Discrete-time Ruijsenaars-Schneider system and Lagrangian 1-form structure
aa r X i v : . [ n li n . S I] J un Discrete-time Ruijsenaars-Schneider system andLagrangian 1-form structure
Sikarin Yoo-Kong † , , Frank Nijhoff ‡ , † Theoretical and Computational Physics (TCP) Group, Department of Physics,Faculty of Science, King Mongkut’s University of Technology Thonburi,Thailand, 10140. ‡ Department of Applied Mathematics, School of Mathematics, University of Leeds,United Kingdom, LS2 9JT. [email protected], nijhoff@maths.leeds.ac.uk November 12, 2018
Abstract
We study the Lagrange formalism of the (rational) Ruijsenaars-Schneider (RS) system,both in discrete time as well as in continuous time, as a further example of a Lagrange 1-formstructure in the sense of the recent paper [28]. The discrete-time model of the RS systemwas established some time ago arising via an Ansatz of a Lax pair, and was shown to leadto an exactly integrable correspondence (multivalued map)[17]. In this paper we consideran extended system representing a family of commuting flows of this type, and establish aconnection with the lattice KP system. In the Lagrangian 1-form structure of this extendedmodel, the closure relation is verified making use of the equations of motion. Performingsuccessive continuum limits on the RS system, we establish the Lagrange 1-form structurefor the corresponding continuum case of the RS model.
The Ruijsenaars-Schneider (RS) system [23, 24], i.e., the relativistic version of the Calogero-Moser (CM) system, is integrable both in the classical and quantum regimes. The classicalmodel was discovered in [23] by considering the Poincar´e Poisson algebra associated with sine-Gordon solitons, and was motivated by the discovery in the late 1970s of explicit soliton-typeS-matrices for some relativistic two-dimensional quantum field theories (such as the massiveThirring model, the quantum sine-Gordon theory and the O ( N ) σ -models). For reasons elu-cidated below, we are interested in Lagrangian aspects of the RS model, which have hardlyreceived attention. An apparent reason for this is that the Hamiltonian description corre-sponding to the system is not of Newtonian form, and hence the usual connection betweenthe Hamiltonian and the Lagrangian description through the Legendre transformation be-comes quite convoluted. At the same time we are interested in the integrable time-discreteversion of the RS system, which was proposed and studied in [17], where the Lagrangiandescription is more natural than the Hamiltonian one, because the finite-step time-iteratecan be naturally viewed as a canonical transformation where the Lagrangian plays the roleof its generating function. In [17] the corresponding discrete-time Lagrangian was found, butthe continuum limits were not considered so far. As we shall show, the latter can be usedto derive a natural Lagrangian description for the continuous RS model as well, but in thecontext of what we call a Lagrangian 1-form structure. We will now explain what we meanby this latter notion. ecently, a novel point of view was developed on the role of the Lagrangian structure inintegrable systems, cf. [9], where it was proposed that the fundamental property of multidi-mensional consistency can be made manifest in the Lagrangians by thinking of the latter ascomponents of a difference (or differential) “Lagrange-form” when the flows are embedded ina multidimensional space-time. A new variational principle was formulated which involvesnot only variations with respect to the dependent variables of the theory, but also with re-spect to the geometry in the space of independent (discrete or continuous) variables. In [9],this was laid out in the case of two-dimensional lattice equations, whilst in [8] it was ex-tended also to the case of the 3-dimensional bilinear Kadomtsev–Petviashvili (KP) equation(Hirota’s equation). Furthermore, in [27] a universal Lagrangian structure was establishedfor quadrilateral affine-linear lattice equations as well as for their corresponding continuouscounterparts, the so-called generating PDEs of the system. The key property in all these sys-tems, in which in a sense the integrability of the system resides, is that the Lagrangian formis closed on solutions of the equations of the motion (but not identically closed for arbitraryfield values). This can be viewed as a manifestation of the multidimensional consistency ofthe system under consideration on the Lagrangian level.In the case of integrable systems of ODEs, like system of equations of motion of integrablemany-body systems, the Lagrangian form structure is that of Lagrange 1-forms. Recently,in collaboration with S Lobb, the authors studied a first example of such a Lagrange 1-formstructure, namely the case of the discrete-time (rational) Calogero-Moser (CM) system,[28, 30]. The multidimensional consistency of the system in that case is represented by theco-existence of two or more independent commuting discrete-time flows in the case of threeor more particles. Starting with the discrete-time case, we furthermore established the La-grange structure of the corresponding continuous case by performing systematic continuumlimits on the discrete-time equations and Lagrangians. Of course, these systems exhibit alsoa multi-time Hamiltonian structure, where the various time-flows generated by the Hamil-tonians, which are in involution with respect to a canonical Poisson structure, commute.However, it is not the case that one can perform naively a Legendre transformation on eachof these Hamiltonians separately to yield a proper Lagrangian structure that makes senseas a coherent system. In fact, the higher-order Lagrangians emerging from such a naiveapproach would yield rather complicated algebraic expressions which seem unsuitable forfurther study. However, as we have shown in [28], a proper Lagrangian 1-form structure canbe defined for the CM system, in which the components of the form are mixed Lagrangians,of polynomial form in the time-derivatives, obeying the crucial closure property , expressingthe commutativity of the flows, on solutions of the equations of the motion . To derive theseLagrangians, the connection between the semi-discrete KP equation and the discrete-timeCM system, which arises as the pole-reduction of the former, was instrumental in order toguide the proper choice of higher-order continuum limits obtained by systematic expansionsperformed on the discrete-time model, thus leading to the Lagrangians in the continuumcase. (Unfortunately, we do not know at this stage a Lagrangian of the semi-discrete KPequation in the relevant form, which would have allowed us to do the pole-reduction on theLagrangians directly.)In the present paper, we proceed in the same spirit as in the paper [28], to establishthe Lagrange 1-form structure of the discrete-time rational RS system. However, comparedto the the case of the discrete-time CM system where there is a direct connection betweenthe Lax matrices and the relevant Lagrangians, and where the closure relation is a directconsequence of a zero-curvature condition, such a direct connection seems absent in the RScase. Thus, in the latter case, the establishment of the closure relation for the Lagrangianswhich essentially were provided in [17], has to be verified by an explicit computation, andseems to be governed by a different mechanism. It is this aspect that makes the study of theRS system a worthwhile addition to the emerging theory of Lagrangian multi-form structures,confirming that the latter is universal structure underlying integrable systems. Furthermore,whereas the discrete-time CM system arises from the pole-reduction of a semi-discrete KPequation (with one continuous and two discrete independent variables), the discrete-timeRS is connected by an analogous reduction to the fully discrete KP equation (with threediscrete independent variables). Thus, it is evident that (discrete-time) RS case, viewed as relativistic analogue of the corresponding CM system, is richer than the CM case of [28],containing an additional (deformation) parameter which can be viewed as the reciprocalof the speed of light. It is observed that this non-relativistic limit also corresponds to aparticular continuum limit on the lattice KP system.Although our focus in this paper is on the rational case of the RS model, most of ourresults on the Lagrangian structure can be extended straightforwardly to the trigonomet-ric/hyperbolic case and even the elliptic case, however, as in the case of the CM system,we prefer all the statements we make to be backed up by the explicit solution of the equa-tions of motion that can be constructed in the rational case. For instance, an importantingredient in the structure is what we call ”constraint equations”, which in addition to theone-dimensional equations of the motion can be verified explicitly for the solution obtained.These constraint equations involve the dynamics in two discrete variables (corresponding totrajectories in the space of independent variables which involve corners). Since, in contrastto the paper [28], the starting point in the present paper is an Ansatz of a Lax pair ratherthan a reduction from a KP system (the connection with the lattice KP is established aposteriori ), the verification that all relations (equations of the motion, constraint equationand closure relation) hold true for a nontrivial family of solutions backs up the consistencyof the whole structure of this system.The organization of the paper is as follows. In Section 2, we review the construction of thesingle-flow discrete-time RS system and its exact solution in terms of a secular problem. Next,we extend the system by imposing additional commuting discrete flows in additional timedirections, and derive the conditions (i.e., the constraint relations) for their compatibility. InSection 3, the Lagrangian 1-form structure of the discrete multi-time RS system is studied,and we verify the relevant closure relation by a direct computation involving the equations ofmotion as well as the constraints. Thus, we establish that this system possesses a Lagrangian1-form structure in the sense of the paper [28]. In Section 4, a “skew” continuum limit istaken, guided by the exact solution of the discrete-time RS system system, yielding whatwe call the semi-continuous RS system . The latter, in fact, acts as a generating systemfor the continuous RS system, thus allowing us in Section 5 to derive the full continuumlimit, by which we recover the continuous-time RS hierarchy albeit in a form involvingmixed derivatives w.r.t. the multiple times of the RS hierarchy. Applying the same limitto the Lagrangian of the semi-continuous RS system, we obtain the Lagrangian componentsof the relevant continuous higher-time flows of the RS system, in a form (namely involvingmixed higher-time derivatives) which is suitable for the interpretation as a Lagrangian 1-formstructure, where the Lagrangian components obey the (continuous) closure relation subjectto the solutions of the equations of the motion. In Section 6, the connection to the latticeKP system is presented, showing that the exact solution of the discrete multiple-time RSmodel leads to solutions of the lattice KP dependent variable as function of these multipletimes. In particular, the characteristic polynomial associated with the exact solution can beidentified with the corresponding lattice KP τ -function. Finally in Section 7 summary anddiscussion will be presented along with some open problems. In this Section we review the discrete-time RS system which has been introduced in [17].This gives us an occasion to introduce appropriate notation which we will use throughoutthe paper. Furthermore, we derive the exact solution of the discrete equations of the motionand identify the constraint relations on commuting flows that can coexist in the system. .1 The single-flow RS system Following [17] the discrete-time RS model is obtained on the basis of a Lax pair of the form: e φ = M κ φ , L κ φ = ζ φ , (2.1a)for a vector function φ and an eigenvalue ζ , in which the matrices L κ and M κ are given by L κ = hh T κ + L , (2.1b) M κ = e hh T κ + M , (2.1c)where in the rational case L = N X i,j =1 h i h j x i − x j + λ E ij , and M = N X i,j =1 e h i h j e x i − x j + λ E ij . (2.2)In (2.1) the x i are the particle positions, whilst the h i are auxiliary variables which will bedetermined later. The tilde e is a shorthand notation for the discrete-time shift, i.e. for x i ( n ) = x i , and we write x i ( n + 1) = e x i , and x i ( n −
1) = x i e . The variable κ is the additionalspectral parameter, whereas λ is a parameter of the system related to the non-relativisticlimit. The matrix E ij are the standard elementary matrices whose entries are given by( E ij ) kl = δ ik δ jl .The compatibility condition of the system (2.1): e L κ M κ = M κ L κ ⇒ e h e h T κ + e L ! e hh T κ + M ! = e hh T κ + M ! (cid:18) hh T κ + L (cid:19) (2.3)From the coefficients of 1 /κ we derive the conservation law Tr e L κ = Tr L κ leading to N X j =1 e h j = N X j =1 h j , (2.4)and furthermore, the coefficients of 1 /κ give e L e hh T + e h e h T M = M hh T + e hh T L , (2.5)and the rest produces the equation e L M = M L . (2.6)(2.5) and (2.6) produce the relations N X j =1 e h j e x i − e x j + λ − h j e x i − x j + λ ! = N X j =1 h j x j − x l + λ − e h j e x j − x l + λ ! , (2.7)for all i, j = 1 , , ..., N . Consequently, both sides of (2.7) must be independent of the externalparticle label. Thus, we find a coupled system of equations in terms of the variables h i , and x i of the form: N X j =1 e h j e x i − e x j + λ − h j e x i − x j + λ ! = − p , ∀ i , (2.8a) X j =1 h j x j − x l + λ − e h j e x j − x l + λ ! = − p , ∀ l , (2.8b)where the quantity p = p ( n ) does not carry a particle label, but could still be a function of n . In order to derive a closed set of equations of motion for the variables x i we have to deter-mine the variables h i in terms of the x i and their time-shifts. To do this most effectively,weuse the Lagrange interpolation formula, which is given in the following lemma: Lemma 2.1.
