Integrable Hamiltonian Hierarchies and Lagrangian 1-Forms
aa r X i v : . [ m a t h - ph ] J u l Integrable Hamiltonian Hierarchies and Lagrangian1-Forms
Chisanupong Puttarprom † , + , Worapat Piensuk − , Sikarin Yoo-Kong ∗ † Theoretical and Computational Physics (TCP) Group, Department of Physics,Faculty of Science, King Mongkut’s University of Technology Thonburi, Thailand, 10140. − Department of Mechanical Engineering, Faculty of Engineering,King Mongkut’s University of Technology Thonburi, Thailand, 10140 + Theoretical and Computational Science Center (TaCS),Faculty of Science, King Mongkut’s University of Technology Thonburi, Thailand, 10140. ∗ The Institute for Fundamental Study (IF),Naresuan University, Phitsanulok, Thailand, 65000.
July 3, 2019
Abstract
We present further developments on the Lagrangian 1-form description for one-dimensionalintegrable systems in both discrete and continuous levels. A key feature of integrability in thiscontext called a closure relation will be derived from the local variation of the action on thespace of independent variables. The generalised Euler-Lagrange equations and constraint equa-tions are derived directly from the variation of the action on the space of dependent variables.This set of Lagrangian equations gives rise to a crucial property of integrable systems knownas the multidimensional consistency. Alternatively, the closure relation can be obtained fromgeneralised Stokes’ theorem exhibiting a path independent property of the systems on the spaceof independent variables. The homotopy structure of paths suggests that the space of indepen-dent variables is simply connected. Furthermore, the N¨oether charges, invariants in the contextof Liouville integrability, can be obtained directly from the non-local variation of the action onthe space of dependent variables.
Keywords:
Lagrangian 1-form structure, Generalised Stokes’ theorem, Closure relation, N¨oethercharges, Commuting Hamiltonian flows, Homotopy, Liouville integrability.
It is well-known that integrable systems are exceptionally rare since most (nonlinear) differen-tial equations exhibit chaotic behaviour and no explicit solutions can be obtained. Informally,integrability is the property of a system that allows the solution to be solved in finite stepsof operations (or integrations). In more simply speaking, integrability enables us to solve theset of equations in a closed form or in terms of quadratures (ordinary integrals). In classicalmechanics, especially Hamiltonian systems with N degrees of freedom, integrability has a direct onnection to action-angle variables. It is well-understood that the choice of coordinates onphase space is not unique since one can transform an old set of coordinates to a new set of coor-dinates through the canonical transformation preserving the Hamilton’s equations. Of course,action-angle coordinates are special since the Hamiltonian depends solely on action variableswhich are constants of motion and, consequently, the angle variables are cyclic, evolving linearlyin time. With this feature, the complexity of the problem will be reduced and explicit solutionscan be determined by quadratures. In this sense, the existence of the action-angle variablesguarantees the integrability of the system. However, finding action-angle coordinates is nottrivial in practice. Then, the notion of integrability is rather defined in terms of existence of theinvariants known as Liouville integrability [1]. In this notion, the Hamiltonian system with N degrees of freedom, whose evolution is on 2 N -dimensional manifold embedded in phase space,is integrable if there exists N invariants, which are normally treated as Hamiltonians, that areindependent and in involution as the Poisson brackets for every pair of invariants vanish. Withinvolution of invariants, all evolutions belong to the same level set and are mutually commute,known as commuting Hamiltonian flows which are considered to be the main feature for inte-grable Hamiltonian systems. Furthermore, there exists a canonical transformation such thata set of invariants is nothing but a set of action variables. This means that 2 N -dimensionalmanifold can be foliated into a N -dimensional invariant torus in which the angle variables arenaturally the periodic coordinates on this torus.In discrete world, the differential equations are generalised to their discrete counterparts knownas the difference equations. Under the continuum limit, it happens that many distinct differ-ence equations could be reduced to the same differential equation. In other words, the discreteanalogue for a single differential equation is not unique. This would make a difficulty to sortout what is a preferable difference equation for a particular physical system. At this point,the notion of integrability may help to choose the right equation, which is a solvable equation,from many non-solvable ones. For the Hamiltonian systems, the discrete analogue of the Li-ouville integrability can be naturally constructed [2]. However, there are several notions onwhat integrability would mean in the discrete level, e.g. existence of a Lax pair [3], singu-larity confinement [4] and algebraic entropy [5]. One of the remarkable aspects of integrablemany-dimensional discrete systems is a multidimensional consistency [6]. This feature allows usto embed the difference equation in a multidimensional lattice consistently, i.e., there exists aset of (infinite) compatible equations defined in each subspace corresponding to the number ofindependent variables [7–11]. Two-dimensional lattice system embedded in a three-dimensionalspace is said to be consistent in such way if the quadrilateral equations describing three side-to-side connected surfaces of a cube can be solved and yield the coincide result with given initialvalues, as discussed in [12]. This consistency is called the consistency-around-the-cube (CAC) or3D-consistency, which has led Adler, Bobenko and Suris to classify the quadrilateral equationsof the two-dimensional lattice systems into the remarkable ABS list, see [13].In Liouville integrability, the Hamiltonian is a main object. However, alternative object calledthe Lagrangian can be chosen to work with resulting the same physics. According to the leastaction principle, the system evolves along the trajectory on the configuration space that