that induces the correct dynamics. Wediscuss how a partial dimensional reduction of this metricyields the Eisenhart metric of section III. We conclude insection VII with a summary and open questions. II. THE TODA CHAIN
The non-periodic Toda system with n particles is de-fined by the Hamiltonian H ( p, q ) = n X i = i p i V ( q ) = n X i = i p i n − X i =1 g i e q i − q i +1 ) . (2.1)It describes a system of particles that interact with theirnearest neighbour via an exponential potential. Theequations of motion are˙ q i = p i , ˙ p = − g e q − q ) , ˙ p i = − g i e q i − q i +1 ) + g i − e q i − − q i ) , i > . (2.2)Flaschka in [13] displayed a Lax pair for the system.We are going to use here an alternative version as per [4], given by the matrices L ij = δ ij p j + g i − δ i,j +1 + g i e q i − q i +1 ) δ i,j − , (2.3) M ij = 2 g i e q i − q i +1 ) δ i,j − . (2.4)This means that the equations of motion can be rewrittenin the form ˙ L = [ L, M ]. This equation has solution givenby L ( t ) = A ( t ) L (0) A − ( t ), where the evolution matrix A ( t ) is determined by the equation dAdt = − M A . (2.5)Therefore, if I ( L ) is a function of L invariant under conju-gation L → ALA − , then I ( L ( t )) is a constant of motion.In particular one can construct n independent conservedquantities I i = i T rL i , i = 1 , . . . , n , where I = P p i , I = H . The Toda system is also superintegrable [23]. Itis important noticing that the properties of integrabilityand existence of the Lax pair are valid for all choices ofthe coupling constants. III. THE EISENHART LIFT
As mentioned in the introduction, the Eisenhart lift ofthe Toda chain is given by an n + 1 dimensional spacewith coordinates q µ = { q , . . . , q n , y } and metric g ( E ) = n X i dq i + dy V . (3.1)To see the relation to the original Toda chain of sectionII one can consider the lifted Lagrangian L = 12 g ( E ) µν ˙ q µ ˙ q ν = 12 n X i =1 ˙ q i + ˙ y V , (3.2)which yields the momentum p y = ˙ y V , and the Hamilto-nian H = n X i = i p i p y V ( q ) . (3.3)In particular, since the metric (3.1) does not depend ex-plicitly on y , then p y is conserved and if we set p y = 1then we recover the trajectories of the original system.Considering trajectories with p y = 1 and non-zero isequivalent to working with a Toda chain with all thecoupling constants rescaled by a factor p y . Thereforethe Eisenhart lift space corresponds to a collection ofToda chain systems, one related to the other by an over-all rescaling of the coupling constants, and the originalmotion lifts to geodesic motion in dimension higher byone.Now we can define a lifted Lax pair L ij = δ ij p j + p y g i − δ i,j +1 + p y g i e q i − q i +1 ) δ i,j − , M ij = 2 p y g i e q i − q i +1 ) δ i,j − . Since the original Lax pair was defined for all values ofthe coupling constants, and since p y is constant, then itwill still be the case that ˙ L = [ L , M ], and that we canbuild invariants according to I i = i T r L i , i = 1 , . . . , n .However, this time the I i are polynomials in the mo-menta p µ = { p , . . . , p n , p y } of degree i , and thereforethey must correspond to rank i Killing tensors K µ ...µ i ( i ) , µ = 1 , . . . , n + 1, according to the formula I i = 1 i ! K µ ...