Lagrange interpolation formula : Consider N noncoinciding complexnumbers x k and y k , where k = 1 , , ..., N . Then the following formula holds true: N Y k =1 ( ξ − x k )( ξ − y k ) = 1 + N X k =1 ξ − y k ) Q Nj =1 ( y k − x j ) Q Nj =1 ,j = k ( y k − y j ) . (2.9a) As a consequence − N X k =1 x i − y k ) Q Nj =1 ( y k − x j ) Q Nj =1 ,j = k ( y k − y j ) , i = 1 , ..., N , (2.9b) which is obtained by substituting ξ = x i into (2.9a) . Applying the Lagrange interpolation formula to the following rational function of the in-determinate variable ξ : N Y j =1 ( ξ − x j + λ )( ξ − e x j − λ )( ξ − x j )( ξ − e x j ) , (2.10)we obtain h i = − p Q Nj =1 ( x i − x j + λ )( x i − e x j − λ ) Q Nj = i ( x i − x j ) Q Nj =1 ( x i − e x j ) , (2.11a) e h i = p Q Nj =1 ( e x i − x j + λ )( e x i − e x j − λ ) Q Nj = i ( e x i − e x j ) Q Nj =1 ( e x i − x j ) , (2.11b)for i = 1 , , ..., N which we obtain the following system of equations pp e N Y j =1 j = i ( x i − x j + λ )( x i − x j − λ ) = N Y j =1 ( x i − e x j )( x i − x e j + λ )( x i − x e j )( x i − e x j − λ ) . (2.12)Eq. (2.12) can be considered to be the product version of (2.11) which is a system of N equations for N + 1 unknowns, x , ..., x N and p . There is so far no separate equation for p ,which amounts to a freedom in determining the centre of mass motion, and fixing a specificchoice of the time-evolution for p we would get a closed set of equations (for more details,see [17]). For simplicity we will often take, in what follows, p to be constant as a function ofthe discrete-time variable n .The exact solution of the equations of motion can be derived in a way similar to thecontinuous case of the rational RS model, cf. [17], cf. also [23, 22] using the explicit form ofthe Lax matrices (2.1b), and is given by the following statement. Proposition 2.1.
Let Λ be a constant diagonal matrix, and let p ( n ) be a given function ofthe discrete-time variable n such that p ( n ) I + Λ is invertible for all n ≥ , and let the N × N matrix function of the discrete variable n , Y ( n ) , be given by Y ( n ) = " n − Y k =0 ( p ( k ) I + Λ ) − Y (0) " n − Y k =0 ( p ( k ) I + Λ ) − n − X k =0 p ( k ) λp ( k ) I + Λ , (2.13a) The factors in the first term of (2.13a) come out directly from the computation but they can be removed byconjugation. For clarity, we write the inverses of (diagonal) matrices such as p I + Λ in fractional form, where itdoes not lead to ambiguity. ubject to the following condition on the initial value matrix [ Y (0) , Λ ] = − λ Λ + rank 1 , (2.13b) then the eigenvalues x i ( n ) of the matrix Y ( n ) coincide with the solutions for particle positionof the discrete-time RS system, i.e. they solve the discrete equations of motion (2.12) . The details of the proof are given in are Appendix A. In fact, the matrix Λ can beidentified with the (diagonal) matrix of eigenvalues of the matrix L , cf. (A.3), whichcoincide with the eigenvalues of L (0) at the initial value, since we are dealing with anisospectral problem. However, as far as the above proposition is concerned, both Λ and theinitial value matrix Y (0) can be chosen in lieu of posing initial conditions on the particlepositions .The functions p ( n ) determine the centre of mass motion, which can be separated fromthe relative motion of the particles. In the special case of constant p : p = e p (which amountsto choosing a frame in which the centre of mass remains stationary) the expression for thematrix Y ( n ) becomes Y ( n ) = ( p I + Λ ) − n Y (0)( p I + Λ ) n − npλp I + Λ . (2.14)As a corollary, we have that the solutions can be found from the secular problem for thematrix: Y (0) − npλ ( p I + L (0)) − , (2.15)i.e., the eigenvalues of Y ( n ) are the values of the particle positions at discrete-time n . Following the construction in [28] we introduce another temporal Lax matrix N κ whichgenerates a shift b in an additional discrete time direction. Thus, we impose for the samevector function φ as before, also the system of equations: b φ = N κ φ , L κ φ = ζφ , (2.16)where N κ = b hh T κ + N , where N = N X i,j =1 b h i h j e x i − x j + λ E ij . (2.17)This describes the flow in terms of an additional discrete-time variable m , where hat isa shorthand notation for the discrete-time shift, i.e. for x i ( n, m ) = x i , and we write x i ( n, m + 1) = b x i , and x i ( n, m −
1) = x b i .Obviously, the compatibility relations for (2.16) can be analysed in a very similar man-ner as to the ones for (2.1). Thus, we find a coupled system of equations in terms of thevariables h i , and x i in the form N X j =1 b h j b x i − b x j + λ − h j b x i − x j + λ ! = − q , ∀ i , (2.18a) N X j =1 h j x j − x l + λ − b h j b x j − x l + λ ! = − q , ∀ l , (2.18b) In fact, specifying x i (0) and x i (1), i = 1 , , ..., N , the matrices Y (0) and Λ can be computed from the initialvalues by using the h i from (2.11a) and the matrix L from (2.2) at n = 0, where for simplifity we assume thatthe latter can be diagonalised. here the quantity q = q ( m ) does not carry a particle label, but may still be a function of m .The system (2.18) can be resolved once again by using the Lagrange interpolation formula(2.9b) yielding the resolution: h i = − q Q Nj =1 ( x i − x j + λ )( x i − b x j − λ ) Q Nj = i ( x i − x j ) Q Nj =1 ( x i − e x j ) , (2.19a) b h i = q Q Nj =1 ( b x i − x j + λ )( b x i − b x j − λ ) Q Nj = i ( b x i − b x j ) Q Nj =1 ( b x i − x j ) , (2.19b)for i = 1 , , ..., N , and from which we obtain the following system of equations qq b N Y j =1 j = i ( x i − x j + λ )( x i − x j − λ ) = N Y j =1 ( x i − b x j )( x i − x b j + λ )( x i − x b j )( x i − b x j − λ ) . (2.20)The product version (2.20) of (2.19), thus yields a system of N equations for N +1 unknowns, x , ..., x N and q . There is again no equation for q separately, and thus it should be a priorigiven in order to get a closed set of equations. Thus far, so similar.Assuming now that the dependent variables depend simultaneously on both discrete timevariables n and m , then to obtain a univalent solution of the equations of motion we mustrequire that both flows, in the “ e ” direction and the “ b ” direction, commute. If so, thenwe can fix a value for n and solve the equations in the “ b ” direction similarly as before,leading to the matrix Y which depends on n and m as follows: Proposition 2.2.
Let the N × N matrix function of the discrete variable m , Y ( m ) , be givenby Y ( m ) = " m − Y k =0 ( q ( k ) I + Λ ) − Y (0) " m − Y k =0 ( q ( k ) I + Λ ) − m − X k =0 q ( k ) λq ( k ) I + Λ , (2.21a) subject to the following condition on the initial value matrix [ Y (0) , Λ ] = λ Λ + rank 1 . (2.21b) Then eigenvalues x i ( m ) of the matrix Y ( m ) coincide with the solutions for particle positionof the discretetime RS system, i.e. they solve the discrete equations of motion (2.20) . From now on we will restrict ourselves for simplicity to the case of constant q : q = b q leading to what we would like to call the discrete-time RS system corresponding to the “ b ”direction in terms of the discrete-time variable m and the matrix Y ( m ) becomes Y ( m ) = ( q I + Λ ) − m Y (0)( q I + Λ ) m − mqλq I + Λ . (2.22)In order for this scenario to work there must be further constraints on the flows. Thiswill lead to a system of “constraints” which can be readily obtained from the compatibilitybetween Lax pairs (2.1c) and (2.17) pq = N Y j =1 ( x i − b x j − λ )( x i − e x j )( x i − e x j − λ )( x i − b x j ) , (2.23a) pq = N Y j =1 ( x i − x b j + λ )( x i − x e j )( x i − x e j + λ )( x i − x b j ) . (2.23b)We will refer to relations (2.23a) and (2.23b) as the constraint equations which guarantee thecommutativity between the discrete-time flows with shifts “ e ” and “ b ” in the variables n nd m respectively, and will play a major role in the discrete-time variational principle (seeAppendix B). Equating (2.23a) with (2.23b), we get N Y j =1 ( x i − e x j )( x i − x e j + λ )( x i − x e j )( x i − e x j − λ ) = N Y j =1 ( x i − b x j )( x i − x b j + λ )( x i − x b j )( x i − b x j − λ ) , (2.24)which is a consequence of (2.12) and (2.20), and which expresses the compatibility with theset of O∆Es. Proposition 2.3.
The eigenvalues x ( n, m ) , . . . , x N ( n, m ) of the N × N matrix Y ( n, m ) = ( p I + Λ ) − n ( q I + Λ ) − m Y (0 , p I + Λ ) n ( q I + Λ ) m − npλ ( p I + Λ ) − − mqλ ( q I + Λ ) − (2.25a) in which the initial value matrix Y (0 , is subject to the condition [ Y (0 , , Λ ] = λ Λ + rank 1 , (2.25b) obey simultaneously both the discrete-time Ruijsenaars-Schneider systems given by (2.12) and (2.20) as well as the systems of constraint equations given by (2.23a) and (2.23b) . In order to make a connection with an initial value problem, we mention that the initialvalue matrix Y (0 ,
0) can be obtained from the diagonal matrix of initial values X (0 ,
0) by asimilarity transformation with a matrix U (0 ,
0) which is an invertible matrix diagonalizingthe initial Lax matrix L (0 , x i (0 , x i (1 , x i (0 , i = 1 , . . . , N ) . We note that the secular problem can, hence, be reformulatedas one for the following matrix X ( n, m ) = X (0 , − npλ ( p I + L (0 , − − mqλ ( I + K (0 , − , (2.26)and hence the solution is provided by the roots of the characteristic equation: P X ( x ) = det( x I − X ( n, m )) = N Y i =1 ( x − x i ( n, m )) . (2.27) Remark 1:
We would like to mention that in the more general case where p and q maydepend nontrivially on n and m , they should be subject to compatibility relations between(2.13a) and (2.21a), which produces the conditions b pq = e qp and b p + q = e q + p . From these two equations, the only possible answers would be p = p ( n ) and q = q ( m )implying that p and q can only depend on the discrete variable associated with their ownrespective directions. Remark 2:
In order to understand why we can regard the model described in this sec-tion as relativistic version of the discrete-time Calogero-Moser system, we perform the non-relativistic limit which is obtained by letting the parameter λ →
0. To implement the limit,we note that as a function of λ the variable h i as given in (2.11a) behave as: h i → − pλ (cid:2) λ p i + O ( λ ) (cid:3) , (2.28)where p i are momenta for the discrete-time Calogero-Moser system [15] given by p i = N X j =1 j = i x i − x j − N X j =1 x i − e x j . (2.29) The description of the initial value problem can be imposed the same fashion with the CM case [28] he spatial Lax matrix L κ given in (2.1b) becomes L κ → − p I − pλ (cid:18) κ E + L CM (cid:19) + O ( λ ) , (2.30)in which E = P i,j E ij and where L CM is the spatial Lax matrix for the discrete-timeCalogero-Moser system [15] given by L CM = N X i =1 p i E ii + N X i = j E ij x i − x j . (2.31)The temporal Lax matrix M κ given in (2.1c) expands as M κ → − pλ (cid:18) κ E + M CM (cid:19) + O ( λ ) , (2.32)where M CM is the temporal Lax matrix for the discrete-time Calgoero-Moser system [15]given by M CM = − N X i,j =1 E ij e x i − x j . (2.33)Thus, as λ →
0, the compatibility (2.3) produces e L CM E κ + E κ M CM + e L CM M CM = M CM E κ + E κ L CM + M CM L CM (2.34)consequently implying e L CM M CM = M CM L CM , (2.35) (cid:16) e L CM − M CM (cid:17) E = E ( M CM − L CM ) . (2.36)These two equations give what we know as the discrete-time equations of motion correspond-ing to the Calogero-Moser system [15].With the suitable choices of Λ → − e − λ Λ CM , p → e − λp CM and q → e − λq CM , up to order O ( λ ), the exact solution (2.25a) becomes Y ( n, m ) → Y (0 , − np CM I + Λ CM − mq CM I + Λ CM , (2.37)and in oder O ( λ ), (2.25b) yields[ Y (0 , , Λ CM ] = I + rank 1 . (2.38)These two equations are just identical to the defining relations for the exact solution forthe discrete-time Calogero-Moser system [28]. Thus, both the discrete Lax representation aswell as the solution for the discrete CM model is obtained through the above limits from thediscrete-time Ruijsenaars-Schneider model. In the continuum case the non-relativistic limitof the Ruijsenaars model was discussed in [4]. In this section we consider the Lagrange formulation of the discrete multi-time RS modeland show that it possesses a Lagrange 1-form structure. In the CM case [28], we obtainedLagrangians 1-form structure through the connection of the Lax representation. Here wealso have the Lagrangian 1-form structure for the RS system, but the establishment is moredifficult as connection through the Lax representation is no longer relevant. In this Section,we will first derive the Lagrangian 1-form for the discrete-time RS system and then establish he closure relation. It is easy to show that the actions corresponding to equations of motion(2.12) and (2.20) are given by S ( n ) = X n L ( n ) ( x ( n ) , x ( n + 1)) , (3.1a) S ( m ) = X m L ( m ) ( x ( m ) , x ( m + 1)) , (3.1b)where L ( n ) = N X i,j =1 ( f ( x i − e x j ) − f ( x i − e x j − λ )) − N X i,j =1 j = i ( f ( x i − x j + λ )+ f ( e x i − e x j + λ )) − ln | p | (Ξ − e Ξ) , (3.2a) L ( m ) = N X i,j =1 ( f ( x i − b x j ) − f ( x i − b x j − λ )) − N X i,j =1 j = i ( f ( x i − x j + λ )+ f ( b x i − b x j + λ )) − ln | q | (Ξ − b Ξ) , (3.2b)with Ξ = P Ni =1 x i and the function f ( x ) is given by f ( x ) = x ln( x ). We assume in thisand the following sections that the parameters p and q are constant. The discrete-timeEuler-Lagrange equations read ∂ L ( n ) ∂ e x i + ^ ∂ L ( n ) ∂x i = 0 , and ∂ L ( m ) ∂ b x i + \ ∂ L ( m ) ∂x i = 0 , which lead to (2.12) and (2.20), respectively.The additional terms ln | p | (Ξ − e Ξ) in (3.2a) and ln | q | (Ξ − b Ξ) in (3.2b), containing thedifferences of the centre of mass, are needed in order to account for the constraint equa-tions (2.23a) and (2.23b) as they arrive from the Euler-Lagrange (EL) equations on discretecurves, which is a connected collection of line segments (i.e. elementary links on the lattice)with or without end points (i.e. closed or non-closed), involving corners (vertices connectingline segments with different directions).
