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Discrete-time Calogero-Moser system and Lagrangian 1-form structure

We study the Lagrange formalism of the (rational) Calogero-Moser (CM) system, both in discrete time as well as in continuous time, as a first example of a Lagrange 1-form structure in the sense of the recent paper [19]. The discrete-time model of the CM system was established some time ago arising as a pole-reduction of a semi-discrete version of the KP equation, and was shown to lead to an exactly integrable correspondence (multivalued map). In this paper we present the full KP solution based on the commutativity of the discrete-time flows in the two discrete KP variables. The compatibility of the corresponding Lax matrices is shown to lead directly to the relevant closure relation on the level of the Lagrangians. Performing successive continuum limits on both the level of the KP equation as well as of the CM system, we establish the proper Lagrange 1-form structure for the continuum case of the CM model. We use the example of the three-particle case to elucidate the implementation of the novel least-action principle, which was presented in [19], for the simpler case of Lagrange 1-forms.


Introduction
The Calogero-Moser (CM) model [1,2] is an integrable one-dimensional many-particle system with long-range interactions, originally given as a continuous system with pairwise inverse square potential, but which also has been generalized to the trigonometric [3] and the elliptic case [4], cf. also [6], and later to a relativistic model (the Ruijsenaars-Schneider model [7,8]). The model has been extensively studied in both the classical and quantum cases [9], and is integrable on both levels. From a physical perspective the emphasis is on the dynamics of real particles subject to pairwise repulsive potentials. However, in recent years the study of CM models has attained a wider mathematical significance, from the seminal paper [10] considering the symplectic geometry associated with the model, to more recently its role in the representation theory of Lie and quantum algebras [11]. In this context, the investigation of CM systems with attractive potential, allowing particles to collide, has led to interesting new perspectives from the point of view of algebraic geometry, cf. e.g. [15].
An integrable discrete-time version of the (rational) CM model was presented in [18], where it was obtained from a semi-discrete Kadomtsev-Petviashvili (KP) type equation with two discrete and one continuous independent variable. The construction, which we summarize in Section 2, follows closely Krichever's pole-reduction of the continuous KP equation leading to a connection with the CM system [13]. In the discrete case, this leads to a rather complicated system of ordinary difference equations (O∆Es) which constitutes an integrable Lagrangian correspondence, i.e., a multivalued symplectic map [16]. The construction provides a Lax pair, whilst the classical R matrix is esentially the same as for the continuous case (as given by [14]), from which the Liouville integrability (in the sense of [16]) follows. The structure of the Lax matrices was used to obtain the solution of the initial-value problem in an adaptation of the standard way, cf. [9,10]. It was shown that a "naive" continuum limit of the discrete equations reduces them to the equations of motion of the continuous CM system in the case of an attractive, rather than repulsive, potential. In spite of the fact that this makes the discrete-time CM model perhaps less relevant from a physical perspective, it is certainly of interest from the mathematical point of view in that it provides us an insight into the basic structures underlying discrete integrability. Thus, in the same vein, we will use the model here to understand better the novel insights from our recent paper [19] of the Lagrangian multi-form structure underlying systems which are integrable in the sense of multidimensional consistency.
Our focus in the present paper on Lagrangian structures has many motivations, most notably the possibility of quantizing the discrete-time model via a path integral formalism. Furthermore, whilst many integrable discrete systems admit a Lagrangian description, in the discrete-time case the Hamiltonian no longer provides such a natural framework as it does in the continuous-time case. Recently, two of the authors observed a new fundamental property of Lagrangians for integrable (in the sense of multidimensionally consistent) systems which reflects the multidimensional consistency of the lattice systems [19]. This is given by a closure relation which holds for Lagrangians when the system is embedded in a higherdimensional lattice (meaning that the same equations hold in a multitude of sublattices of the multidimensional lattice together with their corresponding Lagrangians), and leads to the interpretation of the Lagrangians as closed forms on the extended lattice. This observation, based on the elaboration of various explicit examples, prompted the proposal of a new variational principle for integrable systems which involves explicitly the geometries in the space of independent variables. In [19] we focused on establishing and interpreting the closure relation of Lagrangian 2-forms for cases of integrable discrete equations with two independent variables, as well as of their continuous analogues, for lattice equations in the ABS list [20]. Subsequently, the closure property was also established for Lagrangians describing multicomponent lattice systems in the so-called lattice Gel'fand-Dikii hierarchy, [21], and for Lagrangian 3-forms in the case of the discrete bilinear KP equation in [22], whilst in [25] it was shown that all equations in the ABS list possessed this property (cf. also [31] for a "universal" Lagrange multi-form description for quadrilateral affine linear equations and their continuous counterparts). An important case which has not yet been covered is that of Lagrangian 1-forms, which applies to the case of integrable ODEs. Natural candidates for exhibiting such a structure are integrable finite-dimensional many-body systems, such as the Calogero-Moser systems.