µ i ( i ) p µ . . . p µ i . (3.4)These provide to our knowledge new examples of non-trivial higher rank Killing tensors.Interestingly, the same logic can be applied to create anew lifted Lax pair where each coupling constant g i liftsto a different momentum ˜ p i g i . Then it is guaranteed onecan build conserved quantities for the new Hamiltonian H gen ( p, q ) = n X i = i p i n − X i =1 ˜ p i g i e q i − q i +1 ) , (3.5)and these quantities will be homogeneous in the mo-menta. Since the Hamiltonian is quadratic in the mo-menta it can be interpreted in terms of geodesic motionwith respect to an appropriate metric, and therefore wecan also construct Killing tensors with respect to the newmetric. We call this new metric a generalised Eisenhartmetric . In the next section we will show that the descrip-tion of the Toda system given in [4] in terms of geodesicmotion in a symmetric space is exactly the motion withrespect to the generalised Eisenhart metric. IV. THE OLSHANETSKY PERELOMOV LIFT
Olshanetsky and Perelomov [4] showed that theToda chain equations of motion can also be obtainedfrom geodesic motion on a different space, X − = SO ( n ) \ SL ( n, R ), whose elements are symmetric positive-definite n × n matrices x with real components and unitdeterminant, det x = 1. We broadly follow here the no-tation they used.Any real symmetric positive definite matrix x ∈ X − admits a Cholesky U DU decomposition, that is it can bewritten as x = Zh Z T , (4.1)where h is diagonal and with positive elements, and Z ∈Z ⊂ SL ( n, R ), the subgroup of upper triangular matriceswith units on the diagonal. We can parameterise thediagonal part as h = diag [ e q , . . . , e q n ] , (4.2)the q i being ’dilaton’ fields, which will be identified withthe positions of the particles in the Toda chain (2.1). They are restricted by the condition det h = 1, whichimplies n X a =1 q a = 0 , (4.3)which is a condition on the position of the center of massof the system.The equation of motion on X − is given by ddt (cid:0) ˙ x ( t ) x − ( t ) (cid:1) = 0 . (4.4)In other words it is an equation for auto-parallel curveswith tangent vector V such that DV /Dt = 0, where theconnection is given by the right connection of SL ( n, R ).In [4] it is shown that if one chooses specific geodesics forwhich Z − ˙ Z = M , (4.5)with M given by (2.4), then (4.4) implies the Lax equa-tions ˙ L = [ L, M ].It is worth noticing that the condition (4.5), which se-lects specific geodesics, is equivalent to say that Z − sat-isfies the evolution equation (2.5). Then with the currentchoice of Lax pair we have an evolution equation givenby upper triangular matrices. In the work by Flaschkainstead a different Lax pair is used and the evolution isgiven by an orthogonal transformation.In order to show that the Olshanetsky Perelomov spacecorresponds to a generalised Eisenhart lift of the Todachain, that reduces naturally to the Eisenhart lift dis-cussed in sec.III, we first discuss in the n = 2 case as auseful example, and then approach the general n case. V. THE n = 2 CASE
The Lax pair matrices can be written as L = (cid:18) p g e q − q ) g p (cid:19) , (5.1) M = (cid:18) g e q − q ) (cid:19) . (5.2)We consider the space X − = SO (2) \ SL (2 , R ), given bysymmetric positive-definite 2 × x = (cid:18) a cc b (cid:19) (5.3)and unit determinant, det x = 1. If we set a = ξ + ξ , b = ξ − ξ and c = ξ then det x = 1 becomes ξ − ξ − ξ = 1,which determines the Lobachevsky plane H . If Z is thesubgroup in SL (2 , R ) of upper triangular matrices withunits on the diagonal then the matrix x in (5.3) can bewritten as x = Zh Z T , (5.4)with Z = (cid:18) z (cid:19) , (5.5)and det x = det h . As mentioned above we set h = (cid:18) e q e q (cid:19) , (5.6)where det h = 1 implies q + q = 0.The auto-parallel equation (4.4) is right invariant on SO (2) \ SL (2 , R ), under the action of the full group SL (2 , R ). We can build right invariant forms, and fromthese a right invariant metric. Then, auto-parallel curveswill be the same thing as geodesics. To build right in-variant forms ρ A we need to calculate dx x − = ρ A M A , (5.7)where the matrices M A form a basis for the three-dimensional Lie algebra of SL (2 , R ) that acts on X − . Sowe start by anaysing the right invariant term ˙ x ( t ) x − ( t )that appears in the auto-parallel equation (4.4), writingit as ˙ x ( t ) x − ( t ) = 2 Z (cid:20) Z − ˙ Z + (cid:18) ˙ q