Theorem 3.1.
For the Lagrangians (3.2a) and (3.2b) , the closure relation \ L ( n ) ( x , e x ) − L ( n ) ( x , e x ) − ^ L ( m ) ( x , b x ) + L ( m ) ( x , b x ) = 0 , (3.3) holds on the solutions of the equations of motion (2.12) and (2.20) as well as the constraintequations (2.23a) and (2.23b) .Proof. (3.3) can be written in the form \ L ( n ) ( x , e x ) − L ( n ) ( x , e x ) − ^ L ( m ) ( x , b x ) + L ( m ) ( x , b x )= N X i,j =1 b x i ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b x i − be x j b x i − be x j − λ b x i − x j + λ b x i − x j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ln (cid:12)(cid:12)(cid:12)(cid:12) b x i − b x j + λ b x i − b x j − λ (cid:12)(cid:12)(cid:12)(cid:12)! − N X i,j =1 e x i ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e x i − be x j e x i − be x j − λ e x i − x j + λ e x i − x j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ln (cid:12)(cid:12)(cid:12)(cid:12) e x i − e x j + λ e x i − e x j − λ (cid:12)(cid:12)(cid:12)(cid:12)! + N X i,j =1 be x i ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e x i − be x j − λ e x i − be x j e x i − be x j e x i − be x j − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − x i ln (cid:12)(cid:12)(cid:12)(cid:12) x i − e x j x i − e x j − λ x i − b x j − λx i − b x j (cid:12)(cid:12)(cid:12)(cid:12)! + (ln | q | − ln | p | ) (cid:18)e Ξ − be Ξ − Ξ + b Ξ (cid:19) + λ N X i,j =1 ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b x i − be x j − λ e x i − be x j − λ x i − b x j − λx i − e x j − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ln (cid:12)(cid:12)(cid:12)(cid:12) b x i − b x j + λ e x i − e x j + λ (cid:12)(cid:12)(cid:12)(cid:12)! , (3.4a) sing (2.12), (2.20), (2.23a) and (2.23b), we have \ L ( n ) ( x , e x ) − L ( n ) ( x , e x ) − ^ L ( m ) ( x , b x ) + L ( m ) ( x , b x )= N X i =1 ( b x i + e x i − be x i − x i ) ln (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:12) (cid:18)e Ξ − be Ξ − Ξ + b Ξ (cid:19) + λ N X i,j =1 ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b x i − be x j − λ e x i − be x j − λ x i − b x j − λx i − e x j − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ln (cid:12)(cid:12)(cid:12)(cid:12) b x i − b x j + λ e x i − e x j + λ (cid:12)(cid:12)(cid:12)(cid:12)! . (3.4b)Using the fact that the last line of (3.4b) vanishes on the exact solution (2.25a) and e Ξ − be Ξ − Ξ + b Ξ = 0, then we have \ L ( n ) ( x , e x ) − L ( n ) ( x , e x ) − ^ L ( m ) ( x , b x ) + L ( m ) ( x , b x ) = 0 . (3.4c) n i n j Γ n i n j Γ ′ Figure 1: Deformation of the discrete curve Γ.
In [28] we described what we mean by the Lagrangian 1-form, but let us reiterate thishere for the sake of self-contained of this paper.
Definition . Let e i represent the unit vector in the lattice direction labeled by i and letany position in the lattice be identified by the vector n , so that an elementary shift in thelattice can be created by the operation n n + e i . Since the Lagrangian depends on x andits elementary shift in one discrete direction, it can be associated with an oriented vector e i on a curve Γ i ( n ) = ( n , n + e i ) , and we can treat these Lagrangians as defining a discrete1-form L i ( n ) L i ( n ) = L i ( x ( n ) , x ( n + e i )) , (3.5) which satisfies the following relation L i ( x ( n + e j ) , x ( n + e i + e j )) − L i ( x ( n ) , x ( n + e i )) − L j ( x ( n + e i ) , x ( n + e j + e i )) + L j ( x ( n ) , x ( n + e j )) = 0 . (3.6) Equation (3.6) represents the closure relation of the Lagrangian -form for the RS systemand it can be explicitly shown holding on the level of the equations of motion, and as well asconstraints. Choosing a discrete curve Γ consisting of connected elements Γ i , we can define an actionon the curve by summing up the contributions L i from each of the oriented links Γ i in thecurve, to get S ( x ( n ); Γ) = X n ∈ Γ L i ( x ( n ) , x ( n + e i )) . (3.7) he closure relation (3.6) is actually equivalent to the invariance of the action under localdeformations of the curve. To see this, suppose we have an action S evaluated on a curveΓ, and we deform this (keeping end points fixed) to get a curve Γ ′ on which an action S ′ isevaluated, such as in Figure 1.Then S ′ is related to S by the following: S ′ = S − L i ( x ( n + e j ) , x ( n + e i + e j )) + L i ( x ( n ) , x ( n + e i ))+ L j ( x ( n + e i ) , x ( n + e j + e i )) − L j ( x ( n ) , x ( n + e j )) . (3.8)Equation (3.8) shows that the independence of the action under such a deformation is locallyequivalent to the closure relation. The invariance of the action under the local deformationis a crucial aspect of the underlying variational principle.The basic relations constituting the discrete multi-time EL equations were first given in2011 in [30] and arise as the EL equations for actions on a set of basic curves given in Fig.2. We now use the Lagrangian in (3.2a) L ( n ) and the Lagrangian in (3.2b) L ( m ) and derivethe full set of EL equations for these basic curves and associated actions, and apply them tothe case at hand of the discrete RS model. nm x e x be x L n f L m (a) The discrete curve E Γ of a lowercorner nm x b x be x c L n L m (b) The discrete curve E Γ of an uppercorner nm x e x ee x L n f L m (c) The discrete curve E Γ of thestraight horizontal line nm x b x bb x L n c L m (d) The discrete curve E Γ of thestraight vertical line Figure 2: Simple discrete curves for n and m variables. Chapter 3
The variational principle for Lagrangian 1-form of [30] provides the system of actions on theelementary discrete curves as indicated in Fig. 2, together with the corresponding EL equations. In a later paper[25], which appeared after these results were presented at the SIDE X (2012) meeting by the first author, theseequations were restated as a Theorem and applied to a discrete-time Toda system. ase (a) : The action for a discrete curve in Fig. 2(a) is S [ x ] = L p ( x , e x ) + f L q ( e x , be x ) . (3.9)We now vary the variable x x + δ x with the end points fixed: δ x = 0 and δ be x = 0. Thenthe variation of the action is δS = L p ∂ x δ x + L p ∂ e x δ e x + f L q ∂ e x δ e x + f L q ∂ be x δ be x = L p ∂ e x + f L q ∂ e x ! δ e x . (3.10)The first and last terms vanish according to the condition on end points. The δS = 0 oncethe coefficient of δ e x is zero yielding L p ( x , e x ) ∂ e x + f L q ( e x , be x ) ∂ e x = 0 . (3.11)Using (3.2a) and (3.2b), (3.11) gives pq = N Y j =1 ( x i − x b j + λ )( x i − x e j )( x i − x e j + λ )( x i − x b j ) . (3.12)which is the constraint equations given in (2.23a). Case (b) : The action for a discrete curve in Fig. 2(b) is S [ x ] = L q ( x , b x ) + c L p ( b x , be x ) . (3.13)We now vary the variable x x + δ x with the end points fixed: δ x = 0 and δ be x = 0. Thenthe variation of the action is δS = L q ∂ x δ x + L q ∂ b x δ b x + c L p ∂ b x δ b x + c L p ∂ be x δ be x = L q ∂ b x + c L p ∂ b x ! δ b x . (3.14)The first and last terms vanish according to the condition end points. The δS = 0 once thecoefficient of δ e x is zero yielding L q ( x , b x ) ∂ b x + c L p ( b x , be x ) ∂ b x = 0 . (3.15)Using (3.2a) and (3.2b), (3.15) gives pq = N Y j =1 ( x i − b x j − λ )( x i − e x j )( x i − e x j − λ )( x i − b x j ) , (3.16)which is the constraint equations given in (2.23b). Case (c) : The action for a discrete curve in Fig. 2(c) is S [ x ] = L p ( x , e x ) + f L p ( e x , ee x ) . (3.17)We now vary the variable x x + δ x with the end points fixed: δ x = 0 and δ ee x = 0. Thenthe variation of the action is δS = L p ∂ x δ x + L p ∂ e x δ e x + f L p ∂ e x δ e x + f L p ∂ ee x δ be x = L p ∂ e x + f L p ∂ e x ! δ e x . (3.18)The first and last terms vanish according to the condition end points. The δS = 0 once thecoefficient of δ e x is zero yielding L p ( x , e x ) ∂ e x + f L p ( e x , ee x ) ∂ e x = 0 . (3.19) sing (3.2a) and (3.2b), (3.19) gives N Y j =1 j = i ( x i − x j + λ )( x i − x j − λ ) = N Y j =1 ( x i − e x j )( x i − x e j + λ )( x i − x e j )( x i − e x j − λ ) . (3.20)which is the equations of motion given by (2.12). Case (d) : The action for a discrete curve in Fig. 2(d) is S [ x ] = L q ( x , b x ) + c L q ( b x , bb x ) . (3.21)We now vary the variable x x + δ x with the end points fixed: δ x = 0 and δ bb x = 0. Thenthe variation of the action is δS = L q ∂ x δ x + L q ∂ b x δ b x + c L q ∂ b x δ b x + c L q ∂ bb x δ bb x = L q ∂ b x + c L q ∂ b x ! δ b x . (3.22)The first and last terms vanish according to the condition end points. The δS = 0 once thecoefficient of δ b x is zero yielding L q ( x , b x ) ∂ b x + c L q ( b x , bb x ) ∂ b x = 0 . (3.23)Using (3.2a) and (3.2b), (3.23) gives N Y j =1 j = i ( x i − x j + λ )( x i − x j − λ ) = N Y j =1 ( x i − b x j )( x i − x b j + λ )( x i − x b j )( x i − b x j − λ ) . (3.24)which is the equations of motion given by (2.20).In conclusion, the discrete variational principle which comprises the basic set of equations(3.11), (3.15), (3.19) and (3.23), produces the system of EL equations and constraints forthe rational discrete-time RS model. Furthermore, the closure relation (3.3) expresses thecompatibility of these four basic equations, and as a consequence on the solutions of thevariational system the action is stationary under deformations of any discrete curve (withfixed end points) such as indicated in Fig. 1. In Appendix B, we demonstrate how to deriveexplicitly the discrete Euler-Lagrange equation for some specific discrete curves. In this Section, we study a continuum analogue of a previous construction in Section 2 byconsidering a particular semi-continuous limit. Since the exact solution (2.25a) contains twodiscrete variables n and m , we could perform a continuum limit on one of these variablesseparately, while leaving the other discrete variable intact, and thus obtain a semi-continuousequation with one remaining discrete and two continuous independent variables. Alterna-tively, we can first perform a change of independent variables on the lattice and subsequentlyperform the limit on one of the new variables. The advantage of the latter approach overthe former is that it often leads in a more direct way to a hierarchy of higher order flows .Adopting the latter approach in this section, we use a new discrete variable N := n + m , andperform the transformation on the dependent variables by setting x ( n, m ) x ( N , m ) =: x ,which leads to the following expressions for the shifted variables: x = x ( n + 1 , m ) x ( N + 1 , m ) =: e x , b x = x ( n, m + 1) x ( N + 1 , m + 1) =: be x , e x = x ( n + 1 , m + 1) x ( N + 2 , m + 1) =: bee x . earranging the