Thus, in this paper we set out to establish the Lagrange 1-form structure of the discretetime CM system in the rational case. Whilst some results demonstrating that the considerations extend to the hyperbolic/trigonometric and elliptic case as well are given in Appendix B, we prefer to work with this simplest case, which is most transparent, and for which we have exact solutions at our disposal. Starting with the discrete-time case, we will establish the Lagrange structure also for the continuous case by continuum limits, rather than using the (known) Hamiltonians for the higher CM flows. This is necessary because, as it turns out, the proper Lagrangians cannot be obtained by performing Legendre transformations separately on each of the Hamiltonians. This leads, beyond the second-order flow, to Lagrangians containing rather complicated algebraic expressions of the higher-time-derivatives. Instead, we obtain a hierarchy of mixed polynomial Lagrangians in terms of these higherorder derivatives from the discrete-time case by systematic expansions (indicating en passant that Legendre transformations should probably not be implemented order-by-order in the flows but rather altogether in one stroke). In all these considerations, the link with the underlying KP system is instrumental in deciding how to perform the higher-order continuum limits by systematic expansions and in terms of the proper parameters.
The organization of the present paper is as follows. In Section 2, we present the full solution of the semi-discrete KP equation, which requires commuting flows in different discrete-time directions, and thus generalizes the one-dimensional discrete-time CM flow in [18]. In Section 3, we establish the closure relation of the discrete-time Lagrangian 1-form of the discrete-time CM model through the compatibility of the time-part Lax matrices, which are of Cauchy type. In Section 4, we investigate the semi-continuum limit, or skew limit, of the semi-discrete KP equation. This limit leads to an equation which we refer to as the semicontinuous KP equation (defined in terms of two continuous and one discrete independent variable), from which we obtain by pole-expansion the corresponding differential-difference system which we coin the semi-continuous CM system. In Section 5, the latter, which acts as a generating system for the CM hierarchy, is a Lagrangian system in its own right obeying a closure relation of differential-difference type with the original discrete-time Lagrangian of the CM system. In Section 6, the full continuum limit is performed, and we thus recover the usual fully continuous CM hierarchy in mixed form, together with continuous Lagrangians exhibiting a closed 1-form structure in Section 7. We concentrate in Section 8 on the 3particle case, governed by the CM flow and its first higher-order counterpart, in order to implement the geometric variational principle which was formulated in [19]. Thus, we show how the principle leads in this case to the derivation both of the closure relation as well as the pertinent Euler-Lagrange (EL) equations. We end the paper with some discussion of open problems and possible extensions.

Pole reduction of the semi-discrete KP equation
In this section we review the connection between the semi-discrete KP equation, following on from [12,17], and the discrete-time CM model as in [18,24]. This gives us an occasion to introduce appropriate notation which we will use throughout the paper. Furthermore, it helps us to identify the commuting flows needed to establish the corresponding nontrivial solution of the semi-discrete KP equation.
The semi-discrete KP equation that was used in [18] is where p and q are two lattice parameters, u is the classical field, and u represents the discrete shift of u corresponding to a transition in the "time" direction, while u represents the discrete shift of u corresponding to a translation in the "spatial" direction. There are two discrete variables n, m to which the lattice parameters p, q and shifts u, u correspond respectively, and one continuous variable ξ.
We observe that eq. (2.1) is a consequence of the compatibility condition of a scalar Lax pair which reads Taking u to have the form we find that the corresponding solution of the linear Lax equations is Introducing the vectors b = (b 1 , b 2 , ..., b N ) T , and e = (1, 1, ..., 1) T , and the Lax matrices where E ij are the generators of GL N , i.e., matrices with entries (E ij ) kl = δ ik δ jl , we can now rewrite equations (2.7) and (2.8) as and since this must holds for arbitrary value of k we deduce the discrete-time equations of motion of an N -particle system: in which the under-shift x i is short-hand for a shift in the variable n over one unit in the nagative direction, i.e., x i (n, m) = x i (n − 1, m). Furthermore, if (2.10) holds, both the following matrix relations are satisfied: and consequently the inhomogeneous Lax system (2.9) leads to an isospectral problem in terms of the Lax matrices L and M . As was shown in [18] eq. (2.10) is an integrable symplectic correspondence, in the sense of Liouville, [16], whose exact solutions x i (n, m), which denote the position of the particles x i at the nth time step (with fixed m), are obtained by a linearization procedure implementing the Lax pair (for details, see Appendix A). The resulting solution of the initial value problem, imposing initial values x i (0, m) and x i (1, m) = x i (0, m), can be obtained by solving the secular problem given by the characteristic equation of the matrix subject to the constraint on the initial value matrix [Y (0, m) , Λ L ] = I + rank 1 , (in which I denotes the N × N unit matrix). The eigenvalues of Y (n, m) given by (2.12) are the particle positions x i (n, m), and they are determined up to permutations of the particles, which accounts for the multivaluedness of the system of O∆Es. Here the Λ L denotes the diagonal matrix of eigenvalues of the Lax matrix L and the matrix Y (n, m) is related to the the diagonal matrix X(n, m) = N i=1 x i (n, m)E ii of the particle positions by a similarity transformation. It follows from (2.11) that the matrix Λ L is invariant under time shifts, i.e., We note that (2.12) resolves only the n-dependence of the solution of the KP equation, and to provide the full solution we need also to consider the dependence on m. Obviously, the latter should come from the consideration of (2.2b), which can be treated in much the same way, and lead to the following relations: where K and N take exactly the same form as L and M , with just the " " shift replaced by the " " shift. The compatibility condition leads to 14) and the equations of the motion can be directly obtained from (2.10) by replacing the " " shift by the " " shift, where the matrix Λ K is invariant under time shift, so that Λ K = Λ K .