00 ˙ q (cid:19) + 12 (cid:18) e q e q (cid:19) (cid:16) Z − ˙ Z (cid:17) T (cid:18) e − q e − q (cid:19)(cid:21) Z − . (5.8)From this we find that ρ A M A = 2 Z (cid:20)(cid:18) dq dz e q − q ) dz dq (cid:19)(cid:21) Z − = 2 Z " dq dz g dzV ( q ) − dq ! Z − , (5.9)where we defined the variables q = q − q and V ( q ) = g e q . These appear naturally in the reduced Hamilto-nian H = 12 p + 2 g e q , (5.10)which can be obtained from (2.1) by splitting the originalHamiltonian into a function of q plus a Hamiltonian forthe center of mass coordinate Q = q + q .We can take the following basis in the Lie algebra of SL (2 , R ): M = (cid:18) (cid:19) , (5.11) M = (cid:18) (cid:19) . (5.12) M = (cid:18) − (cid:19) . (5.13) M and M form a close sub-algebra, [ M , M ] = − M ,and the Iwasawa decomposition for SL (2 , R ) says that foran element of SO (2) \ SL (2 , R ) there is always a represen-tative - different from that displayed in (5.4) - that canbe written as an exponential of M times an exponentialof M , so that we can think of SO (2) \ SL (2 , R ) as gener-ated by M and M through exponentiation. The actionof M can always be re-absorbed by an SO (2) rotation.Expanding eq.(5.9) one gets, modulo constants, thefollowing right-invariant forms: ρ = g V ( q ) dz ,ρ = − zdq + (cid:18) − g z V ( q ) (cid:19) dz ,ρ = dq g z V ( q ) dz , (5.14)and the canonically associated right invariant vectorfields R A , A = 1 , ,
3. The auto-parallel equation (4.4)then implies the existence of three constants of motion: C = g V ( q ) ˙ z ,C = − z ˙ q + (cid:18) − g z V ( q ) (cid:19) ˙ z ,C = ˙ q g z V ( q ) ˙ z . (5.15)In particular setting C = g we get the condition (4.5)with M given by (5.2). Differentiating C one gets theequation of motion for the Toda system upon imposingthe condition for C .One can then build the right invariant metric ds = 4 ρ ⊗ ρ + 4 ρ ⊗ ρ = dq + g V ( q ) dz , (5.16)and setting z = y g one gets exactly the Eisenhart metric ds = dq + dy V ( q ) , (5.17)that is associated to the reduced Hamiltonian (5.10). Wenow consider the general n case. VI. THE GENERAL n CASE
In this section we use the upper and lower triangularmatrices M ab and M ab , and the matrices M a defined inthe appendix.We can parameterise Z as Z = exp X a a + 1. It isconvenient to define ω = 0 = ω n .Using repeatedly (7.2) one finds that for a < bZ ab = 1( b − a )! ω a ω a +1 . . . ω b − (6.3)and Z − ab = ( − b − a Z ab . (6.4)Now we look for right invariant forms on ˜ X , invari-ant under the full group SL ( n, R ), calculating the form dxx − = ρ A M A , A = 1 , . . . , n −
1, using the generaliza-tion of eq.(5.8). There are three terms, arising from thethree terms in square brackets, and we write each oneseparately. The first one is Z − dZ = n − X a =1 dω a M a,a +1 , (6.5)which is expanded along the M ij matrices defined in sec-tion VII. These are right invariant forms of the subgroup Z .The second term from ˙ xx − is2 Zdiag [ dq , . . . , dq n − , dq n ] Z − = 2 Z n − X a =1 dq a M a ! Z − , (6.6)where the matrices M a are defined in the appendix andwe have rewritten dq n using (4.3). Using eq.(7.6) one canwrite the following expansion for a < n : ZM a Z − = M a + X b