theaction functional is extremum. It is known that most integrable discrete systems admit La-grangian description. Therefore, it is quite natural to consider the evolution of the systemthrough the discrete path on the space of dependent variables as well as that on the spaceof independent variables. Since the integrable discrete equations, exhibiting such consistency,are obtained from Lagrangian, it is essential for multidimensional consistency to be encodedon the level of Lagrangians resulting in a new notion of integrability known as the Lagrangian ultiform. The pioneer works on Lagrangian 2-form and 3-form were initiated by Lobb andNijhoff [12]. In this context, the action functional is invariant under the local deformation ofthe discrete path on the space of independent variables leading to a intriguing feature called theclosure relation. This means that there are many discrete paths on the space of independentvariables corresponding to a single discrete path, whose the action is critical, on the space ofdependent variables. The case of Lagrangian 1-form was later developed by Yoo-Kong, Lobband Nijhoff [14] and Yoo-Kong and Nijhoff [15]. Afterwards, the variational formulation ofcommuting Hamiltonian flows was studied by Suris [16]. The Lagrangian 1-form structure ofthe Toda-type systems, along with their relativistic versions, were developed by Boll, Petreraand Suris [17,18]. Further investigation has been made by Jairuk, Yoo-Kong and Tanasittikosolon the Lagrangian 1-form structure of the Calogero’s Goldfish and hyperbolic Calogero-Mosermodels [19–21]. Recently, the variational symmetries of the pluri-Lagrangian systems has beendeveloped by Suris and Petrera [22, 23].In this paper, we provide further developments and a complete picture on Lagrangian 1-formstructure in both discrete and continuous levels. In section 2, we give a short review on action-angle variables as well as the Liouville integrability. A key feature called the commuting Hamil-tonian flows is also discussed. An explicit example, namely rational Calogero-Moser system,of Liouville integrability is provided. In section 3, we first introduce the notion of discreteLagrangian 1-form and the key feature called a discrete closure relation is derived from the vari-ational principle. The multidimensional consistency on the level of discrete Lagrangians is alsodiscussed. A sequence of continuum limits is performed resulting in the semi-discrete and fullcontinuous Lagrangian 1-form. In the continuous level, the variational principle is consideredfor the path on the space of both dependent and independent variables leading to a generalisedEuler-Lagrange equation, a constraint equation and a continuous closure relation. All thesethree equations can be considered as a set of compatible of Lagrangian equations possessingthe multidimensional consistency. Furthermore, we also give a remark on the derivation of theclosure relation from a point of view of the generalised Stokes’ theorem as well as the homotopyof the paths on the space of independent variables. In section 4, the Legendre transformation isintroduced to obtain the Hamiltonian hierarchy. The variational principle on the phase spacewill be considered resulting in generalised Hamilton’s equations and commuting Poisson bracketwhich consequently gives us the commuting Hamiltonian flows. In section 5, N¨oether chargesare directly derived from the variation of action functional of the Lagrangian 1-form. We findthat all N¨oether charges are nothing but all Hamiltonians (invariants) in the system. In the lastsection, the conclusion together with some potential further studies are delivered. In classical mechanics, we have freedom to solve the problem with any set of coordinates andif we choose a right (good) set of coordinates, the problem can be easily solved. On the otherhand, if we choose a wrong (poor) set of coordinates we possibly have to go through a tremen-dous work for obtaining the answer. In Hamiltonian mechanics, for a system with N degreesof freedom, one can transform an old set of coordinates ( p, q ) on the phase space to a newone ( P ( p, q ) , Q ( p, q )), while the Hamilton’s equations are still preserved, through the canonicaltransformation. However, such transformation may not guarantee analytical exact solutions. In any cases, there is a natural choice of coordinates such that { p, q } ⇒ { I, θ } , H ( p, q ) ⇒ K ( I ) , where I is a set of action variables , θ is a set of angle variables and K is a new Hamiltonianwhich is a function of the action variables only since the angle variables are cyclic. With a newset of coordinates, Hamilton’s equations become˙ I = ∂ K ∂θ = 0 , ˙ θ = ∂ K ∂I = ω ( I ) , where ω is a function of action variables. It can be seen that the action variables I are auto-matically constants of motion. Then we have to solve only N first-order differential equationsfor θ ( t ) and, hence, the system is integrable . Action-angle variables define an invariant tori asthe action variables define the surface, while the angle variables provide coordinates on the tori. I θ I θ Figure 1: The cross section of invariant torus: T in which the smaller and bigger radii are the actionvariables I and I , defining the surface. Each point on the torus is defined by the angle variableswhich are the linear function of time. In order to make things more transparent, we provide a simple example as follows. In thecase of 2 degrees of freedom, a set of action-angle variables is ( I , I , θ , θ ) and the invarianttorus T = S × S can be visualised as a doughnut shape, see figure 1. The coordinates onthe surface of the torus are defined by the angles θ and θ which linearly increase with time θ i = ω i t + θ i (0) . The coordinates θ and θ will be called bases of principal vectors corre-sponding to the translations θ → θ + 2 π and θ → θ + 2 π . Then, in the case of N degreesof freedom, a set of action-angle variables ( I , I , ..., I N , θ , θ , ..., θ N ) will form N -dimensionalinvariant tori T N = S × S × .... × S | {z } N . There are N similar closed loops along the basis ofprincipal directions where the coordinates θ i can be chosen to vary from 0 to 2 π . From the previous section, an existence of action-angle variables is a main characteristic forintegrability. However, searching for these special sets of coordinates may not be an easy task.