The n-and m-parts of the solutions of course can only be combined if the corresponding flows are compatible. In particular this requires that the Lax matrices L and K commute, which in turn suggests that these matrices can be simultaneously diagonalized. Furthermore, it requires the compatibility of the flows in the two lattice directions, which leads to the relations Eq. (2.17) leads to the relation which yields in addition which is a consequence of (2.19) and (2.20), guarantees that the zero-curvature condition M N = N M holds, which in turn implies that ( M − N )e = (p − q)e . Equations (2.19) and (2.20), which we will refer to as the constraint equations, guarantee that the discrete flows in the variables n and m commute, and hence that the corresponding linear equations can be simultaneously solved. In fact, the downward shift of (2.20) yields which is also a consequence of (2.10) and (2.15), and which expresses this compatibility with the sets of O∆Es. It is, thus, the consistency of the equations (2.12) and (2.16), that lead us to the complete solution of the semi-discrete KP equation. In particular, from eq. (2.19), it follows that the diagonal parts of the matrices L and K differ by (p − q)I, and hence we can identify Λ L = pI + Λ , Λ K = qI + Λ , with common diagonal matrix Λ. Using this identification we can now combine the solution given by (2.12) for the CM system of O∆Es in the n-direction and the solution (2.16) in the m-direction into a simultaneous solution which satisfies also the commutativity constraints (2.19) and (2.20), leading to the following statement: Proposition: The eigenvalues x 1 (n, m), . . . , x N (n, m) of the N × N matrix The proof is presented in Appendix A. In order to make a connection with an initial value problem, we mention that the initial value matrix Y (0, 0) can be obtained from the diagonal matrix of initial values X(0, 0) by a similarity transformation with a matrix U (0, 0) which is an invertible matrix diagonalizing the initial Lax matrices L(0, 0) and K(0, 0). To find the latter, we need the initial values x i (0, 0), x i (1, 0) and x i (0, 1), (i = 1, . . . , N ). We note that the secular problem can, hence, be reformulated as one for the following matrix and hence the solution is provided by the roots of the characteristic equation: Returning now to the solution of the semi-discrete KP equation (2.1), inserting the roots obtained from (2.25) into the expression (2.3) we obtain the required pole solutions. Furthermore, for the solution of the corresponding eigenfunction φ(n, m) as given in (2.4) for the eigenvalue problem, the vector b(n, m) must be determined. This is obtained, for arbitrary value of the spectral parameter k, from (2.9a), or equivalently (2.13a), and is given by Plugging all the results into (2.4), we obtain the explicit form of the eigenfunction φ(n, m) solving eqs. (2.2a) and (2.2b). Furthermore, eliminating the function u from the latter, we obtain the following nonlinear differential-difference equation for φ itself: whose pole-solutions are given by the construction outlined above.

The Lagrangian 1-form and its closure relation
In this section, in order to derive the Lagrangian 1-form structure, we first establish a direct connection between the Lagrangian of the discrete-time CM model, as was presented in [18], and the temporal part of the Lax representation. We will show that this connection, through the compatibility of the matrices M and N of the previous section, will immediately lead to a closure relation for the corresponding Lagrangians.
Let x = (x 1 , x 2 , . . . , x N ). Equation (2.10) can be computed from the variation of a discrete action given in [18] as where L (n) (x, x) is the Lagrangian corresponding to the " " direction, and the sum over n represents the sum over all discrete-time " " iterates. Similarly, the action corresponding to the " " direction takes the form where L (m) (x, x) is the Lagrangian corresponding to the " " direction, and the sum over m represents the sum over all discrete-time " " iterates.
The matrices M and N are Cauchy-type matrices, and so the exact forms of the determinant can be written as: and we also have that Remarkably, the action (3.1) can be obtained by considering the infinite chain product of the matrix M in the following way The same procedure can performed on the matrix N in order to get the action (3.2): We note that the connection between the time-part Lax matrix and the Lagrangian also exists in the cases of the elliptic and trigonometric discrete-time CM systems (see Appendix B).
We now consider the compatibility relation M N = N M , which we recall is satisfied if both (2.19) and (2.20) hold, and we rewrite this as where we define Ξ = N i=1 x i to be the centre of mass and The last terms in (3.10a) and (3.10b) are the total derivative terms involving centre of mass motion which can be separated from the relative motion. The extra equation comes directly from (3.9) which holds on the solutions.
We would now like to interpret (3.9) as the closure relation for a Lagrangian 1-form. Let e i represent the unit vector in the lattice direction labeled by i and let any position in the lattice be identified by the vector n, so that an elementary shift in the lattice can be created by the operation n → n + e i . Since the Lagrangian depends on x and its 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 discrete 1-form L i (n) (3.12) Proposition: As functions on the two-dimensional lattice, the discrete-time Lagrangians given in eq. (3.10) satisfy the following relation Equation (3.9) represents the closure relation of the Lagrangian 1-form. The point is that the closure relation (3.9) is derived from the Lax equation (the compatibility of the matrices M and N ), together with the property of determinants of Cauchy matrices. In contrast, in those cases of Lagrangian 2-forms and 3-forms in [19,21,22], the equation of the motion must be invoked in order to show that the closure relation holds. Also in those cases the Euler-Lagrange equation stemming from the variational principle was not exactly the equation of motion needed to verify the closure relation; rather it was a discrete derivative, or a sum of copies, of the original equation of motion.