1. It ishowever useful to define an n -th form ρ n , which is linearlydependent on the other ones, by setting a = n in eq.(6.18)above: ρ n = 2 dq n − e − q n − − q n ) ω n − dω n − = − n − X a =1 ρ a . (6.20)Lastly, there are the forms of type ρ ab . We don’t needto write all of them, the ones we will use are ρ a,a +1 = 2 ω a ( dq a +1 − dq a ) + dω a + ω a h(cid:16) ω a +1 e − q a +1 − q a +2 ) dω a +1 − ω a e − q a − q a +1 ) dω a (cid:17) − (cid:16) ω a e − q a − q a +1 ) dω a − − ω a − e − q a − − q a ) dω a − (cid:17)i . (6.21)Now we build a right invariant metric that will reduceto the Eisenhart lift metric. An explicit calculation showsthat g = n X a =1 (cid:16) ρ a (cid:17) + n − X a =1 ρ a +1 ,a ρ a,a +1 n − X a =1 (cid:16) ρ a (cid:17) + n − X a =1 ρ a ! + n − X a =1 ρ a +1 ,a ρ a,a +1 n X a =1 dq a + 12 n − X a =1 e − q a − q a +1 ) dω a . (6.22)We can define a new variable y = n − X a =1 g a ω a . (6.23)On allowed trajectories it satisfies˙ y = n − X a =1 g a e q a − q a +1 ) = 2 V ( q ) , (6.24)where the potential V is given in (2.1). On the otherhand, on trajectories we also have12 n − X a =1 e − q a − q a +1 ) ˙ ω a = 2 V ( q ) , (6.25) so we have the equality between forms12 n − X a =1 e − q a − q a +1 ) dω a = dy V ( q ) (6.26)on all allowed trajectories. Then this identity holds onthe whole span of the trajectories and we can re-writethe metric as g = n X a =1 dq a + dy V ( q ) , (6.27)which is the Eisenhart metric.It is worthwhile noticing that the free particle Hamil-tonian associated to the metric (6.22) is H = n X a =1 p q a n − X a =1 p w a e q a − a a +1 ) , (6.28)where p w a = e − q a − a a +1 ) ˙ ω a is equal to g a on allowedtrajectories, according to eq.(6.17). This thus corre-sponds to a multi-particle Eisenhart lift, where for eachcoupling constant g a one adds a variable ω a . Then theisometries of the metric g , given by the Killing vectors as-sociated to the right invariant forms of SL ( n, R ), becometransformations that leave H unchanged, and hence aretransformations that mix q and ω coordinates while leav-ing the energy (2.1) of the original Toda chain unchanged.For example for the conserved quantities explicitly cal-culated here one has the following cases. First c a +1 ,a = 2 p ω a (6.29)correspond to trivial translations of the ω variables.Next, the quantities λ = p q + p ω ω ,λ a = p q a + p ω a ω a − p ω a − ω a − , a > , (6.30)which give transformations of the kind δq = ǫ ,δω = ǫω , (6.31) δ p ω = − ǫp ω , (6.32)and δq a = ǫ ,δω a = ǫω a , δω a − = − ǫω a − (6.33) δ p ωa = − ǫp ω a , δ p ωa − = ǫp ω a − , (6.34)where ǫ is an infinitesimal parameter. These correspondto the fact that one can make a constant shift of one ofthe variables q a and at the same time a constant rescalingof the g a and g a − couplings, while keeping the energyunchanged. Already the conserved quantities associatedto the conserved forms of equation (6.21) give rise to morecomplicated transformations where δp ω a is not constant.Summarising, the symmetries of the metric (6.22) giveall the dynamical transformations between the q and p ω a ∼ g a variables that leave the energy unchanged. Inparticular, the transformations that are of lowest orderin the momenta are those given by the Killing vectors,which form an SL ( n, R ) algebra. VII. CONCLUSIONS