terms in (2.25a), we have Y ( N , m ) = ( p I + Λ ) − N (cid:18) q I + Λ p I + Λ (cid:19) − m (cid:20) Y (0 , − N pλp I + Λ + mλ (cid:18) pp I + Λ − qq I + Λ (cid:19)(cid:21) ( p I + Λ ) N (cid:18) q I + Λ p I + Λ (cid:19) m . (4.1)We perform the limit n → −∞ , m → ∞ , ε → N fixed and setting ε = p − q ,such that εm = τ remains finite. Focusing on the penultimate factor in (4.1) we have thatlim m →∞ ε → εm → τ (cid:18) − εp I + Λ (cid:19) m = lim m →∞ (cid:18) − τm ( p I + Λ ) (cid:19) m = e − τp I + Λ , (4.2)so that the exact solution takes the form Y ( N , τ ) = ( p I + Λ ) − N e τp I + Λ (cid:20) Y (0 , − N pλp I + Λ + τ λ Λ ( p I + Λ ) (cid:21) ( p I + Λ ) N e − τp I + Λ . (4.3)This equation represents the full solution after taking the skew limit. The position of theparticles x i ( N , τ ) can be determined by computing the eigenvalues of (4.3). We first rewrite the equations of motion (2.12), taking p to be constant , in terms of thevariables ( N , m ) as follows N X j =1 j = i (ln(x i − x j + λ ) − ln(x i − x j − λ )) = N X j =1 (cid:16) ln(x i − eb x j ) − ln(x i − x eb j + λ )+ ln(x i − x be j ) − ln(x i − eb x j − λ ) (cid:17) , (4.4)Introducing the notations b x = x( N , τ + ε ) and x b = x( N , τ − ε ) with the use of the Taylorexpansion, we obtainx( N , τ ± ε ) = x( N , τ ) ± ε ∂ x( N , τ ) ∂τ + ε ∂ x( N , τ ) ∂τ ± ... . (4.5a)Collecting terms in order O ( ε ), we have the equations of motion for the RS system corre-sponding to the “ N ” variable N X j =1 j = i [ln(x i − x j + λ ) − ln(x i − x j − λ )] = N X j =1 [ln(x i − e x j ) − ln(x i − x e j + λ )+ ln(x i − x e j ) − ln(x i − e x j − λ )] , (4.6)and O ( ε ), we have N X j =1 (cid:20) ∂ e x j ∂τ (cid:18) i − e x j − λ − i − e x j (cid:19) + ∂ x e j ∂τ (cid:18) i − x e j + λ − i − x e j (cid:19)(cid:21) = 0 . (4.7)which are the equations of motion for the RS system corresponding to the “ τ ” variable.Similarly, changing the variables x ( n, m ) x(N , τ ) in constraints (2.23a) and (2.23b)and collecting terms in order O ( ε ), we have − p = N X j =1 ∂ e x j ∂τ (cid:18) i − e x j − λ − i − e x j (cid:19) , (4.8a)1 p = N X j =1 ∂ x e j ∂τ (cid:18) i − x e j + λ − i − x e j (cid:19) , (4.8b) Next, to obtain the continuum limit of the action, we proceed exactly with the same stepsas in [28]. First, we observe that eq. (4.6) can be once again be obtained by implementingthe usual variational principle on the following action S ( N ) given by S ( N ) = X N L ( N ) = X N N X i,j =1 ( f (x i − e x j ) − f (x i − e x j − λ )) − N X i,j =1 j = i f (x i − x j + λ ) − N X i,j =1 j = i f ( e x i − e x j + λ ) − ln | p | N X i =1 (x i − e x i ) , (4.9)where now the Lagrangian L ( N ) involves variables e x i shifted in the discrete variable N insteadof the original variable n , and the corresponding discrete Euler-Lagrange equation reads: ^ ∂ L ( N ) ∂ x i + (cid:18) ∂ L ( N ) ∂ e x i (cid:19) = 0 , (4.10)yielding (4.6).Second, we observe that eq. (4.7) can be once again be obtained by implementing theusual variational principle on the following action S ( τ ) given by S ( τ ) = Z τ τ dτ L ( τ ) (cid:18) x ( N − , τ ) , ∂ x ( N , τ ) ∂τ (cid:19) , (4.11)which is obtained by taking the skew limit together with anti-Taylor expansion of (3.1b) and L ( τ ) = N X i,j =1 (cid:18) ∂ e x j ∂τ (ln | x i − e x j − λ | − ln | x i − e x j | ) (cid:19) − N X i,j =1 j = i (cid:18) ∂ e x j ∂τ (ln | e x i − e x j + λ | − ln | e x i − e x j − λ | ) + ∂ e x i ∂τ − ∂ e x j ∂τ (cid:19) + N X i =1 (cid:18) p (x i − e x i ) + ∂ e x i ∂τ ln | p | (cid:19) . (4.12)The Euler Lagrange equations ∂ L ( τ ) ∂ x i − ddτ (cid:18) ∂ L ( τ ) ∂ ( d x i /dτ ) (cid:19) = 0 , (4.13)yield (4.7). In the previous Section, we took the continuum limit on the discrete variable m , leadingto a system of differential-difference equations. The full continuum limit, performed on theremaining discrete variable N as well as τ , will lead to a coupled system of poles in the firstinstance, from which a hierarchy of ODEs can be retrieved, which is the RS hierarchy. How o perform this limit is inspired by the structure of the solutions of (4.3). Performing thefollowing computation, Y ( N , τ ) = (cid:18) I + Λ p (cid:19) − N e τp ( I + Λ p ) − Y (0 , e − τp ( I + Λ p ) − (cid:18) I + Λ p (cid:19) N − N λ (cid:18) Λ p (cid:19) − + τ λ Λ p (cid:18) Λ p (cid:19) − = e − N ln ( Λ p ) + τp ( Λ p ) − Y (0 , e N ln ( Λ p ) − τp ( Λ p ) − − N λ (cid:18) Λ p (cid:19) − + τ λ Λ p (cid:18) Λ p (cid:19) − . (5.1)We now introduce t = τp + N p , t = − τp − N p , t = 3 τp + N p , ..... , (5.2)and expand (5.1) with respect to variable p . We have Y ( t , t , t , ..., N ) = e − Λ t + Λ t − Λ t + ... Y (0 , , ... ) e Λ t − Λ t + Λ t + ... − N λ + Λ λt + Λ λt + Λ λt + ... , (5.3)which is a function of time variables ( t , t , t , ..., N ). The positions of the particles X i ( t , t , t , ..., N )can be computed by looking for the eigenvalues of (5.3). The explicit expression of the so-lution for the RS can be obtained from the secular problem for the matrix X (0 , − ξ + L (0 , λt + L (0 , λt + L (0 , λt , (5.4)where ξ = N λ and X (0 ,
0) = U − (0 , Y (0 , U (0 ,
0) and L (0 ,
0) = U − (0 , Λ U (0 , N -time flows for the RS system. The next solutions in the hierarchycan be generated by pushing further on with the expansion. We now would like to see what would result from taking the limit on the equations of motion(4.7). First, we introduce˙ x i = ∂ x i ∂τ = ∂X i ∂t ∂t ∂τ + ∂X i ∂t ∂t ∂τ + ∂X i ∂t ∂t ∂τ + ... = 1 p ∂X i ∂t − p ∂X i ∂t + 3 p ∂X i ∂t + ... , (5.5)and x i ( N ±
1) = x i ∓ λ ± p ∂X i ∂t + 1 p (cid:18) ∂ X i ∂t ∓ ∂X i ∂t (cid:19) + 1 p (cid:18) ± ∂X i ∂t − ∂ X i ∂t ∂t ± ∂X i ∂t (cid:19) + 1 p (cid:18) ∂ X i ∂t ∓ ∂ X i ∂t ∂t + 12 ∂ X i ∂t + ∂ X i ∂t ∂t (cid:19) + O (1 /p ) . (5.6)Then we expand (4.7) with respect to the variable p together with (5.2). We find that (cid:7) The leading term of order O (1 /p ) gives us ∂ X i ∂t / ∂X i ∂t + N X j =1 ∂X j ∂t (cid:18) X i − X j + λ + 1 X i − X j − λ − X i − X j (cid:19) = 0 , (5.7) hich is the equations of the motion for the usual continuous RS system [2]. (cid:7) The term of order O (1 /p ) gives us2 ∂ X i ∂t ∂t / ∂X i ∂t − ∂ X i ∂t ∂X i ∂t / (cid:18) ∂X i ∂t (cid:19) − λ ∂ X i ∂t + N X j =1 (cid:20) ∂X j ∂t (cid:18) X i − X j + λ + 1 X i − X j − λ − X i − X j (cid:19) + 12 ∂ X j ∂t (cid:18) X i − X j − λ − X i − X j + λ (cid:19) + 12 (cid:18) ∂X j ∂t (cid:19) (cid:18) X i − X j − λ ) − X i − X j + λ ) (cid:19)(cid:21) = 0 . (5.8)This equation represents the next equation of motion beyond the usual continuous RS inthe hierarchy. We will stop at this equation, but we can actually get the higher terms ofthe equation in which the variable t and higher order time-flows must be taken into account.The full limit of the addition between (4.8a) and (4.8b) in the order O (1 /p ) gives2 λ ∂X i ∂t − ∂X i ∂t / ∂X i ∂t + N X j =1 ∂X j ∂t (cid:18) X i − X j + λ − X i − X j − λ (cid:19) = 0 , (5.9)which is the constraint equations for the full limit which will play a crucial role in the nextSubsection, see (5.21).Using (5.9), we can simplify (5.8) into ∂ X i ∂t ∂t / ∂X i ∂t + N X j =1 (cid:20) ∂X j ∂t (cid:18) X i − X j + λ + 1 X i − X j − λ − X i − X j (cid:19) − ∂X i ∂t ∂X j ∂t (cid:18) X i − X j − λ ) − X i − X j + λ ) (cid:19)(cid:21) = 0 . (5.10)Note that (5.10) can be obtained directly from the full limit in order O (1 /p ) from thecombination of the relations − p = N X j =1 ∂ x j ∂τ (cid:18) e i − x j − λ − e i − x j (cid:19) , (5.11)1 p = N X j =1 ∂ x j ∂τ (cid:18) e x i − x j − λ − e x i − x j (cid:19) , (5.12)which are the backward shift and forward shift of (4.8a) and (4.8b), respectively. We will follow the steps in [28] in order to obtain the full limit on the action. We now takethe action to be of the form S [ x ( N , τ ); Γ] = Z τ τ dτ L ( τ ) ( x ( N , τ ) , ˙ x ( N , τ )) + X N L ( N ) ( x ( N , τ ) , x ( N + 1 , τ )) , (5.13)where the first term belongs to the vertical part and the second term belongs to the horizontalpart of the curve Γ. sing anti-Taylor expansion, the action now becomes S [ x ( N , τ ); Γ] = Z τ τ dτ L ( τ ) ( x ( N , τ ) , ˙ x ( N , τ )) + Z N N d N L ( N ) ( x ( N , τ ) , x ( N + 1 , τ )) , (5.14)where we do not need to take into account the boundary terms coming from the expansion,because they are constant at the end points and do not contribute to the variational process.We now perform a change of variables ( τ, N ) ( t , t ) by using (5.2), dτ = − p dt − p dt , (5.15a) d N = p dt + 2 pdt , (5.15b)and also expand the Lagrangians with respect to variable p . We obtain S [ X ( t , t ); Γ] = Z t (2) t (1) dt L ( t ) (cid:18) X ( t , t ) , ∂ X ( t , t ) ∂t (cid:19) + Z t (2) t (1) dt L ( t ) (cid:18) X ( t , t ) , ∂ X ( t , t ) ∂t , ∂ X ( t , t ) ∂t (cid:19) , (5.16)where L ( t ) and L ( t ) are given by L ( t ) = N X i =1 ∂X i ∂t ln (cid:12)(cid:12)(cid:12)(cid:12) ∂X i ∂t (cid:12)(cid:12)(cid:12)(cid:12) − N X i = j ∂X j ∂t (ln | X i − X j − λ | − ln | X i − X j | ) , (5.17)which first appeared in [4] and the Euler-Lagrange equation ∂ L ( t ) ∂X i − ∂∂t ∂ L ( t ) ∂ ( ∂X i ∂t ) ! = 0 , (5.18)gives exactly eq. (5.7) and L ( t ) = N X i =1 ∂X i ∂t ln (cid:12)(cid:12)(cid:12)(cid:12) ∂X i ∂t (cid:12)(cid:12)(cid:12)(cid:12) − λ (cid:18) ∂X i ∂t (cid:19) + 3 ∂X i ∂t ! − N X i = j (cid:20) ∂X j ∂t (ln | X i − X j − λ | − ln | X i − X j | ) + 12 ∂X i ∂t ∂X j ∂t X i − X j + λ (cid:21) . (5.19)We see that the Lagrangian L ( t ) contains derivatives with respect to two time flows t and t . We observe that the equations of motion (5.10) require the Euler-Lagrange equation inthe form ∂ L ( t ) ∂X i − ∂∂t ∂ L ( t ) ∂ ( ∂X i ∂t ) ! = 0 . (5.20)Furthermore, we find that ∂ L ( t ) ∂ ( ∂X i ∂t ) = 2 λ ∂X i ∂t − ∂X i ∂t / ∂X i ∂t + N X j =1 ∂X j ∂t (cid:18) X i − X j + λ − X i − X j − λ (cid:19) = 0 , (5.21)which is continuum analogue of discrete constraints (5.9) in order O (1 /p ). Actually thereare more constraints from the expansion which will play a major role when we consider thehigher Lagrangians in the hierarchy.Here we obtained the hierarchy of continuous Lagrangians associated with the discrete-time RS-model through the full continuum limit. Obviously, higher Lagrangians in thefamily can be generated by pushing further on with the expansion. Interestingly, theseLagrangians as well as the constraints also satisfy the variational principle for Lagrangian1-form presenting in [28, 30]. We here restrict ourselves with the first two flows for simplicity. .3 The full limit on the closure relation We take the full limit on the discrete closure relation (3.3) leading to
Theorem 5.1.