Choosing a discrete curve Γ consisting of connected elements Γ i , we can define an action on the curve by summing up the contributions L i from each of the oriented links Γ i in the curve, to get S(x(n); Γ) = n∈Γ L i (x(n), x(n + e i )). (3.14) The closure relation (3.9) is actually equivalent to the invariance of the action under local deformations of the curve. To see this, suppose we have an action S evaluated on a curve Γ, and we deform this (keeping end points fixed) to get a curve Γ ′ on which an action S ′ is evaluated, such as in Figure 1. Figure 1: Deformation of the discrete curve Γ.
Then S ′ is related to S by the following: Equation (3.15) shows that the independence of the action under such a deformation is locally equivalent to the closure relation. The invariance of the action under the local deformation is a crucial aspect of the underlying variational principle. Figure 2: The effect of changing variables on the discrete curve I.
Next, it is interesting to investigate further how to derive the discrete Euler-Lagrange equation from the variational principle. The system given in Fig. (1) is difficult to work with because of a matter of notation. For general discrete curves it is cumbersome to implement the variational principle because of the notation it would require. We will, however, demonstrate how the principle works for a few simple cases: (a) the curve shown in Fig  and so we work with the curve given in Fig. (2b). The action evaluated on this curve can be written in the form The minus sign in (3.16) indicates the reverse direction of the Lagrangian L (N) along the horizontal links. Performing the variation x → x + δx, we have We now obtain the Euler-Lagrange equations   Figure 3: The effect of changing variables on the discrete curve II.  Fig. (3b). The action evaluated on the curve Γ ′ reads Performing the variation x → x + δx, we have We now obtain the Euler-Lagrange equations which are again equivalent to (2.19) and (2.20), respectively.
By working with the specific type of curves given in Fig. (2) and Fig. (3), we can perform the variational principle with either the new variables (N, m) or (N ′ , m). Even though we have these two possibilities, the natural choice is to work with (N, m). This is because the variable N ′ leads to difficulties in performing the skew limit in the next section. This can, in fact, be seen from the structure of the plane wave solution (2.4).
It is not yet clear whether the processes of varying the curve of the independent variables, from which we get the closure relation, and that of varying the dependent variables, from which we get Euler-Lagrange equations, commute. We defer consideration of this point to a future publication.

The semi-continuum limit: skew limit
In this section, we study a continuum analogue of a previous construction in Section 2 by considering a particular semi-continuous limit. Since the semi-discrete KP equation (2.1) contains two discrete variables n and m, we could perform a continuum limit on one of these variables separately, while leaving the other discrete variable intact, and thus obtain a semi-continuous equation with one remaining discrete and two continuous independent variables. Alternatively, we can first perform a change of independent variables on the lattice and subsequently perform the limit on one of the new variables. The advantage of the latter approach over the 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, and perform the transformation on the dependent variables by setting u(n, m) → u(N, m) =: u, which leads to the following expressions for the shifted variables: Rearranging the discrete exponential factor in (2.4), we have We perform the limit n → −∞, m → ∞, ε → 0 while keeping N fixed and setting ε = p − q, such that εm = τ remains finite. Focusing on the penultimate factor in (4.1) we have that so that the solution to the Lax equations (2.2a) and (2.2b) takes the form We would now like to see the effect of this limit on the semi-discrete KP equation. We rewrite eq. (2.1) as Taking the limit, we have Setting τ = τ 0 + mε, where ε 0 is a background constant of the continuous variable, and applying the Taylor expansion, we get Eq. (4.7), which contains two continuous and one discrete variable, we call the semicontinuous KP equation (to distinguish it from the semi-discrete KP equation, which contains one continuous and two discrete variables).
Remark: The connection to the Toda system. Introducing the variable 1 +u =: exp(v) we can write eq. (4.7) in the form Introducing next another variable y through the relation v = y − y, we obtain from which we can identify This is the well-known two-dimensional Toda lattice, cf. [29]. A Lagrangian exists for this system is given by To follow the reductions of the Lagrangian structure for the CM system, it would be desirable to have a Lagrangian for the semi-discrete KP equation (2.1) and for the semi-continuous KP (4.7) equation, but these are strangely elusive. Obviously, the fully continuous KP equation does possess a Lagrangian structure which is easy to establish by inspection, and furthermore the 2D Toda lattice (4.10) reduces to the KP equation through some specific continuum limits, see e.g. [23]. Thus it is conceivable that CM lagrangians could be established through the "Toda route", but we will not pursue this line in the present paper. We would like to mention also Krichever's elliptic analogue of the one-dimensional Toda lattice [32], which remarkably contains a term corresponding to the elliptic discrete-time CM given in [24]. We will give some results on the Lagrangian 1-form structure of the discrete-time elliptic CM model in Appendix B, but our main concern in the present paper is to study the Lagrangian 1-form structure in the simplest possible case, for the sake of transparency.