In this work we have examined the relation betweenthe Eisenhart lift of the Toda chain and its higher di-mensional description as geodesic motion in a symmetricspace. We have found that the symmetric space is ageneralised Eisenhart lift of the Toda chain, that dimen-sionally reduces to the standard Eisenhart lift. In partic-ular, the SL ( n, R ) algebra of isometries of the symmet-ric space describes transformations that mix the coordi-nates q with the coupling constants g while keeping theenergy constant. We have also constructed new higherrank Killing tensors for the Eisenhart lift metric as a by-product of our analysis.The generalised Eisenhart lift is a rather general con-struction that can be performed for any Hamiltonian sys-tem that is well defined when its parameters vary in anappropriate open set. In particular, if the system is in-tegrable for all values of the parameters then the gen-eralised lift space should be parallelisable. The conceptof transforming coupling constants into dynamical enti-ties resonates in a way with what happens in String/M-Theory, where the expectation values of the dilatons andmoduli behave as coupling constants. It would be in-teresting to know whether this concept can be useful incontexts different from the Toda chain discussed here.A novel mathematical aspect of this work is that liftingthe original Toda Lax pair we have been able to constructhigher rank Killing tensors for the lift metrics. This con-struction seems rather general and can be applied to anyintegrable system whose Lax pair has homogeneous en-tries in the momenta and coupling constants, for examplethe systems described in [4]. Therefore we expect that itcan be used to significantly expand the number of knownexamples of non-trivial higher rank Killing tensors.An open question that is left unanswered in this workis what is the relation between the Eisenhart lift andthe symmetric space lift for the other types of systemsdiscussed in [4]. From the analysis done there it seemsthat no generalised Eisenhart lift will be involved, sinceonly one coupling constant is appears in the Hamilto-nian, but at the same time the dimension of the symmet-ric space is strictly higher than that of the Eisenhart lift.Therefore, one might wonder whether a correspondencebetween the two types of lifts still exists, and in positivecase whether a completely different mechanism is respon-sible for it. Answering this question might help clarifythe conjecture mentioned in the introduction about therelation between one-dimensional integrable systems andsymmetric spaces. APPENDIX: USEFUL IDENTITIES
Upper unitriangular matrices Z form a subgroup of SL ( n, R ), with Lie algebra Z generated by the strictlyupper triangular matrices M ab , a, b = 1 , . . . , n , a < b ,with components ( M ab ) ij = δ ia δ jb . (7.1)From the definition it follows that M ab = 0. The productof two M matrices is given by M ab M cd = δ bc M ad . (7.2)Given a diagonal matrix D , with elements ( D ) ij = d i δ ij (no sum), one gets the product rule M ab D = d b M ab , DM ab = d a M ab . (7.3)We also define the following diagonal, zero trace ma-trices M a , a = 1 , . . . , n − M a = diag [0 , . . . , , . . . , , − , (7.4)with the 1 term in the a -th position. It is convenient forformulas in the main text to define M n = 0.Similarly we introduce the lower diagonal matrices M ab , a, b = 1 , . . . , n , a > b , with components (cid:0) M ab (cid:1) ij = δ ia δ jb . (7.5)These also satisfy the product rule M ab D = d b M ab , DM ab = d a M ab . (7.6)Then we can parameterise the Lie algebra of SL ( n, R )with the set { M ab , M ab , M a } .Another useful product rule is M ab M cd = δ bc (cid:0) M ad + M ad + δ ad I a (cid:1) , (7.7)where I a is the diagonal matrix with a 1 in the a -th ele-ment and zero otherwise. [1] E. Fermi, J. Pasta, S. Ulam, Studies of NonlinearProblems , Collected Works of Enrico Fermi, Universityof Chicago Press, Vol.II, 978- 988, 1965. Theory,Methods, and Applications, 2nd ed., Marcel Dekker,New York, 2000.[2] M. Toda,
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