We find that the continuous version of the closure relation between t and t ∂ L ( t ) ∂t = ∂ L ( t ) ∂t , (5.22) which holds on the equations of motion and constraint.Proof. : We find that ∂ L ( t ) ∂t = N X i =1 (cid:18) ∂ X i ∂t ∂t ln (cid:12)(cid:12)(cid:12)(cid:12) ∂X i ∂t (cid:12)(cid:12)(cid:12)(cid:12) + ∂ X i ∂t ∂t (cid:19) − N X i = j (cid:18) ∂ X j ∂t ∂t [ln | X i − X j − λ | − ln | X i − X j | ]+ ∂X j ∂t ∂X i ∂t (cid:20) X i − X j − λ − X i − X j (cid:21) + ∂X j ∂t ∂X j ∂t (cid:20) X i − X j − λ − X i − X j (cid:21)(cid:19) , (5.23a)and ∂ L ( t ) ∂t = N X i =1 (cid:18) ∂ X i ∂t ∂t ln (cid:12)(cid:12)(cid:12)(cid:12) ∂X i ∂t (cid:12)(cid:12)(cid:12)(cid:12) + ∂X i ∂t ∂ X i ∂t / ∂X i ∂t − λ ∂X i ∂t ∂ X i ∂t + 3 ∂ X i ∂t ∂t (cid:19) − N X i = j (cid:18) ∂ X j ∂t ∂t [ln | X i − X j − λ | − ln | X i − X j | ]+ ∂X i ∂t ∂X j ∂t (cid:20) X i − X j − λ − X i − X j (cid:21) + ∂X j ∂t ∂X j ∂t (cid:20) X i − X j − λ − X i − X j (cid:21) + 12 ∂ X j ∂t ∂X i ∂t (cid:20) X i − X j + λ − X i − X j − λ (cid:21) − (cid:18) ∂X j ∂t (cid:19) ∂X i ∂t (cid:20) X i − X j − λ ) − X i − X j + λ ) (cid:21)! . (5.23b)We find that ∂ L ( t ∂t = ∂ L ( t ∂t gives − N X i =1 ∂ X i ∂t ∂t + ∂X j ∂t ∂X i ∂t (cid:20) X i − X j − λ − X i − X j (cid:21) = N X i =1 (cid:18) ∂X i ∂t ∂ X i ∂t / ∂X i ∂t − λ ∂X i ∂t ∂ X i ∂t + 3 ∂ X i ∂t ∂t (cid:19) + N X i = j (cid:18) − ∂X i ∂t ∂X j ∂t (cid:20) X i − X j − λ − X i − X j (cid:21) + 12 ∂ X j ∂t ∂X i ∂t (cid:20) X i − X j + λ − X i − X j − λ (cid:21) − (cid:18) ∂X j ∂t (cid:19) ∂X i ∂t (cid:20) X i − X j − λ ) − X i − X j + λ ) (cid:21)! . (5.23c) ividing (5.23c) by ∂X i ∂t we find that ∂ L ( t ) ∂t − ∂ L ( t ) ∂t = N X i =1 ∂X i ∂t ∂ X i ∂t t / ∂X i ∂t + ∂ X i ∂t ∂X i ∂t / (cid:18) ∂X i ∂t (cid:19) − λ ∂ X i ∂t − N X j =1 (cid:20) ∂X j ∂t (cid:18) X i − X j + λ + 1 X i − X j − λ − X i − X j (cid:19) + 12 ∂ X j ∂t (cid:18) X i − X j + λ + 1 X i − X j − λ (cid:19) − (cid:18) ∂X i ∂t (cid:19) (cid:18) X i − X j − λ ) − X i − X j + λ ) (cid:19)(cid:21)! . (5.23d)Using (5.8), (5.23d) becomes ∂ L ( t ) ∂t − ∂ L ( t ) ∂t = N X i =1 ∂X i ∂t ∂ X i ∂t ∂X i ∂t / (cid:18) ∂X i ∂t (cid:19) − N X j =1 ∂X j ∂t (cid:18) X i − X j + λ + 1 X i − X j − λ − X i − X j (cid:19)(cid:19) . (5.23e)Inserting (5.7), we have now ∂ L ( t ) ∂t − ∂ L ( t ) ∂t = − N X i,j =1 (cid:18) ∂X j ∂t ∂X i ∂t + ∂X i ∂t ∂X j ∂t (cid:19) (cid:18) X i − X j + λ + 1 X i − X j − λ − X i − X j (cid:19) (5.23f)The second term of (5.23f) is the antisymmetric function, hence vanishes. We now have ∂ L ( t ) ∂t − ∂ L ( t ) ∂t = 0 . (5.23g) In [28], we set out the key principles of the new variational calculus associated with themulti-time Lagrangian 1-form structure. These principles were discovered on the basis ofthe careful analysis of the rational CM model, which formed the “laboratory” for studyinghow this new least-action principle should work. For simplicity, we focused on the case ofLagrangian 1-forms in the 2-time case, but the general principles apply to the case of thegeneral multi-time case in an obvious manner . Let us summarize here our findings.In the 2-time case the action is defined by S [ x ( t , t ); Γ] = Z Γ (cid:0) L ( t ) dt + L ( t ) dt (cid:1) = Z s s (cid:18) L ( t ) dt ds + L ( t ) dt ds (cid:19) ds . (5.24)where Γ is an arbitrary curve in the space of the two time-variables t and t , which isparametrised by Γ : ( t , t ) = ( t ( s ) , t ( s )) with the parameter s ∈ [ s , s ], see Fig. 3(a). L ( t ) and L ( t ) are the components of the Lagrangian 1-form: L ( t ) = L ( t ) ( x ( t , t ) , x t ( t , t ) , x t ( t , t )) , (5.25) L ( t ) = L ( t ) ( x ( t , t ) , x t ( t , t ) , x t ( t , t )) , (5.26) This was done in [30] as well as in a recent preprint, [25], adopting a somewhat different point of view. n which x t = ∂ x /∂t x t = ∂ x /∂t . The dependent variable x = ( x , x , ..., x N ) isthe vector of the position coordinates of the particles. The action should, in our point ofview, be considered as a functional of both the dependent variables x ( t , t ) as well as ofthe curve Γ, i.e., of the functions t ( s ) = ( t ( s ) , t ( s )). Thus, the least-action principle for1-forms is the implementation of the demand for criticality of the action under (infinites-imal) variations of the curve Γ in the space of independent variables, as well as of the evaluation curve E Γ in the space of dependent variables x ( t ), as indicated in Fig. 3(a).Thus, we have to apply the principle of criticality of the action (5.24) under: i) variations t x t t ( s ) , t ( s )ΓΓ ′ t ( s ) , t ( s ) x ( t ( s ) , t ( s )) x ( t ( s ) , t ( s )) E ′ Γ E Γ ( a ) t x t ( t ( s ) , t ( s ))Γ Γ Γ ( t ( s ) , t ( s ))( b )Figure 3: (a) The deformation of the curves Γ → Γ ′ and E Γ → E ′ Γ in the x − t configuration.(b) The deformation of the curve Γ on the space of independent variables. t ( s ) → t ( s ) + δ t ( s ) of the independent variables parametrising the curve Γ, as well as ii) variations x ( t , t ) → x ( t , t ) + δ x ( t , t ) of the dependent variables on an arbitrary eval-uation curve E Γ . i) Requiring criticality of the action w.r.t. variations of the curve δS = S [ x ( t + δt , t + δt ); Γ ′ ] − S [ x ( t , t ); Γ] = 0 with the condition δ t ( s ) = δ t ( s ) = 0, leads to the continuous closure relation : ∂ L ( t ) ∂t = ∂ L ( t ) ∂t , (5.27a) ii) Criticality of the action under variations of the dependent variables requires to considertwo types of variations: variations of the variables x ( t ) and its derivatives tangential to thecurve, and variations w.r.t. derivatives of the variables x ( t ) transversal to the curve. Thelatter should be treated as independent variations, whilst the former give rise to integrationby parts in the integral over the variable “ s ”. This leads to the system of Euler-Lagrange(EL) equations: ∂ L ( t ) ∂ x dt ds + ∂ L ( t ) ∂ x dt ds − dds (cid:26) k d t /ds k × "(cid:18) dt ds (cid:19) ∂ L ( t ) ∂ x t + (cid:18) dt ds (cid:19) (cid:18) dt ds (cid:19) (cid:18) ∂ L ( t ) ∂ x t + ∂ L ( t ) ∂ x t (cid:19) + (cid:18) dt ds (cid:19) ∂ L ( t ) ∂ x t = 0 , (5.27b)together with ∂ L ( t ) ∂ x t (cid:18) dt ds (cid:19) + (cid:18) ∂ L ( t ) ∂ x t − ∂ L ( t ) ∂ x t (cid:19) dt ds dt ds − ∂ L ( t ) ∂ x t (cid:18) dt ds (cid:19) = 0 . (5.27c)Here (5.27c) could be considered to be a system of constraints whilst (5.27b) are EL equations long the curve Γ. It is a conceptually novel point, put forward in [28] and [9], that the entire set of gener-alized EL equations (5.27) should be considered not only as a system producing equationsof the motion for a given Lagrange function, but should actually be considered as a systemof equations for the Lagrangians themselves. The solutions of the system, which are the admissable
Lagrangians, are necessarily the ones associated with integrable systems.Since the equations (5.27) must hold on an arbitrary curve, the system must hold inparticular on curves made out of straight segments along the t - and t -axes. Thus, invokingthe closure relation (5.27a), we can deform an arbitrary curve Γ to a simpler curve : Γ → Γ + Γ as shown in Fig. 3(b). On the curve Γ , where the time variable t is “frozen”, theconstraint equations arises from variations of the derivative x t , whereas on Γ the constraintarises from variations of the derivative x t . Thus, in the case that L ( t ) is independent ofthe latter derivatives (as in the example of the RS system) we get the system of equations: ∂ L ( t ) ∂ x − ∂∂t (cid:18) ∂ L ( t ) ∂ x t (cid:19) = 0 , ∂ L ( t ) ∂ x t = 0 , (5.28a) ∂ L ( t ) ∂ x − ∂∂t (cid:18) ∂ L ( t ) ∂ x t (cid:19) = 0 , ∂ L ( t ) ∂ x t = 0 . (5.28b)Thus, we recover the system of EL equations and closure as given in subsection 5.3. In contrast to the CM case [28], where we started with a semi-discrete KP equation, andapplied a pole-reduction to it to a yield a compatible CM system, here we start from RSsystem and reconnect it to the fully discrete lattice KP systems. In [17], the connectionbetween the RS system and the KP system was established for the trigonometric case, buthere we will focus on the (simpler) rational case as it clarifies the situation more clearly,We will develop now a scheme along the lines of the papers [30, 10, 11, 29]. Starting fromthe “solution matrix” Y ( n, m ) given in (A.13), we will introduce the relevant τ -function asits characteristic polynomial: τ ( ξ ) = det( ξ I − Y ) , (6.1)where Y = Y ( n, m, h ) is now a function of three discrete variables obeying the shift relations: e Y − Y + µ I = e rs T , (6.2a) b Y − Y + η I = b rs T , (6.2b) Y − Y + ν I = rs T , (6.2c)where r and s depend on the discrete variables via the following shift relations (see (A.5)):( p I + Λ ) · e r = r , s T · ( p I + Λ ) = e s T , (6.3a)( q I + Λ ) · b r = r , s T · ( q I + Λ ) = b s T , (6.3b)( r I + Λ ) · r = r , s T · ( r I + Λ ) = s T . (6.3c)As explained in Appendix A, we have introduced in (6.2) a slight generalization by introduc-ing the parameters µ , η and ν instead of all three being equal to λ . The shifts e and