Theorem 1 ( Liouville-Arnold [1]) . Suppose that there is a set of N functions ( f , f , ..., f N ) ≡ f in involution, i.e., { f i , f j } = 0 , where i, j = 1 , , .., N , on a symplectic N -dimensionalmanifold. Let M c := { ( p, q ) ∈ M ; f k ( p, q ) = c k } , c k = constant , k = 1 , , ..., N (2.1) be a level set of the functions f i which are independent. Then M c is a smooth manifold, invariant under the phase flow with Hamiltonian H = f . • If M c is compact and connected, there exists a diffeomorphism from M c to a torus T N ≡ S × S × ... × S | {z } N , and, in a vicinity of this torus, the action-angle coordinates ( I, θ ) = ( I , I , ..., I N , θ , θ , ..., θ N ) , ≤ θ k ≤ π, (2.2) can be constructed such that angles θ k are coordinates on M c and actions I k = I k ( f , ..., f N ) are first integrals. • The canonical Hamilton’s equations are linearised ˙ I k = 0 , ˙ θ k = ω k ( I , I , ..., I N ) , k = 1 , , ..., N . (2.3) Therefore, the systems are solvable by quadratures. qp M c T N Figure 2: A 2 N -dimensional phase space M c can actually be foliated to N -dimensional tori T N . The set of invariants is normally treated as a set of Hamiltonians called the Hamiltonianhierarchy: ( I , I , .., I N ) ≡ ( H , H , ..., H N ). Therefore, the evolution on phase space is associ-ated with N time variables ( t , t , ..., t N ). For any function F ( p, q ) defined on the phase space,we find that d F d t j = { H j , F } , (2.4)where t j is the time variable associated with the Hamiltonian H j . Equation (2.4) describes thetime evolution (flow) of function F along the surface that H j is constant. We also have anotherHamiltonian H k such that d F d t k = { H k , F } , (2.5)where t k is the time variable associated with the Hamiltonian H k . We find that ∂∂t j ∂F∂t k = ∂∂t k ∂F∂t j ∂∂t j { H k , F } = ∂∂t k { H j , F }{ H j , { H k , F }} − { H k , { H j , F }} = 0 {{ H j , H k } , F } = 0 , (2.6)since { H j , H k } = 0. The relation (2.6) represents an interesting feature, known as commutingHamiltonian flows , which is a main feature for integrable Hamiltonian system telling us thatthe flows extend to all possible of time variables t i and fill the whole manifold M c . .3 Liouville integrability: The discrete-time case For the discrete case, the notion of Liouville integrability can be naturally constructed.
Theorem 2 ( Liouville-Arnold-Veselov [2, 24]) . If a canonical map is integrable, then anycompact non-singular level set M c = { ( p, q ) ∈ M : I k ( p, q ) = c k , k = 1 , , .., N } is a discon-nected union of tori on which the dynamic takes place according to regular shifts. A discrete system is completely integrable if there exists a set of functions { I , I , ..., I N } satisfying the following requirements: • The functions are invariant, I ( p, q ) = I ( T p, T q ), under discrete map: ( p, q ) → ( T p, T q ),where T is the discrete-time shift. The Hamiltonian is one of them. • The functions are in involution with respect to the Poisson bracket { I i , I j } = N X k =1 (cid:26) ∂ I i ∂p k ∂ I j ∂q k − ∂ I j ∂p k ∂ I i ∂q k (cid:27) = 0 . • The functions are functionally independent throughout the phase space.In order to show that I i ( p, q ) is invariant under discrete-time shift, suppose there exists adiscrete symplectic map: ( p, q ) ( T p, T q ) such that T p j = g j ( q, p ) , T q j = f j ( q, p ) , j = 1 , , ..., N , (2.7)where g j and f j are some functions of coordinates ( q, p ). Equation (2.7), equipped with thePoisson bracket structure { T q j , T q k } = 0 , { T p j , T p k } = 0 , { T q j , T p k } = δ jk , j, k = 1 , , .., N , (2.8)can be considered as a canonical transformation from an old set of coordinates ( p, q ) to a newset of coordinates ( T p, T q ). If the Jacobian | ∂f j /∂p i | is nonzero, we introduce a generatingfunction H ( q, T p ) [25] in which (2.7) becomes T q j − q j = ∂H ( q, T p ) ∂ T p j , (2.9a) T p j − p j = − ∂H ( q, T p ) ∂q j , (2.9b)where (2.9) is sometimes referred to a set of discrete-time Hamilton’s equations . On the otherhand, if the Jacobian | ∂g j /∂q i | is nonzero, we may introduce another generating function H ′ ( T q, p ) in which (2.7) becomes T q j − q j = ∂H ′ ( T q, p ) ∂p j , (2.9c) T p j − p j = − ∂H ′ ( T q, p ) ∂ T q j . (2.9d)Next, we would like to consider a canonical transformation between old variables ( q, p ) and newvariables ( Q, P ) such that P j = P j ( q, p ) , Q j = Q j ( q, p ) , j = 1 , , ..., N , (2.10) ith the Poisson bracket structure { Q j , Q k } = 0 , { P j , P k } = 0 , { Q j , P k } = δ jk , j, k = 1 , , .., N . (2.11)Therefore, the discrete symplectic map for new variables reads T P j = G j ( Q, P ) , T Q j = F j ( Q, P ) , j = 1 , , ..., N , (2.12)where G j and F j are some functions of the coordinates ( Q, P ). Again, (2.12), equipped withthe Poisson bracket structure { T Q j , T Q k } = 0 , { T P j , T P