Thus, we obtain Using a Taylor expansion as in the previous case, we can write (4.13) as Inserting the special form of the solution, eq. (4.3), into eq. (4.12), we recover the set of equations (2.5) and (2.6), obviously with the replacements of x i (n, m) by x i (N, m) and with x i = x i (N + 1, m). For the sake of selfcontainedness we write the Lax corresponding matrices (which are essentially the same as before) in the new notation as follows: The compatibility between the matrices L and M produces the equations of motion implying as before the system of equations (2.15), but now viewed as part of a semicontinuous Calogero-Moser system. This set of equations will be complemented by equations involving derivatives with respect to the variable τ , which are obtained by inserting the solution (4.3) into eq. (4.15) and obtaining the complementary equations for b i . Equating to zero the resulting coefficients of (ξ − Introducing the matrix together with the matrix M as given above, we can rewrite equations (4.19) and (4.20) as This, using (4.21), implies that we also have 25) and thus, combining these two relations, we get: We observe that if we use (4.25), we recover (4.21) from (4.26). Furthermore, we can show that eq. The skew limit on the solution: we finish this section by performing the skew limit on the full solution (2.16). Let us observe that due to the fact that the matrices L and K are related linearly through the constraint (2.19) (this relation amounts to a shift over (p− q) times the unit matrix) we can identify the diagonal matrices of eigenvalues as follows: where Λ is the matrix which is independent of both p and q (i.e., independent of the direction of the lattice). Then the full solution (2.16) can be expressed in the form Using the definitions of the variable N = n + m and p − q = ε, we obtain Taking the limit, we have This equation represents the full solution after taking the skew limit. The positions of the particles x i (N, m) can be determined by computing the eigenvalues of (4.31).

Semi-continuous Lagrangian and closure relation
Having obtained, in the previous section, the skew limit of the Lax equation and the corresponding differential-difference system comprising eqs. (4.21) and (4.25), together with eq. (4.18), we now proceed to present the corresponding Lagrange form. First, we observe that eq. (4.18) can be once again be obtained by implementing the usual variational principle on the following action S (N) given by where now the Lagrangian L (N) involves variables x i shifted in the discrete variable N instead of the original variable n, and the corresponding discrete Euler-Lagrange equation reads: yielding (4.18). Next we observe that the combined differential-difference equation (4.26) can be obtained from the two-dimensional action yielding (4.26) from the two-dimensional EL equations However, we will argue that this is not the correct point of view to take in the context of the CM systems, because they consist after all of ODE's rather than PDE's or differentialdifference equations. In fact, this action, containing a summation over the variable N and an integration over the variable τ can never be obtained as a continuum limit of the onedimensional discrete actions presented in Section 3, which contain one single summation.
The skew limit on the action: We note that the actions in (5.1) and (5.3) are obtained directly from the corresponding equations of motion (4.18), (4.21) and (4.25), respectively. An interesting question is whether we can obtain the mixed discrete and continuous 1-form structure directly by taking the skew limit on the fully discrete 1-form action (3.14). This will be rather tricky because it will sensitively depend on parametrization of the curve. Furthermore, in order to perform a systematic expansion on the action functional one would need to employ not only a Taylor expansion on the relevant lattice shifts, but at the same time also the opposite of the Taylor expansion on the sums. For the latter we can employ the following "anti-Taylor" expansion of the sum of a discrete function f (n) =: F (τ ), where τ = τ 0 + nε with τ 0 fixed: the formula which guarantees that the analogue of the fundamental theorem of integration holds, namely: We now would like to pursue this task by considering a simple discrete curve Γ given in Fig.  (2a). The action corresponding to the curve has been given by (3.14). Performing the skew limit, the curve has been transformed to Fig. (2b) with the new variables (N, m). Using the anti-Taylor expansion, the action (3.16) becomes We also use the fact that the Lagrangian is an antisymmetric function under interchange of the arguments together with expansion with respect to ε, then the action becomes where L (τ ) is given in (5.8). By using a specific choice of curve given in Fig. (2), we can perform the skew limit on the discrete action leading to what is called the "semi-continuous" action in (5.6).
The skew limit on the discrete-time closure relation: We would like to consider the closure relation of the semi-continuous Lagrangians. Taking, as we did in the equations of motion, the skew limit on the discrete-time closure relation eq. (3.9), implementing a change of variables on the lattice first, we obtain: Equating the coefficients of ε gives us where the Lagrangian L (τ ) is obtained by performing the skew limit on the Lagrangian L (m) in eq. (3.10b) to give which produces the semi-continuous equations of motion. The derivation of (5.8) is given in Appendix C.
Furthermore, the variational derivatives with respect to x(τ ) and x (τ ) yield: which yield the equations which are identical to eqs. (4.21) and (4.25).