b are,as before, lattice shifts associated with the lattice parameters p and q respectively, whereasthe shift in the third variable is indicated by and is associated with a lattice parameter r . In [28] eq. (5.27b) was given in a slightly different form, using a different basis of decomposition of the deriva-tives of x along and transversal to the curve, whereas this particular form uses an orthogonal basis as suggestedby [25]. Although the form of [28], which was obtained using a non-orthogonal basis for the decomposition, hasthe slight disadvantage that it is not well-defined for points on the curves where dt /dt becomes singular, bothforms are equivalent when viewed as part of the system containing both (5.27b) and (5.27c), the latter beinginvariant under the choice of basis. o derive the equations directly from the resolvent of the matrix Y , we proceed as follows.First, we perform the simple computation e τ ( ξ ) = det( ξ + µ − Y − e rs T ) , = det(( ξ + µ − Y )(1 − e rs T ( ξ + µ − Y ) − )) , = τ ( ξ + µ )(1 − s T ( ξ + µ − Y ) − e r ) , then we have e τ ( ξ ) τ ( ξ + µ ) = 1 − s T ( ξ + µ − Y ) − ( p + Λ ) − r = v p ( ξ + µ ) , (6.4)in which the function v p is given byv a ( ξ ) := 1 − s T ( ξ − Y ) − ( a + Λ ) − r (6.5)for a general parameter a setting a = p .The reverse formula to Eq. (6.4) can be obtained by a similar computation: τ ( ξ ) = det( ξ − µ − e Y + e rs T ) , = det(( ξ − µ − e Y )(1 + e rs T ( ξ − µ − e Y ) − )) , = e τ ( ξ − µ )(1 + s T ( ξ − µ − e Y ) − e r ) , then we have τ ( ξ ) e τ ( ξ − µ ) = 1 + e s T ( p + Λ ) − ( ξ − µ − e Y ) − e r = e w p ( ξ − µ ) , (6.6)in which the function w p is given byw a ( ξ ) := 1 + s T ( a + Λ ) − ( ξ − Y ) − r (6.7)for again a general parameter a setting a = p .From (6.4) and (6.6), we have the relation τ ( ξ ) e τ ( ξ − µ ) = e w p ( ξ − µ ) = 1 v p ( ξ ) . (6.8)The same type of the relation for the other discrete directions can be obtained in the forms τ ( ξ ) b τ ( ξ − η ) = b w q ( ξ − η ) = 1 v q ( ξ ) , (6.9a) τ ( ξ ) τ ( ξ − ν ) = w r ( ξ − ν ) = 1 v r ( ξ ) . (6.9b)In order to derive discrete KP equations for τ ( ξ ), w and v , we introduce the N-componentvectors u a ( ξ ) = ( ξ − Y ) − ( a + Λ ) − r , (6.10a) t u b ( ξ ) = s T ( b + Λ ) − ( ξ − Y ) − , (6.10b)as well as the scalar variables S ab ( ξ ) = s T ( b + Λ ) − ( ξ − Y ) − ( a + Λ ) − r . (6.11)We now consider (6.10a) which can be written in the form u a ( ξ ) = ( p − a ) e u a ( ξ − µ ) + v a ( ξ ) e u ( ξ − µ ) , (6.12) ith u ( ξ ) = ( ξ − Y ) − r .The same process can be applied to (6.10b) and we obtain f t u b ( ξ ) = ( p − b ) f t u b ( ξ + µ ) + e w b ( ξ ) f t u ( ξ + µ ) , (6.13)with t u ( ξ ) = s T ( ξ − Y ) − .Another type of relation can be obtained by multiply e s T ( b + Λ ) − on the left hand sideof (6.12). We have e s T ( b + Λ ) − u a ( ξ ) = ( p − a ) e s T ( b + Λ ) − e u a ( ξ − µ )+ v a ( ξ ) e s T ( b + Λ ) − e u ( ξ − µ ) , s T ( p + Λ )( b + Λ ) − u a ( ξ ) = ( p − a ) e S ab ( ξ − µ ) + v a ( ξ ) e w b ( ξ − µ ) , v a ( ξ ) e w b ( ξ − µ ) = 1 + ( p − b ) S ab ( ξ ) − ( p − a ) e S ab ( ξ − µ ) . (6.14)Similarly, multiplying the right hand side of (6.13), we obtain e w b ( ξ ) v a ( ξ + µ ) = 1 + ( p − b ) S ab ( ξ + µ ) − ( p − a ) e S ab ( ξ ) . (6.15)By proceeding the similar steps, we can derive the relations in other discrete-time directions,namely v a ( ξ ) b w b ( ξ − η ) = 1 + ( q − b ) S ab ( ξ ) − ( q − a ) b S ab ( ξ − η ) , (6.16a) v a ( ξ ) w b ( ξ − ν ) = 1 + ( r − b ) S ab ( ξ ) − ( r − a ) S ab ( ξ − ν ) , (6.16b)Using the identity e w b ( ξ − µ − η ) v a ( ξ − µ ) b w b ( ξ − µ − ν ) v a ( ξ − µ ) = e w b ( ξ − µ − ν ) e v a ( ξ − µ ) b w b ( ξ − µ − η ) b v a ( ξ − µ ) be w b ( ξ − µ − η ) b v a ( ξ − η ) be w b ( ξ − ν − η ) e v a ( ξ − ν ) , (6.17)we can derive 1 + ( p − b ) S ab ( ξ − ν ) − ( p − a ) e S ab ( ξ − µ − ν )1 + ( q − b ) S ab ( ξ − ν ) − ( q − a ) b S ab ( ξ − ν − η )= 1 + ( r − b ) e S ab ( ξ − µ ) − ( r − a ) e S ab ( ξ − µ − ν )1 + ( q − b ) e S ab ( ξ − µ ) − ( q − a ) be S ab ( ξ − µ − η ) × p − b ) b S ab ( ξ − η ) − ( p − a ) be S ab ( ξ − µ − η )1 + ( r − b ) b S ab ( ξ − η ) − ( r − a ) b S ab ( ξ − η − ν ) , (6.18)which is a three-dimensional lattice equation which appeared first (in a slightly differentform) in [13]. Effectively, this is the Schwarzian lattice KP equation which in its canonicalform was first given in [5], cf. also [29].We now multiply e s T on the left hand side of (6.12) leading to e s T u a ( ξ ) = ( p − a ) e s T e u a ( ξ − µ ) + v a ( ξ ) e s T e u ( ξ − µ ) , s T ( p + Λ ) u a ( ξ ) = ( p − a )(1 − e v a ( ξ − µ )) + v a ( ξ ) e s T e u ( ξ − µ ) . (6.19)Introducing u ( ξ ) = s T ( ξ − Y ) − r (6.20)(6.19) can be written in the form( p + e u ( ξ − µ )) v a ( ξ ) − ( p − a ) e v a ( ξ ) = a + s T Λ u a ( ξ ) . (6.21) nother two relations related to the “ b ” and “ ¯ ” directions can be automatically obtained( q + b u ( ξ − η )) v a ( ξ ) − ( q − a ) b v a ( ξ − η ) = a + s T Λ u a ( ξ ) , (6.22a)( r + u ( ξ − ν )) v a ( ξ ) − ( r − a ) v a ( ξ − ν ) = a + s T Λ u a ( ξ ) . (6.22b)Eliminating the term s T Λ u a ( ξ ), we can derive the relations( p − q + e u ( ξ − µ ) − b u ( ξ − η )) v a ( ξ ) = ( p − a ) e v a ( ξ − µ ) − ( q − a ) b v a ( ξ − η ) , (6.23a)( p − r + e u ( ξ − µ ) − u ( ξ − ν )) v a ( ξ ) = ( p − a ) e v a ( ξ − µ ) − ( r − a ) v a ( ξ − ν ) , (6.23b)( r − q + u ( ξ − ν ) − b u ( ξ − η )) v a ( ξ ) = ( r − a ) v a ( ξ − ν ) − ( q − a ) b v a ( ξ − η ) . (6.23c)We now set p = a then (6.23a) and (6.23b) become p − q + e u ( ξ − µ ) − b u ( ξ − η ) = − ( q − p ) b v p ( ξ − η ) v p ( ξ ) , (6.24a) p − r + e u ( ξ − µ ) − u ( ξ − ν ) = − ( r − p ) v p ( ξ − ν ) v p ( ξ ) , (6.24b)The combination of (6.24a) and (6.24b) gives p − q + e u ( ξ − µ ) − b u ( ξ − η ) p − r + e u ( ξ − µ ) − u ( ξ − ν ) = p − q + e u ( ξ − µ − ν ) − b u ( ξ − η − ν ) p − r + be u ( ξ − µ − η ) − b u ( ξ − η − ν ) , (6.25)which is the “ lattice KP equation ”, [13], cf. also [12].From the definition of the function v p ( ξ ) in (6.4), (6.24a) and (6.24b) can be written interms of the τ -function p − q + e u ( ξ − µ ) − b u ( ξ − η ) = − ( q − p ) be τ ( ξ − µ − η ) b τ ( ξ − η ) τ ( ξ ) e τ ( ξ − µ ) , (6.26a) p − r + e u ( ξ − µ ) − u ( ξ − ν ) = − ( r − p ) e τ ( ξ − µ − ν ) τ ( ξ − ν ) τ ( ξ ) e τ ( ξ − µ ) . (6.26b)From (6.23c), if we set r = a we also have r − q + u ( ξ − µ ) − b u ( ξ − η ) = − ( q − r ) b τ ( ξ − η − ν ) b τ ( ξ − η ) τ ( ξ ) τ ( ξ − ν ) . (6.27)The combination of (6.26a) (6.26b) (6.27) yields( p − q ) be τ ( ξ − µ − η ) τ ( ξ − ν ) + ( r − p ) e τ ( ξ − µ − ν ) b τ ( ξ − η )+( r − q ) b τ ( ξ − η − ν ) e τ ( ξ − µ ) = 0 , (6.28)which is the bilinear lattice KP equation, (originally coined DAGTE, cf. [7]).To summarize, we have established in this section a direct connection between thediscrete-time Ruijsenaars model, embedded in a multi-time space, and well-known latticesystems of KP type. This shows that the rational discrete-time RS model yields a spe-cial class of rational solutions of the KP equation through the exploitation of the matrix Y ( n, m, h ), whose eigenvalues are the RS particle positions and which at the same time actsas a kernel for the lattice KP solutions. In this way we obtain solutions for all membersof the family of KP lattices as classified in [1], cf. also [ ? ]. In the trigonometric case ofthe discrete-time RS system the corresponding solutions are of soliton type, cf. [17]. Theconnection between the (continuous-time) RS system and solitons has also been discussed in[21]. emark: We note that in the non-relativistic limit λ → p → e − λp CM and q → e − λq CM ofthe lattice parameters, we have as a consequence that p − q → − λ ( p CM − q CM ) + O ( λ ) .Hence, the non-relativistic limit λ → In this paper we have studied the Lagrangian structure for the Ruijsenaars-Schneider system,and shown that similarly to the Calogero-Moser system, which was treated in [28], it possessesa Lagrangian 1-form structure, both on the discrete-time level as well as in the continuous-time case. Thus, this is the second example of a system of ODEs which exhibits a Lagrangianmulti-form structure in the sense of [9] but in a lower-dimensional situation. The presentexample is important, because in contrast to the CM case where the Lagrange structure isclosely related to the Lax representation (and hence inherit the closure relation from thezero-curvature condition), here the relation between the Lax matrices and the Lagrangiansis less obvious, and the validity of the closure relation has to be verified by a separatecalculation and is therefore more surprising. Thus, we believe that these results seem toconfirm once again that these Lagrangian form structures are fundamental and ubiquitousamong integrable systems.It is well known that the classical RS system is Liouville-integrable in the continuous-time case, [23, 24] and formally so in the discrete-time case [17, 18] as well. With regardto the continuous-time model, the Lagrangian 1-form