k } = 0 , { T Q j , T Q k } = δ jk , j, k = 1 , , .., N , (2.13)can be considered as a canonical transformation from an old set of variables ( P, Q ) to a new setof variables ( T P, T Q ). If it happens to be that T P j = P j , one can obtain F j ( Q, P ) = Q j + ν j ( P )which immediately leads to Q j ( n ) = nν j ( P ) + Q j (0). The original discrete evolution can berecovered by means of (2.12). Here, ν j are the frequencies and can be obtained as follows.Suppose there is a generating function W ( q, P ) such that p j = ∂W ( q, P ) ∂q j , Q j = ∂W ( q, P ) ∂P j . (2.14a)Integrating the first equation of (2.14a), we have W ( q, P ) = N X k =1 Z q k ( n ) q k (0) p k ( P, q ′ ) d q ′ k , (2.14b)and then the second equation of (2.14a) gives Q j ( n ) = N X k =1 Z q k ( n ) q k (0) ∂p k ( P, q ′ ) ∂P j d q ′ k . (2.14c)Since Q j ( n ) is a function of the invariants P , the corresponding frequencies ν j are given by ν j ( P ) = N X k =1 Z T q k q k ∂p k ( P, q ′ ) ∂P j d q ′ k . (2.15)An above structure gives us a symplectic map, i.e., ( Q, P ) ( T Q, T P = P ) such that T P j − P j = 0 , (2.16a) T Q j − Q j = ∂S∂P j , (2.16b)where S is nothing but the action of the system given by S = N X k =1 Z T q k q k p k ( P, q ′ ) d q ′ k . (2.17)Then, a set of coordinates ( Q, P ) is of course the action-angle variables in the discrete case andthe existence of these variables, together with interpolation discrete map (2.16a), implies theintegrability of the system. .4 Lax pair: The continuous case The main object in Liouville integrability is a set of invariants. As we have mentioned earlier,searching for these invariants may turn out to be a difficult task. However, with the help of Laxmatrices, all invariants are possible to be computed. Let ( L , M ) be a Lax pair where L is thespatial part and M is the temporal part. These two matrices are functions on the phase spaceof the system satisfying L Ψ = λ Ψ , (2.18) M Ψ = d Ψ d t , (2.19)where λ is a fixed parameter, t is a time variable and Ψ is an auxiliary function. Compatibilitybetween (2.18) and (2.19) gives dd t ( L Ψ) = dd t ( λ Ψ) d L d t Ψ + L d Ψ d t = λ d Ψ d t d L d t Ψ + LM Ψ =
M L Ψ d L d t Ψ = ( LM − M L )Ψ ⇒ d L d t = − [ L , M ] , (2.20)which is called the “Lax equation” producing the equations of motion.Furthermore, the solution of (2.20) is in the form L ( t ) = U ( t ) L (0) U − ( t ) , (2.21)where U is an invertible matrix. The time derivative of (2.21) yields d L d t = d U ( t ) d t L (0) U − ( t ) − U ( t ) L (0) d U − ( t ) d t = d U ( t ) d t U − ( t ) U ( t ) L (0) U − ( t ) − U ( t ) L (0) U − ( t ) d U ( t ) d t U − ( t ) . (2.22)Comparing (2.22) with (2.20), we find that the matrix M ( t ) takes the form of M ( t ) = d U ( t ) d t U − ( t ) . (2.23)From structure of (2.21), we find that T r ( L ( t )) = T r ( L (0)) and, of course, T r ( L l ( t )) = T r ( L l (0)). This suggests that I l ≡ T r L l , where l = 1 , , ..., N , are nothing but the invari-ants of the system. Then, the existence of Lax pair allows us to construct all invariants. The same procedure on the construction of conserved quantities in continuous case can beapplied in the discrete case. Suppose L and M are Lax matrices for the discrete systemsatisfying L Ψ = ΛΨ , (2.24) M Ψ = T Ψ , (2.25) here Λ is a fixed parameter and Ψ is an auxiliary function. The operator T is the discrete-timeshift such that T Ψ( n ) = Ψ( n + 1), where n is a discrete-time variable. Performing the shiftoperator on (2.24), we obtain T ( L Ψ) = T (ΛΨ) T L T Ψ = Λ T Ψ . (2.26)Substituting (2.25) into (2.26), the equation turns into( T L ) M = M L , (2.27)which is a discrete-time analogue of (2.20) and it gives us the discrete-time equation of motionof the system. Suppose that Lax matrix L can be factorised as L ( n ) = U ( n ) L (0) U − ( n ) , (2.28)where U is an invertible matrix. Applying shift operator T on (2.28) and inserting into (2.27),we find that the matrix M ( n ) can be expressed in the form of M ( n ) = T ( U ( n )) U − ( n ) . (2.29)From the structure in (2.28), we can see immediately that T r ( L l ( n )) = T r ( L l (0)). Then, I l ≡ T r ( L l ( n )), where l = 1 , , ..., N , are the invariants in the discrete case. The same conclusion,with those in the continuous case, can be drawn that the existence of discrete Lax pair allowsus to construct all invariants for the discrete system. In this section, we provide an explicit example of the integrable system. We choose a one-dimensional N -body system with pairwise interaction called the rational Calogero-Moser system[26, 27] which is a well-known example of the finite-dimensional integrable Hamiltonian system.The equations of motion are given by [28]¨ X i = g N X j =1 i = j X i − X j ) , i = 1 , , ..., N , (2.30)where g is a coupling constant and X i = X i ( t ) is the position of the i th particle. In thecontinuous-time case, Lax