The relation (5.7) represents a closure relation between the semi-continuous Lagrangian L (τ ) and the semi-discrete Lagrangian L (N) . This points again to a Lagrangian 1-form structure but now of a mixed, part discrete and part continuous, type where the total action is a combination of a discrete part of the action S (N) and continuous part S (τ ) defined on a semidiscrete curve in the space of the discrete independent variable N and the the continuous variable τ . This is most easily visualized in the case where we have a piecewise linear curve as in Fig. 4, where in the horizontal direction we have step-wise jumps over units, while vertically the curve is continuous. The closure relation guarantees, nevertheless, that any such semi-discrete curve may be "locally" deformed without changing the action functional consisting of sums of the form (5.1) over the horizontal iterates and of the form (5.6) in the vertical segments . Variational principle for a semi-continuous 1-form structure: In (5.7), we obtain the semi-continuous closure relation. It is interesting to see how the variational principle works in this case. We will end this section with a scheme for deriving the closure relation. Now, suppose we have a curve Γ living on the semi-continuous (N − τ )-configuration shown in Fig. (5). The action associated with the curve Γ can be written in the form where the first term is the Lagrangian living on the vertical (continuous) line and the second term corresponds to the Lagrangian living on the horizontal (discrete) line of the curve Γ.
We now impose the condition that δS = S ′ − S = 0, yielding Using the Taylor expansion with respect to the variable τ , we have N + ∆N)) . (5.14) The term inside the bracket is required to be zero according to the condition δS = 0, yielding the semi-continuous closure relation.

The full continuum limit
In the previous section, we took the continuum limit on the discrete variable m, leading to a system of differential-difference equations. The full continuum limit, performed on the remaining discrete variable N, will lead to a coupled system of poles in the first instance, from which a hierarchy of ODEs can be retrieved, which is the CM hierarchy. How to perform this limit is inspired by the structure of the solutions of (4.3). Performing the following computation, we can identify the full continuum variables as the coefficients of the various powers of k, namely where η = N p . This reduces the KP eigenfunction given in (2.4) to the following form.
The corresponding reduction to CM removes the centre of mass, which can be identified with the t 1 flow, from the system of equations. Since X i = X i (t 1 , t 2 , t 3 , ...) and we take the assumption that ∂Xi ∂t1 = 0, we find thaṫ We also find that (6.5) and The full limit on the solution: Next, we would like to perform the full limit on the exact solution. We now rewrite (4.31) in the form Using the definitions of t 1 , t 2 , and t 3 in (6.2), we now have the solution in the form which is a function of time variables (t 1 , t 2 , t 3 , ..). The positions of the particles X i (t 1 , t 2 , t 3 , ...) can be computed by looking for the eigenvalues of (6.8). Since, the term t 1 − ξ represents the centre of mass flow, the explicit expression of the solution for the CM can be obtained from the secular problem for the matrix where X(0, 0) = U −1 (0, 0)Y (0, 0)U (0, 0) and L(0, 0) = U −1 (0, 0)ΛU (0, 0). We note that (6.9) is identical to what was obtained in [9] up to the time variable t 2 (involving a scaling of the variable). The solution (6.8) involves N -time flows for the CM system. The next solutions in the hierarchy can be generated by pushing further on with the expansion.
The full limit on the equations of motion: We now would like to see what would result from taking the limit on the equations of motion. We know that the combination of eqs. (4.21) and (4.25) gives us the equations of motion in the skew limit case. We look at the full limit for each of these. Using eqs. (6.4), (6.5) and (6.6), we find that eq. (4. 21) gives and (4.25) yields The difference between (6.10) and (6.11) gives the following: The leading term of order O(1/p 3 ) gives us which is the equation of the motion for the continuous CM system [18].
We now use eq. (6.12) to simplify eq. (6.13), and we get This equation represents the next equation of motion beyond the usual continuous CM in the hierarchy. We will stop at this equation, but we can actually get the higher terms of the equation in which the variable t 4 and higher order time-flows must be taken into account.
Note finally that summing (6.10) and (6.11) gives the following equations These equations are the constraints for the full limit case which will play an important role in the next section. The occurrence of these constraints seems to be related to the superintegrability of the CM system, cf. [5], and their validity can be checked on solutions.

The continuous Lagrangian 1-form and its closure relation
In the previous section, we obtained directly the equations of motion of the continuous CM from the full limit. In this section, we obtain the continuous Lagrangians from the full limit as well. We will restrict ourselves to the case of 3-particles involving time flows t 2 and t 3 , to establish the simplest example of a Lagrangian 1-form for the CM system. Both the Lagrangians L (τ ) and L (N) produce the hierarchy of the continuous CM Lagrangians corresponding to the equations of motion in the previous section. Rather than performing the full limit on the individual Lagrangians, we prefer to take the limit on the action itself.
The full limit on the action: In Section 5, we performed the skew limit on the discrete action. In the present section, we will continue to perform the full continuum limit on the semi-continuous action. We now take the action to be of the form where the first term belongs to the vertical part and the second term belongs to the horizontal part of the curve given in Fig. (4).
Using anti-Taylor expansion, the action now becomes 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, η) → (t 2 , t 3 ) by using (6.2) and also expand the Lagrangians with respect to variable p. We obtain where L (t2) and L (t3) are given by of which the Euler-Lagrange equation gives exactly eq. (6.12) and We see that the Lagrangian L (t3) contains derivatives with respect to two time flows t 2 and t 3 . We observe that the equation of motion (6.14) requires the Euler-Lagrange equation in the form Furthermore, we find that which is identical to eq. (6.15a).