structure had to be established in arather indirect way, namely by performing systematic limits on Lagrangians of the discrete-time system. We have already pointed out that establishing these Lagrange structures byLegendre transformation from the known Hamiltonians of the model is complicated, becauseit is not a priori known how these Hamiltonian flows are embedded in a coherent structure,such that we get acceptable Lagrangian components of the 1-form. In this sense the discrete-time model can be viewed as a generating object for such Lagrangians for the continuous-timemodel. On the basis of those results, which in fact establish the proper form of the “kineticterms” of the continuous hierarchy of Lagrangians it is possible to show that the Lagrangian1-form structure precisely selects the general form of the integrable “potentials” when a prioriarbitrary forms for those potentials are fed into the determining equations, cf. [20].In conclusion, let us state that in our view the importance of this new Lagrangian formstructure resides in the understanding that it manifests the multidimensional consistency,in the sense of the papers [14, 3], at the level of the variational principle: It provides ananswer to the problem of how to find a single Lagrangian framework for a situation wherewe have a multitude of compatible equations imposed on one and the same (possibly vector-valued) function of many independent variables. In the case of ODEs, as is the case dealtwith in the present paper, the structure is that of a Lagrangian 1-form describing systems ofcommuting flows in many time-variables (as many as the number of degrees of freedom of thesystem). It is obvious that for this structure to hold, the relevant Lagrangian components ofthe 1-form should have very specific forms, in order for the closure relation to hold subjectto the equations of the motion. In fact, such admissable Lagrangians can be consideredthemselves to be solutions of the system of equations arising from the variational principle.In the continuous case the constitutive relations arising from this new variational principle,which involves variations not only with respect to the dependent variables but also withrespect to the underlying geometry, were first given in [28, 30]. In a recent paper, [25],Yu. Suris from a slightly different point of view formulated the corresponding Legendre Rather than considering the variations with respect to the geometry [25] inspired by our results, consideredLagrangian 1-forms on arbitrary curves. We argue, however, that posing a least-action principle with respect to ransform. That theories which exhibit structures as exemplified in the present paper, butalso in higher dimensions, always correspond to integrable systems in the sense of other well-known integrability features (such as the applicability of the inverse scattering, existence ofLax pairs and higher symmetries, etc.) is challenging question that we hope to answer infuture work. A The construction of the exact solution
In this Appendix we review the construction of the exact solution for the RS system. Thebasic relations following from the Lax pair (2.1) together with the definitions (2.2) lead to µ M + f XM − M X = e hh T , (A.1a) λ L + XL − L X = hh T , (A.1b)where X = P Ni =1 x i E ii is the diagonal matrix of the particle positions. We have adoptedhere the freedom of making the model slightly more general by introducing in addition tothe (relativistic) parameter λ a new parameter µ replacing λ in the M matrix. On theother hand, from the Lax equation (2.5) and (2.6), we obtain the relations e L M = M L , (A.2a) e L e h − M h = − p e h , (A.2b) h T L − e h T M = − ph T , (A.2c)where (A.2b) and (A.2c) are equivalent to the relations (2.8). We now factorize the Laxmatrices as follows: L = U Λ U − , and M = e U U − , (A.3)where U is an invertible N × N matrix, and where the matrix Λ is constant: e Λ = Λ , asa consequence of (A.2a). (Obviously, if L is diagonalizable Λ is just its diagonal matrix ofeigenvalues). Next, introducing Y = U − XU , r = U − · h , s T = h · U , (A.4)we obtain from (A.2) and (A.3),( p I + Λ ) · e r = r , s T · (p I + Λ ) = e s T , (A.5)where I is the unit matrix, as well as from (A.1a) and (A.1b), we have µ + e Y − Y = e rs T , (A.6) λ Λ + [ Y , Λ ] = rs T . (A.7)Eliminating the dyadic rs T from (A.6) by making use of (A.5), we find the linear equation e Y = ( p I + Λ ) − Y ( p I + Λ ) − pµp I + Λ + ( λ − µ ) Λ p I + Λ , (A.8)which can be immediately solved to give Y ( n, m ) = ( p I + Λ ) − n Y (0 , m )( p I + Λ ) n − npµp I + Λ + n ( λ − µ ) Λ p I + Λ , (A.9)subject to the constraint on the initial value matrix[ Y (0 , m ) , Λ ] = − λ Λ + rank 1 . (A.10) both dependent as well as independent variables is a conceptually important step, forming a new paradigm invariational calculus, cf. also [6], and constitutes potentially a novel principle of fundamental physics. e now consider the matrix N in (2.17) which can be rewritten in the form η N + c XN − N X = b hh T , (A.11)where η is another relativistic parameter associating to the temporal Lax matrix N . Iteratingthe same process as we did before, we find b Y = ( q I + Λ ) − Y ( q I + Λ ) − qηq I + Λ + ( λ − η ) Λ q I + Λ . (A.12)Combining (A.8) and (A.12), we can solve for Y ( n, m ) = ( p I + Λ ) − n ( q I + Λ ) − m Y (0 , q I + Λ ) m ( p I + Λ ) n − npµp I + Λ − mqηq I + Λ + n ( λ − µ ) Λ p I + Λ + m ( λ − η ) Λ q I + Λ . (A.13)If we take λ = µ = η we recover the relation (2.25a).Conversely, we can start from a given N × N diagonal matrix Λ with distinct entries,and an initial value matrix Y (0 ,
0) subject to the condition that[ Y (0 , , Λ ] = − λ Λ + rank 1 , (A.14)where [ , ] represents the matrix commutator bracket. Let U − (0 ,
0) be the matrix thatdiagonalized Y (0 , Y (0 ,
0) = U − (0 , X (0 , U (0 , , X (0 ,
0) = diag( x (0 , , . . . , x N (0 , . (A.15)If the eigenvalues of Y (0 ,
0) are distinct (which we can take as an assumption on the initialcondition) then U − (0 ,
0) is determined up to multiplication from the right by a diagonalmatrix times a permutation matrix of the columns. (Fixing an ordering of the eigenvalues x i (0 , U − (0 ,
0) unique only up to multiplication by a diagonal matrix from the right).We can fix U − (0 ,
0) up to an overall multiplicative factor by demanding that[ Y (0 , , Λ ] = − λ Λ + r (0 , s T (0 , . (A.16)Next, we consider the matrix function given by Y ( n, m ) = ( p I + Λ ) − n ( q I + Λ ) − m Y (0 , q I + Λ ) m ( p I + Λ ) n − npµp I + Λ − mqηq I + Λ + n ( λ − µ ) Λ p I + Λ + m ( λ − η ) Λ q I + Λ . (A.17)Let U ( n, m ) be the matrix diagonalizing Y ( n, m ) by an appropriate choice of an overallfactor (as a function of n and m ) this matrix can be fixed such that it obeys: r ( n, m ) = ( p I + Λ ) − n ( q I + Λ ) − m r (0 , , and s T ( n, m ) = s T (0 , p I + Λ )( q I + Λ ) , (A.18)and [ Y ( n, m ) , Λ ] = − λ Λ + r ( n, m ) s T ( n, m ) . (A.19)From the expression (A.17) we can derive the relations( p I + Λ ) e Y − Y ( p I + Λ ) = − pµ + ( λ − µ ) Λ , (A.20a)( q I + Λ ) b Y − Y ( q I + Λ ) = − qη + ( λ − η ) Λ , (A.20b)with the usual notation for the shifts in n and m over one unit. Together with the relation(A.19) this subsequently yields: e Y − Y = − µ + e rs T , b Y − Y = − η + b rs T . (A.21) eversing these relations by rewriting them in terms of X ( n, m ) = U ( n, m ) Y ( n, m ) U − ( n, m ) (A.22)and now defining the Lax matrix by L := U Λ U − , (A.23)together with M := e U U − , N := b U U − , (A.24)we recover the relations: [ X , L ] = − λ L + hh T , (A.25)and f X M − M X = − µ M + e hh T , (A.26a) c X N − N X = − η N + b hh T , (A.26b)which determine the matrices M and N as functions of the x i ( n, m ) as well as the off-diagonal parts of the matrices L and K .Furthermore, from (A.20) we obtain e L f XM − M XL = ( − p e hh T + ( λ − µ ) e L M , − p e hh T + ( λ − µ ) M L , (A.27a) c K c XN − N XK = ( − q b hh T + ( λ − η ) b L N − q b hh T + ( λ − η ) N L , (A.27b)which, when combined with the relations of (A.25), yield (cid:16)e LM − M L (cid:17) X + (cid:16) e L e h − M h (cid:17) h T = − p e hh T , (A.28a) (cid:16)c KN − N K (cid:17) X + (cid:16)c K b h − N h (cid:17) h T = − q b hh T . (A.28b)On the other hand, using the relations (A.26) we also obtain f X (cid:16) e LM − M L (cid:17) + e h (cid:0) h L − h T M (cid:1) = − p e hh T , (A.29a) c X (cid:16)c KN − N K (cid:17) + b h (cid:0) h K − h T N (cid:1) = − q b hh T . (A.29b)From the relations (A.28) and (A.29) it follows that the Lax equations hold and their formis determined up to the diagonal part of the matrices L and K . B Examples
In this Appendix, we will investigate how to derive the discrete Euler-Lagrange equationfrom the variational principle. For general discrete curves it is cumbersome to implementthe variational principle because of the notation it would require. We will, however, demon-strate how the principle works for a few simple cases: (a) the curve shown in Fig. (4), (b) the curve shown in Fig. (5).