matrices are given by [29] L = N X i =1 p i E ii + N X i,j =1 i = j i a E ij X i − X j , (2.31) M = a N X i,k =1 i = k E ii ( X i − X k ) − a N X i,j =1 i = j E ij ( X i − X j ) , (2.32)where a is a constant related to the coupling constant g defined by g ≡ a − a , p i is themomentum of the i th particle and E ij are entries defined by ( E ij ) kl = δ ij δ kl . The first three onstants of motion are I = T r L = N X i =1 p i , (2.33) I = 12 T r ( L ) = N X i =1 p i N X i,j =1 i = j g X i − X j ) ≡ H , (2.34) I = 13 T r ( L ) = N X i =1 p i g N X i,j =1 i = j p j ( X i − X j ) , (2.35)where (2.34) is the Hamiltonian of the continuous-time rational Calogero-Moser system.In the discrete case, the equations of motion are given by [30]1 x i − T x i + 1 x i − T − x i + N X j =1 i = j (cid:18) x i − T x j + 1 x i − T − x j − x i − x j (cid:19) = 0 , i = 1 , , ..., N , (2.36)where x i = x i ( n ) is the position of the i th particle. The discrete Lax matrices are given by [30] L = N X i =1 p i E ii − N X i,j =1 i = j x i − x j E ij , (2.37) M = − N X i,j =1 T x i − x j E ij , (2.38)where x i is the position of i th particle and the variables p i are given by p i = N X j =1 x i − T x j − N X j =1 i = j x i − x j , i = 1 , , ..., N . (2.39)We see that (2.37) is exactly the same as the continuous case. Therefore, all invariants can beimmediately obtained from I l ≡ T r L l , where l = 1 , , ..., N . In the previous section, the integrability criterion is constructed on the Hamiltonian structureof the system. The Poisson commuting of invariants plays an important role to exhibit thecommuting Hamiltonian flows. In this section, we set out to construct a mathematical relationto indicate the integrability from the Lagrangian point of view. The variational principle is amain mathematical process that will be used throughout the whole section.
In this section, we would like to introduce the notion of the discrete-time Lagrangian 1-form.First, let us define a set of discrete-time generalised coordinates (dependent variables) x ( n ), here x ≡ ( x , x , x , ..., x N ), N is the number of degrees of freedom and n ≡ ( n , n , n , ..., n M )is the set of the discrete-time variables (independent variables). Next, let us define the timeshift operators as follows( T i ) µ ( T j ) ν x ≡ x ( n , n , ..., n i + µ, ..., n j + ν, ...., n M ) , µ , ν ∈ Z , where T j is a time shift operator in j -direction. n j n k n i x T − i x T j x T k T j x T − i T k T j x ΓFigure 3: Discrete curve Γ on the space of discrete independent variables.
Suppose that the system evolves along an arbitrary discrete curve Γ embedded on M -dimensionalspace of discrete independent variables, see figure 3. Therefore, 1-form Lagrangian is a two-point function between two end points of the discrete line element given by L ( i ) ( n i ) = L ( i ) ( x, T i x ) , i = 1 , , ..., M . (3.40)The Lagrangian (3.40) possesses an antisymmetric property under an interchange of arguments,i.e., L ( x, y ) = − L ( y, x ). The action of discrete curve Γ can then be expressed in the form S Γ [ x ( n )] = X σ ( n ) ∈ Γ L ( i ) ( x, T i x ) . (3.41)Equation (3.41) is nothing but the sum of all discrete elements σ ( n ) = { σ i ( n ) = ( n, T i n ) , i =1 , , ..., M } . Now, we consider the discrete curve E Γ along the i -direction, see figure 4, and the actionfunctional is given by S E Γ [ x ( n )] = ... + T − i L ( i ) + L ( i ) + ... , (3.42) The term “1-form” here may be not yet relevant at this discrete level. It will become clear in later section on thecontinuous-time level. However, we can think that the term “1-form” indicates the fact that there is only 1 discretevariable active on each discrete element. The antisymmetric property indicates the direction of evolution on discrete element. i x n i − n i − n i n i + 1 n i + 2 T − i x T − i x xx + δx T i x T i xE Γ E ′ Γ ΓFigure 4: The local deformation at point x on discrete curve E Γ . where T − i L ( i ) ≡ L ( i ) ( T − i x, x ) and L ( i ) ≡ L ( i ) ( x, T i x ). Under local deformation E Γ → E ′ Γ such that x → x + δx , the action functional of a new discrete curve E ′ Γ is given by S E ′ Γ [ x ( n )] = ... + T − i L ′ ( i ) + L ′ ( i ) + ... , (3.43)where T − i L ′ ( i ) ≡ L ( i ) ( T − i x, x + δx ) and L ′ ( i ) ≡ L ( i ) ( x + δx, T i x ). The variation of actionbetween these two actions is S E ′ Γ [ x ( n )] − S E Γ [ x ( n )] = T − i L ′ ( i ) + L ′ ( i ) − T − i L ( i ) − L ( i ) . (3.44)Using Taylor expansion with respect to δx and keeping only the first-order contribution, weobtain S E ′ Γ [ x ( n )] − S E Γ [ x ( n )] ≡ δS E Γ [ x ( n )] = δx ∂ T − i L ( i ) ∂x + ∂ L ( i ) ∂x ! . (3.45)According to the least action principle: δS E Γ = 0, with the conditions δ T − i x = δ T i x = 0, thenwe find that ∂ T − i L ( i ) ∂x + ∂ L ( i ) ∂x = 0 , (3.46)since δx = 0. This is the discrete-time Euler-Lagrange equation . In fact, we could have M ( i = 1 , , , ..., M ) copies of (3.46) for each discrete direction.Next, we consider a bit more complicate discrete curve E Γ that lives on ( N + M )-dimensionalspace of dependent variables, see figure 5.The action functional is given by S E Γ1 [ x ( n )] = L ( j ) ( x, T j x ) + L ( i ) ( T j x, T i T j x ) . (3.47) i n j x E Γ E Γ E ′ Γ E ′ Γ Γ Γ x T i x T i x + δ T i x T i T j x T j x T j x + δ T j x ( n i + 1 , n j )( n i , n j ) ( n i + 1 , n j + 1)( n i , n j + 1)Figure 5: The local deformation of corner curves on the space of dependent variables. Then we consider the local deformation such that T j x → T j x + δ T j x producing a new discretecurve E ′ Γ with the action functional S E ′ Γ1 [ x ( n )] = L ( j ) ( x, T j x + δ T j x ) + L ( i ) ( T j x + δ T j x, T i T j x ) . (3.48)We do the Taylor expansion with respect to δ T j x in (3.48) and the variation of the action δS E Γ1 [ x ( n )] ≡ S E ′ Γ1 [ x ( n )] − S E Γ1 [ x ( n )] is δS E Γ1 [ x ( n )] = δ T j x (cid:18) ∂ L ( j ) ( x, T j x ) ∂ T j x + ∂ L ( i ) ( T j x + , T i T j x ) ∂ T j x (cid:19) . (3.49)Again, the least action principle requires δS E Γ1 = 0 with the end point conditions δx = δ T i T j x =0, resulting in ∂ L ( j ) ( x, T j x ) ∂ T j x + ∂ L ( i ) ( T j x, T i T j x ) ∂ T j x = 0 , i, j = 1 , , , ..., M , (3.50)which are the corner Euler-Lagrange equations . In fact, these Euler-Lagrange equations producethe equations of motion, called the constraint equations , which tell us how the system evolvesfrom discrete i -direction to the discrete j -direction. Similarly, the variation of action functionalof E Γ expressed as S E Γ2 [ x ( n )] = L ( i ) ( x, T i x ) + L ( j ) ( T i x, T i T j x ) (3.51)produces constraint equations in the form of ∂ L ( i ) ( x, T i x ) ∂ T i x + ∂ L ( j ) ( T i x, T i T j x ) ∂ T i x = 0 , i, j = 1 , , , ..., M , (3.52) escribing the evolution from discrete j -direction to discrete i -direction. These corner equationsfirst appeared in [14, 31]. In the previous section, we consider the variational principle of the discrete curve E Γ with re-spect to dependent variables and we obtain two types of discrete-time Lagrangian equations.The first one, Euler-Lagrange equation, tells us how the system interpolates on a certain discretedirection and the second one, constraint equation, tells us how the system changes the discretedirection from one to another.Here, in this section, we will consider the discrete curve Γ which lives on the space of inde-pendent variables, see figure 4. Actually, we can see that the discrete curve Γ is the projectionof the discrete curve E Γ . In figure 5, the action functional of the curve Γ and Γ are given by S Γ [ x ( n )] = L ( j ) ( x, T j x ) + L ( i ) ( T j x, T i T j x ) , (3.53)and S Γ [ x ( n )] = L ( i ) ( x, T i x ) + L ( j ) ( T i x, T i T j x ) . (3.54)We find that the discrete curve Γ can be obtained by locally deforming the curve Γ such that( n i + 1 , n j ) → ( n i , n j + 1), and vice versa. The least action principle, ∆ S Γ = S Γ − S Γ = 0,gives us immediately0 = L ( j ) ( x, T j x ) − L ( i ) ( x, T i x ) − L ( j ) ( T i x, T i T j x ) + L ( i ) ( T j x, T i T j x ) , (3.55)or in short 0 = L ( j ) − L ( i ) − T i L ( j ) + T j L ( i ) , i = j = 1 , , ...M , (3.56)where T i L ( j ) ≡ L ( j ) ( T i x, T i T j x ) and T j L ( i ) ≡ L ( i ) ( T j x, T i T j x ). Equations (3.56) are calledthe . These equations ensure the invariance of action betweentwo arbitrary discrete curves sharing the same end points on the space of independent variables. In this section, we give a concrete example of the Lagrangian 1-form structure of the discrete-time Calogero-Moser system [14]. The Lagrangian is given by L ( j ) ( x, T j x ) = N X m,l =1 log | x m − T j x l | − N X m,l =1 m = l (cid:20) log | x m − x l | + log | T j x m − T j x l | (cid:21) − p j N X m =1 ( x m − T j x m ) , j = 1 , , (3.57)where x m ( n , n ) is the position of the m th particle, N is the number of the particles in thesystem and p j is the lattice parameter. Equation of motion : Suppose that the action of two discrete elements along the j -directionis given by S n j = T − j L ( j ) + L ( j ) , j = 1 , , (3.58) here L ( j ) ≡ L ( j ) ( x, T j x ) and T − j L ( j ) ≡ L ( j ) ( T − j x, x ), resulting in the Euler-Lagrangeequation for j -direction ∂ T − j L ( j ) ∂x m + ∂ L ( j ) ∂x m = 0 . (3.59)Substituting (3.57) into (3.59), we obtain N X l =1 " x m − T j x l + 1 x m − T − j x l − N X l =1 m = l x m − x l = 0 , j = 1 , , m = 1 , , ..., N , (3.60)which are the discrete-time equations of motion for Calogero-Moser system along the j -direction[30]. Constraint equation : There are four different types of discrete trajectories around the cor-ners, see figure 6. T xx T x (a) T − xx T − x (b) T xx T − x (c) T − xx T x (d) Figure 6: Discrete-time evolutions around the corners.