Here we obtained the hierarchy of Lagrangians for the CM-model through the full continuum limit. Obviously, higher Lagrangians in the family can be generated by pushing further on with the expansion.
The full limit on the closure relation: Using the results regarding the limits of the Lagrangians L (τ ) and L (N) in the previous section, and also the relations of eq. (6.2), we can write Substituting eq. (7.10) into the closure relation eq. (5.7) together with the full limit of L (τ ) − L (τ ) , we find that the leading term is of order O(1/p 3 ), and is This equation represents the closure relation for the continuous case of the Lagrangian 1-form.
We are now taking into account not only the usual CM Lagrangian, but also a higher-order Lagrangian; these have up until now not been considered. The verification of the closure relation (7.11) is presented in Appendix D.

The variational principle for forms
In the previous section, we obtained the Lagrangian hierarchy of the CM system along with the corresponding Euler-Lagrange equations. In this section, we are interested in investigating the variational principle for continuous Lagrangian 1-forms. Here we limit ourselves to the first two time flows in the hierarchy, t 2 and t 3 , corresponding to a 3-particle system in order to exhibit the simplest example. Let us define the time variables to be functions of a new variable s, so that t i = t i (s), associated to the curve Γ : (t 2 (s), t 3 (s)) as in Fig. (6a). We now define the Lagrangians to be The action evaluated on the curve Γ can be defined as Figure 6: The deformation of the continuous curve Γ .
We now perform variation on the curve of time variables t 2 and t 3 in the following way. Suppose we have an action S defined on a curve Γ, and the curve is deformed t 2 → t 2 + δt 2 and t 3 → t 3 + δt 3 in Fig. (6b), keeping the end points fixed, so we have the conditions δt 2 (s 0 ) = δt 2 (s 1 ) = 0, δt 3 (s 0 ) = δt 3 (s 1 ) = 0. (8.3) The action S ′ is defined as Extremizing the action, we have δS = 0, where δS is defined by Using the Taylor expansion, we obtain Using the fact that the end points are fixed, the first line of eq. (8.6) is zero. We can also use Here we must have two copies of the closure relation 9) in order that δS = 0. The closure relation (8.9) guarantees that any such continuous curve may be locally deformed without changing the value of the action functional corresponding to the time variables t 2 and t 3 .
Next, we would like to perform a variation on the field variables x. We now fix the curve Γ on the t 2 − t 3 plane and examining the corresponding curve E Γ of the variables x living in the x − t 2 − t 3 space as in Fig. (7a). The action can be expressed in the form We now introduce Figure 7: The curves Γ and E Γ in the x − t configuration.
where y is a new independent variable, and the combination of these two equations yields Performing the variation on the action, we have Integrating by parts the second line in (8.14) with the boundary conditions δx(t 2 (s 0 )) = δx(t 3 (s 0 )) = 0 , δx(t 2 (s 1 )) = δx(t 3 (s 1 )) = 0 , We consider eq. (8.16) to be a generalized Euler-Lagrange equation containing two separate Euler-Lagrange equations, and the equation (8.17) is the constraint resulting from the variational principle. To summarize, performing the variation on the curve Γ with respect to the time variables on the t 2 − t 3 plane, the computation leads to the closure relation (8.9). By fixing the curve Γ and varying the curve E Γ , the computation leads to the Euler-Lagrange equations corresponding to the Lagrangian 1-form. However, the closure relation allows the deformation of the curve Γ to a simpler curve, as in Fig. (8). If we now are on the curve Γ 2 where the time variable t 2 is a constant leading to dt2 ds = 0, we obtain If we are on the curve Γ 1 , where the time variable is a constant leading to dt3 ds = 0, we obtain exactly the same equation (8.18) with interchange of indices 2 ↔ 3. This is entirely consistent with the results obtained in the previous section.
The Euler-Lagrange equations (8.16), the constraint equations (8.17) and the closure relation (8.9) together form a consistent set of equations, representing the multidimensional consistency of the system.

Concluding remarks
In this paper, we present what we believe to be the treatment of the Lagrangian structure for the Calogero-Moser system which best captures the essential integrability characteristics. It is well known that the continuous CM system is Liouville integrable [2] and also superintegrable [5] and this remains the case on the discrete-time level [28,18]. However, in this paper, we focused on another aspect of the integrability, namely multidimensional consistency, meaning in this context of finite-dimensional systems the existence of commuting flows. We believe that in the present paper, we have given convincing arguments for the assertion that the proper Lagrangian structure should be the one in terms of Lagrange 1-forms. Intriguingly, this is most manifest on the discrete-time level where it follows directly from the relevant Lax equations. In the continuous case, the relevant Lagrangians (apart from the obvious Lagrangian for the second order CM flow) are more difficult to establish, but they follow through systematic continuum limits performed on the discrete Lagrangian. However, the latter is quite a subtle computation, and it is here that the connection between the CM system and the KP system (the former arising from the pole reduction of the latter) is essential. The construction in this paper starts from the semi-discrete KP equation, and it is the dependence of this equation on (lattice) parameters which guides the choice of continuum limits. We believe that the CM system forms a first important example for the study of the new variational principle which applies to the case of Lagrangian 1-forms, cf. [19,21,22,31], as preparation for similar structures in the case of higher Lagrangian multiforms.