Case(a) : The curve shown in Fig. (4): We now introduce a new variable N = n + m together with the change of notation x ( n, m ) x ( N , m ) , e x := x ( N + 1 , m ) and b x := x ( N , m + 1) , m (a)Γ ( n , m ) n m N m (b) Γ ′ ( N , m ) N − N m Figure 4: The effect of changing variables on the discrete curve I. and so we work with the curve given in Fig. (4b). The action evaluated on this curve canbe written in the form S [ x ; Γ ′ ] = m − X m = m − L ( N ) ( x ( N − , m ) , x ( N , m )) + m − X m = m L ( m ) ( x ( N − , m ) , x ( N , m + 1)) , (B.1)where L ( N ) ( x , y ) = N X i,j =1 ( f ( y i − x j ) − f ( y i − x j − λ )) − N X i,j =1 j = i f ( y i − y j + λ ) − N X i,j =1 j = i f ( x i − x j + λ ) − ln | p | N X i =1 ( y i − x i ) , (B.2) L ( m ) ( x , y ) = N X i,j =1 ( f ( x i − y j ) − f ( x i − y j − λ )) − N X i,j =1 j = i f ( x i − x j + λ ) − N X i,j =1 j = i f ( y i − y j + λ ) − ln | q | N X i =1 ( x i − y i ) , (B.3)The minus sign in (B.1) indicates the reverse direction of the Lagrangian L ( N ) along thehorizontal links. Performing the variation x x + δ x , we have δS = 0 = m − X m = m (cid:18) − ∂ L ( N ) ( x ( N − , m ) , x ( N , m )) ∂ x ( N , m ) δ x ( N , m ) − ∂ L ( N ) ( x ( N − , m ) , x ( N , m )) ∂ x ( N − , m ) δ x ( N − , m ) (cid:19) + m − X m = m (cid:18) ∂ L ( m ) ( x ( N − , m ) , x ( N , m + 1) ∂ x ( N , m + 1) δ x ( N , m + 1)+ ∂ L ( m ) ( x ( N − , m ) , x ( N , m + 1) ∂ x ( N − , m ) δ x ( N − , m ) (cid:19) . (B.4) e now obtain the Euler-Lagrange equations − ∂ L ( N ) ( x ( N − , m ) , x ( N , m )) ∂ x ( N , m ) + ∂ L ( m ) ( x ( N − , m − , x ( N , m ) ∂ x ( N , m ) = 0 , (B.5a) − ∂ L ( N ) ( x ( N − , m ) , x ( N , m )) ∂ x ( N − , m ) + ∂ L ( m ) ( x ( N − , m ) , x ( N , m + 1) ∂ x ( N − , m ) = 0 , (B.5b)which produceln (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) = N X j =1 (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) x i ( N , m ) − x j ( N − , m ) x i ( N , m ) − x j ( N − , m ) + λ (cid:12)(cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12)(cid:12) x i ( N , m ) − x j ( N − , m −
1) + λ x i ( N , m ) − x j ( N − , m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , ln (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) = N X j =1 (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) x i ( N − , m ) − x j ( N , m + 1) x i ( N − , m ) − x j ( N , m + 1) − λ (cid:12)(cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12)(cid:12) x i ( N − , m ) − x j ( N , m ) − λ x i ( N − , m ) − x j ( N , m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , which are equivalent to (2.23a) and (2.23b), respectively. nm (a)Γ( n , m ) n m N ′ m (b)Γ ′ ( N ′ , m ) N ′ = N ′ + 1 m Figure 5: The effect of changing variables on the discrete curve II.
Case(b) : The curve shown in Fig. (5): Introducing the variable N ′ = n − m , the corre-sponding curve is given in Fig. (5b). The action evaluated on the curve Γ ′ reads S [ x ; Γ ′ ] = m − X m = m L ( N ′ ) ( x ( N ′ , m ) , x ( N ′ + 1 , m ))+ m − X m = m L ( m ) ( x ( N ′ + 1 , m ) , x ( N ′ , m + 1)) , (B.7) here L ( N ′ ) ( x , y ) = N X i,j =1 ( f ( x i − y j ) − f ( x i − y j − λ )) − N X i,j =1 j = i f ( y i − y j + λ ) − N X i,j =1 j = i f ( x i − x j + λ ) − ln | p | N X i =1 ( x i − y i ) , (B.8) L ( m ) ( x , y ) = N X i,j =1 ( f ( x i − y j ) − f ( x i − y j − λ )) − N X i,j =1 j = i f ( x i − x j + λ ) − N X i,j =1 j = i f ( y i − y j + λ ) − ln | q | N X i =1 ( x i − y i ) , (B.9)Performing the variation x x + δ x , we have δS = 0 = m − X m = m (cid:18) ∂ L ( N ′ ) ( x ( N ′ , m ) , x ( N ′ + 1 , m ) ∂ x ( N ′ , m ) δ x ( N ′ , m )+ ∂ L ( N ′ ) ( x ( N ′ , m ) , x ( N ′ + 1 , m ) ∂ x ( N ′ + 1 , m ) δ x ( N ′ + 1 , m ) (cid:19) + m − X m = m (cid:18) ∂ L ( m ) ( x ( N ′ + 1 , m ) , x ( N ′ , m + 1) ∂ x ( N ′ , m + 1) δ x ( N ′ , m + 1)+ ∂ L ( m ) ( x ( N ′ + 1 , m ) , x ( N ′ , m + 1) ∂ x ( N ′ + 1 , m ) δ x ( N ′ + 1 , m ) (cid:19) . (B.10)We now obtain the Euler-Lagrange equations ∂ L ( N ′ ) ( x ( N ′ , m ) , x ( N ′ + 1 , m ) ∂ x ( N ′ , m ) + ∂ L ( m ) ( x ( N ′ + 1 , m − , x ( N ′ , m ) ∂ x ( N ′ , m + 1) = 0 , (B.11a) ∂ L ( N ′ ) ( x ( N ′ , m ) , x ( N ′ + 1 , m ) ∂ x ( N ′ + 1 , m ) + ∂ L ( m ) ( x ( N ′ + 1 , m ) , x ( N ′ , m + 1) ∂ x ( N ′ + 1 , m ) = 0 , (B.11b)which produceln (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) = N X j =1 (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) x i ( N ′ , m ) − x j ( N ′ − , m ) x i ( N ′ , m ) − x j ( N ′ − , m ) + λ (cid:12)(cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12)(cid:12) x i ( N ′ , m ) − x j ( N ′ − , m −
1) + λ x i ( N ′ , m ) − x j ( N ′ − , m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , ln (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) = N X j =1 (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) x i ( N ′ − , m ) − x j ( N ′ , m + 1) x i ( N ′ − , m ) − x j ( N ′ , m + 1) − λ (cid:12)(cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12)(cid:12) x i ( N ′ − , m ) − x j ( N ′ , m ) − λ x i ( N ′ − , m ) − x j ( N ′ , m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , which are equivalent to (2.23a) and (2.23b), respectively.By working with the specific type of curves given in Fig. (4) and Fig. (5), we canperform the variational principle with either the new variables ( N , m ) or ( N ′ , m ). We obtainthe Euler Lagrange equations corresponding to each link of the discrete curve and we obtainconstraint equations (2.23a) and (2.23b) describing the dynamic of the system from onedirection to another direction of the discrete-time (while the equations of motion (2.12) and(2.20) represent the dynamic of the system on one discrete-time direction). cknowledgements S. Yoo-Kong is supported by King Mongkut’s University of Technology Thon-buri Research Grant for a new acdemic staff 2012. F. Nijhoff is supported bya Royal Society/Leverhulme Trust Senior Research Fellowship and would like tothanks Department of Physics, Faculty of Scicence, King Mongkut’s Universityof Technology Thonburi for hospitality as the final state of the paper has beendone. We are grateful to S. Ruijsenaars for useful comments.
References [1] Adler V E, Bobenko A I, Suris Y B 2011,
Classification of integrable discrete equationsof octahedron type , Int Math Res Notices, , pp.1-68.[2] Bruschi M and Calogero F 1987, The Lax representation for an integrable class of rela-tivistic dynamical systems , Commum. Math. Phys. , pp.481-492.[3] Bobenko A I, Suris Y B 2008,
Discrete Differential Geometry: Integrable Structure , Grad-uate Studies in Mathematics, Vol. 98, AMS, Providence, xxiv+404.[4] Braden H W and Sasaki R 1997,
The Ruijenaars-Schneider Model , Progress of TheoreticalPhysics, Vol. 97, pp.1003-1017.[5] Dorfman I Ya, Nijhoff F W 1991,
On a (2+1)-dimensional version of the Krichever-Novikov equation , Phys. Lett. A, , pp.107-112.[6] Gelfand I M and Fomin S V,
Calculus of Variations , (Dover Publications, Mineola, NewYork, 2000) [translated from Russian in 1963].[7] Hirota R 1981,
Discrete Anologue of a Generalised Toda Equation , J. Phys. Soc. Japan, , pp.3785-3791.[8] Lobb S B, Nijhoff F W, and Quispel G R W 2009, Lagrangian multiform structure forthe lattice KP system , J. Phys. A, , 472002.[9] Lobb S B and Nijhoff F W 2009, Lagrangian multiforms and multidimensional consis-tency , J. Phys. A: Math. Theor. , 454013.[10] Nijhoff F W, Atkinson J and Hietarinta J 2009, Soliton Solutions for ABS LatticeEquations: I Cauchy Matrix Approach , J. Phys. A: Special Issue dedicated to the DarbouxDays, , 404005.[11] Nijhoff F W and Atkinson J 2010, Elliptic N-soliton solutions of ABS lattice equations ,Int. Math. Res. Notices, , pp.3837-3895.[12] Nijhoff F W, Capel H W, and Wiersma G L 1985, Integrable lattice systems in twoand three dimensions , In: Geometric Aspects of the Einstein Equations and IntegrableSystems, Ed. R. Martini, Lecture Notes in Physics, Berlin/New York, Springer Verlag,pp.263-302.[13] Nijhoff F W, Capel H W, Wiersma G L, and Quispel G R W 1984,
B¨acklund transfor-mations and three-dimensional lattice equations , Phys. Lett. A, , pp.267-272.[14] Nijhoff F W and Walker A, 2001
The discrete and continuous Painlev ´ e VI hierarchyand the Garnier systems , Glasgow Math. J. , pp.109-123.[15] Nijhoff F W and Pang G D 1994,
A time-discretized version of the Calogero-Mosermodel , Physics Letters , pp.101-107[16] Nijhoff F W, Capel H W and Wiersma G L 1985,
Integrable Lattice Systems in Twoand Three Dimensions , Ed. R. Martini, in: Geometric Aspects of the Einstein Equationsand Integrable Systems, Lecture Notes in Physics, pp.263-302, Berlin/New York, SpringerVerlag.[17] Nijhoff F W, Ragnisco O and Kuznetsov V 1996,
Integrable Time-Discretisation of theRuijsenaars-Schneider Model , Commun. Math. Phys. , pp.681-700.
18] Nijhoff F W, Kuznetsov V, Sklyanin E.K and Ragnisco O 1996,
Dynamical r-matrix forthe elliptic Ruijsenaars-Scneider system , J. Phys. A: Math. Gen. , L333-L340.[19] Quispel G R W, Nijhoff F W, Capel H W and Linden van der J 1984, Linear integralequations and nonlinear difference-difference equations , Physica A, , pp.344-380.[20] Nijhoff F W and Yoo-Kong S,
Lagrangian 1-form structure implies integrability , inpreparation.[21] Ruijsenaars S N M 1997,
Integrable particle systems vs solutions to the KP and 2D Todaequations , Ann. Phys. , pp.226-301.[22] Ruijsenaars S N M 1988,
Action-angle maps and scattering theory for some finite-dimensional integrable systems. I. The pure soliton case , Commun. Math. Phys. ,pp.127-165.[23] Ruijsenaars S N M and Schneider H 1986,
A New Class of Integrable Systems and Itsrelation to Solitons , Ann. Phys, , pp.370-405.[24] Ruijsenaars S N M 1987,
Complete integrability of Relativistic Calogero-Moser systemsand Elliptic Function Identities , Commun. Math. Phys, , pp.191-213.[25] Suris Y B 2012
Variational formulation of commuting Hamiltonian flows: multi-timeLagrangian 1-forms , arXiv:1212.3314.[26] Wilson G 1998,
Collisions of Calogero-Moser particles and an adelic Grassmannian ,Invent. Math. , pp.1-41.[27] Xenitidis P, Nijhoff F and Lobb S 2011,
On the Lagrangian formulation of multidimen-sionally consistent systems , Proc. of the Royal Society A, , pp.3295-3317.[28] Yoo-Kong S, Lobb S and Nijhoff F 2011,
Discrete-time Calogero-Moser system andLagrangian 1-form structure , J. Phys. A: Math. Theor, , 365203.[29] Yoo-Kong S and Nijhoff F 2011, Elliptic ( N, N ′ ) -Soliton Solutions of the lattice KPsystem , arXiv:1111.5366 nlin.SI , submitted to Journal of Mathematical Physics.[30] Yoo-Kong S 2011 Calogero-Moser type systems, associated KP systems, and Lagrangianstructure , Thesis, University of Leeds., Thesis, University of Leeds.