The actions of each configurations around point x are given by S a = L (2) ( T x, x ) + L (1) ( x, T x ) , (3.61) S b = L (1) ( T − x, x ) + L (2) ( x, T − x ) , (3.62) S c = L (1) ( T − x, x ) + L (2) ( x, T x ) , (3.63) S d = L (2) ( T − x, x ) + L (1) ( x, T x ) . (3.64)The local variation at point x of each action gives us ∂ L (2) ( T x, x ) ∂x + ∂ L (1) ( x, T x ) ∂x = 0 , (3.65a) ∂ L (1) ( T − x, x ) ∂x + ∂ L (2) ( x, T − x ) ∂x = 0 , (3.65b) ∂ L (1) ( T − x, x ) ∂x + ∂ L (2) ( x, T x ) ∂x = 0 , (3.65c) ∂ L (2) ( T − x, x ) ∂x + ∂ L (1) ( x, T x ) ∂x = 0 , (3.65d) espectively. Equation (3.65) gives us a set of equations p − p = N X l =1 (cid:18) x m − T x l − x m − T x l (cid:19) , (3.66a) − ( p − p ) = N X l =1 (cid:18) x m − T − x l − x m − T − x l (cid:19) , (3.66b) − ( p − p ) = N X l =1 (cid:18) x m − T − x l + 1 x m − T x l (cid:19) − N X l =1 m = l x m − x l , (3.66c) p − p = N X l =1 (cid:18) x m − T − x l + 1 x m − T x l (cid:19) − N X l =1 m = l x m − x l . (3.66d)Equation (3.66) is a set of constraint equations corresponding to discrete evolutions in figure 6,respectively. Legendre transformation : With the Lagrangian defined in (3.57), we find that their as-sociated momentum variable is given by T j P ml = − ∂ L ( j ) ∂ T j x l = N X m =1 x m − T j x l − N X m =1 m = l T j x m − T j x l − p j . (3.67)We now introduce two extra variables P ml and ρ ml such that T j P ml ≡ x m − T j x l and T j ρ ml ≡ − T j x m − T j x l , (3.68)together with the Legendre transformation H ( j ) ( T j P ml , T j ρ ml , x ) = N X m,l =1 T j P ml ( x m − T j x l ) + 12 N X m,l =1 m = l T j ρ ml ( T j x m − T j x l ) − L ( j ) ( x, T j x ) . (3.69)Obviously, the relation (3.69) is not the same with those in [30] since the extra term appearsin the Lagrangian (3.57). Furthermore, Nijhoff and Pang pointed out that H ( j ) is not theHamiltonian in the usual sense, see [30], but rather be the Hamiltonian generator, see section2.3. However, we insist to use (3.69) as the Legendre transformation. Substituting (3.57) into(3.69), we obtain H ( j ) ( T j P ml , T j ρ ml , x ) = N X m,l =1 log | T j P ml | − N X m,l =1 m = l log | T j ρ ml | + 12 N X m,l =1 m = l log | x m − x l | + p j N X m =1 T j P mm . (3.70) he action can be rewritten in terms of Hamiltonian as S n j = N X m,l =1 T j P ml ( x m − T j x l ) + 12 N X m,l =1 m = l T j ρ ml ( T j x m − T j x l ) − H ( j ) ( T j P ml , T j ρ ml , x ) . (3.71)We consider the variation of action with respect to dependent variables, i.e., T j P ml → T j P ml + δ T j P ml , T j ρ ml → T j ρ ml + δ T j ρ ml and x m → x m + δx m , resulting in δS n j = N X m,l =1 δ T j P ml (cid:20) ( x m − T j x l ) − ∂ H ( j ) ∂ T j P ml (cid:21) + N X m,l =1 m = l δ T j ρ ml (cid:20)
12 ( T j x m − T j x l ) − ∂ H ( j ) ∂ T j ρ ml (cid:21) + N X m =1 δx m N X l =1 ( T j P ml − P ml ) + 12 N X l =1 m = l ( ρ ml − ρ lm ) − ∂ H ( j ) ∂x m . (3.72)Imposing the least action condition: δS = 0, we obtain ∂ H ( j ) ∂ T j P ml = x m − T j x l , (3.73) ∂ H ( j ) ∂ T j ρ ml = 12 ( T j x m − T j x l ) , m = l , (3.74) ∂ H ( j ) ∂x m = N X l =1 ( T j P ml − P ml ) + 12 N X l =1 m = l ( ρ ml − ρ lm ) . (3.75)These equations are the discrete-time Hamilton’s equations. Equations (3.73) and (3.74) giveus back the definition of the extra variables in (3.68). Equation (3.75) gives us precisely theequation of motion (3.60).Using (3.59) and (3.67), all momentum variables are given by ∂ L (1) ∂x l = P ml = N X l =1 x m − T x l − N X l =1 m = l x m − x l − p , (3.76a) ∂ L (2) ∂x l = P ml = N X l =1 x m − T x l − N X l =1 m = l x m − x l − p , (3.76b) − T − (cid:18) ∂ L (1) ∂ T x l (cid:19) = P ml = − N X l =1 x m − T − x l + N X l =1 m = l x m − x l − p , (3.76c) − T − (cid:18) ∂ T L (2) ∂ T x l (cid:19) = P ml = − N X l =1 x m − T − x l + N X l =1 m = l x m − x l − p . (3.76d) lternatively, the constraint equation (3.66a) can also be acquired by equating (3.76a) with(3.76b). Similarly, equating (3.76c) with (3.76d) results in another constraint equation (3.66b).The last two constraint equations (3.66c) and (3.66d) can be obtained by the equating (3.76c)with (3.76b), and (3.76d) with (3.76a), respectively. Closure relation : In [14], it has been shown that the discrete-time rational Calogero-Mosersystem possesses the discrete closure relation0 = L (2) − L (1) − T L (2) + T L (1) . (3.77)A crucial ingredient to show the existence of closure relation is the connection between discreteLagrangian (3.57) and temporal part of the Lax matrices (2.37) L ( j ) ( x, T j x ) = ln | det M j | + p j N X m =1 ( x m − T j x m ) , j = 1 , . (3.78)The compatibility between the matrices M and M results in( T M ) M = ( T M ) M . (3.79)From this equation, we fine thatln | det T M | + ln | det M | = ln | det T M | + ln | det M | . (3.80)We observe that (3.80) is precisely the discrete closure relation with N X m =1 ( T x m − T T x m − x m + T x m ) = 0 , (3.81)which is an extra relation of center of mass of the system. Remark.