We finish with a few observations.
• Since the construction is based on the connection with a semi-discrete KP equation, it would have been more satisfactory if a Lagrangian for that equation was at our disposal. However, at this stage, such a Lagrangian seems to be elusive, even though it exists for the bilinear discrete KP case [22].
• Another issue is the connection with the Hamiltonian hierarchy associated with the Calogero-Moser system. Usually, the connection between the Hamiltonian and the Lagrangian is given through the Legendre transformation, which for the second flow in the CM hierarchy is the standard one. However, when we move to the higher Hamiltonians in the hierarchy, these are no longer of Newtonian type, and hence lead to complicated expressions when implementing the Legendre transformation. What the treatment in of Section 7 reveals , however, is that the higher flows fit naturally in the 1-form structure through mixed, but polynomial, Lagrangians in terms of the higher-time derivative.
• The natural framework for quantization of the discrete-time CM model seems to be that of the path integral, which can be interpreted as a quantum version of the leastaction-principle. One may conjecture that the quantization scheme exploiting the new Lagrange structure could lead to a rigorous approach to constructing the relevant path integral.
Our motivation in this paper was to present the most simple example of a Lagrange 1-form structure, hence our restriction to the rational case. However, most of our results can be readily expanded to the trigonometric and hyperbolic and also elliptic cases. The formulae for those cases are presented in Appendix B. A natural question is how our results extend to the relativistic case, i.e., the Ruijsenaars-Schneider (RS) model. We intend to address that problem in a future paper.

A The construction of the exact solution
In this appendix we review the construction of the exact solution (2.12), which follows a procedure similar to the continuous case, cf. e.g. [9]. The basic relations following from the Lax pair, i.e., (2.11) together with the definitions (2.7) and (2.8), lead to: x i (n, m)E ii is the diagonal matrix of particle positions. We concentrate first on the part of the Lax pairs associated with the dynamics in terms of the variable n. From the Lax equation (2.11), we may write where U = U (n, m) is the matrix used to diagonalize L. Then (A.1) becomes where Y = U −1 XU , which leads to the following expression Noting the invariance of Λ L under the discrete-time shift, we can easily solve (A.4), and its general solution is with Y (0, m) determined from the initial data X(0, m). A similar analysis can be applied to create the solution associated with the " " shift. 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 If the eigenvalues of Y (0, 0) are distinct (which we can take as an assumption on the initial condition) then U −1 (0, 0) is determined up to multiplication from the right by a diagonal matrix times a permutation matrix of the columns. (Fixing an ordering of the eigenvalues x i (0, 0), U −1 (0, 0) unique only up to multiplication by a diagonal matrix from the right). We can fix U −1 (0, 0) up to an overall multiplicative factor by demanding that Next we consider the matrix function given by with Λ L = pI + Λ and Λ K = qI + Λ. Let U −1 (n, m) be the matrix diagonalizing Y (n, m); by an appropriate choice of an overall factor (as a function of n and m) this matrix can be fixed such that it obeys: From the expression (A.9) we can now derive the relations with the usual notation for the shifts in n and m over one unit. Together with the relation (A.11) this subsequently yields:

B The elliptic and trigonometric discrete-time Calogero-Moser
In this section, we show that there is a connection between the time-part Lax matrix and the Lagrangian for the trigonometric/hyperbolic and elliptic cases of the discrete-time CM system, similar to that which we have established for the rational case.

B.1 The elliptic case
The Lax matrices in this case read [24]  The determinant of the matrix M elliptic is The last term in eq. (B.7) is related to the motion of the centre of mass of the system and it can be separated from the relative motion.

B.2 The trigonometric case
The Lax matrices in this case read [24]  Again the last term in eq. (B.14) is related to the center of mass motion of the system, and it is not going to effect the equations of the relative motion.
C The derivation of the Lagrangian L (τ ) In this section, we show the derivation of the Lagrangian L (τ ) (5.8) from the skew limit of the Lagrangian L (m) (3.10b). We recall the Lagrangian L (m) again here L (m) = log | det(N )| + q(Ξ − Ξ) . (C.1) Performing the skew limit, we obtain The inverse of the matrix M is given by . (C.5b) We now can show that We now consider the identity which we differentiate with respect to ξ at a given value ξ = x j , yielding (C.8) We now differentiate (C.8) with respect to x j , yielding (C.9) Using (C.9), we can rewrite (C.6) in the form x j ∂ ∂ x j ln |Ψ( x j )| , (C.10) then the Lagrangian L (τ ) can be expressed in the form D The direct proof of the closure relation (7.11) In this section, we show that the closure relation (7.11) holds on the equations of motion.
Here we recall the equations of motion for the time variables t 2 and t 3 The Lagrangians corresponding to (D.1) and (D.2) take the form respectively.
We find that and We see that the first terms of eq. (D.5) and eq. (D.6) are identical. The remaining work is to show that the last term in eq. (D.6) is zero. Using eq. (D.1), we may rewrite the last term in eq. (D.6) with the help of the identities: (D. 7) We see that the first term is an antisymmetric function and hence vanishes, while the second term (D.8) The first and third terms are the antisymmetric functions, hence vanish leaving the middle term which is actually the opposite of the term on the left hand side, i.e., (D.9)