Discrete multidimensional consistency: We knew that the compatibility between dif-ferent temporal Lax matrices (3.79) gives the constraint equations (3.66a) and (3.66b) . Thisrelation tells us that it does not matter whether the system will go from point x to point T T x ,see figure 5, through point T x in curve E Γ , or point T x in curve E Γ . Then, (3.79) ex-hibits the multidimensional consistency on the level of dependent variables. Furthermore, theprojection of E Γ and E Γ gives Γ and Γ , respectively, on the space of independent variables.Therefore, these two curves Γ and Γ relate to each other by the closure relation which ex-hibits the multidimensional consistency on the level of independent variables. Then, this seemsto suggest that the existence of (3.79) implies the existence of (3.55) , and vice versa through (3.80) Suppose a set of generalised coordinates is a function of discrete and continuous variables x ≡ x( n, τ ) = (x , x , ..., x N ), where ( n, τ ) = ( n , n , ..., τ , τ , .... ). The definition of discrete shiftoperators in the previous section is still applicable. Now, we consider curve E Γ that consists ofone discrete element and one continuous element shown in figure 7. Then, the action functionalassociated with this curve is given by S E Γ [x( n, τ )] = X n i L ( i ) (x , T i x) + X i =1 Z τ ′′ τ ′ d τ i L ( τ i ) (cid:18) x , T k x , ∂ x ∂τ , ∂ x ∂τ , ... (cid:19) , (3.82)where L ( i ) is the discrete-time Lagrangian defined in the previous section and L ( τ i ) is the La-grangian associated with the continuous time τ i containing both discrete shifts and derivatives. .2.1 Variation on dependent variables The situation now is that we have a curve constituted from discrete and continuous elements.Then, it is possible to consider the variation on discrete and continuous elements separately.Of course, if we consider only the discrete elements, we can obtain the result in the previoussection and we won’t repeat it here. We now start with this action functional S [x( n, τ )] = X i =1 Z τ ′′ τ ′ d τ i L ( τ i ) (cid:18) x , T k x , (cid:26) ∂ x ∂τ j (cid:27)(cid:19) , j = 1 , , ... . (3.83)The variations of the variables x → x + δ x, T k x → T k x + δ T k x and ∂ x ∂τ j → ∂ x ∂τ j + δ ∂ x ∂τ j , result ina curve, see figure 7, with the action functional S ′ [x( n, τ )] = X i =1 Z τ ′′ τ ′ d τ i L ( τ i ) (cid:18) x + δ x , T k x + δ T k x , (cid:26) ∂ x ∂τ j + δ ∂ x ∂τ j (cid:27)(cid:19) , j = 1 , , ... . (3.84)We then do the Taylor expansion of (3.84) and keep only the first two terms in the expansion.The variation of action between (3.83) and (3.84) is δS = N X i =1 Z τ ′′ τ ′ d τ i (cid:20) δ ( T k x) ∂ L ( τ i ) ∂ T k x + δ x ∂ L ( τ i ) ∂ x (cid:21) + N X i =1 Z τ ′′ τ ′ d τ i δ (cid:18) ∂ x ∂τ i (cid:19) ∂ L ( τ i ) ∂ (cid:16) ∂ x ∂τ i (cid:17) + N X i,j =1 i = j Z τ ′′ τ ′ d τ i δ (cid:18) ∂ x ∂τ j (cid:19) ∂ L ( τ i ) ∂ (cid:16) ∂ x ∂τ j (cid:17) . (3.85)Integrating by parts the second term and imposing the end-point conditions δ x( τ ′ ) = δ x( τ ′′ ) = 0,we obtain δS = N X i =1 Z τ ′′ τ ′ d τ i (cid:20) δ ( T k x) ∂ L ( τ i ) ∂ T k x (cid:21) + N X i =1 Z τ ′′ τ ′ d τ i ∂ L ( τ i ) ∂ x − ∂ L ( τ i ) ∂ (cid:16) ∂ x ∂τ i (cid:17) δ x+ N X i,j =1 i = j Z τ ′′ τ ′ d τ i δ (cid:18) ∂ x ∂τ j (cid:19) ∂ L ( τ i ) ∂ (cid:16) ∂ x ∂τ j (cid:17) . (3.86)The least action principle, δS = 0, gives us ∂ L ( τ i ) ∂ x − ∂∂τ i ∂ L ( τ i ) ∂ (cid:16) ∂ x ∂τ i (cid:17) = 0 , i = 1 , , ... , (3.87)which are the Euler-Lagrange equations. Furthermore, we also have extra equations ∂ L ( τ i ) ∂ T k x = 0 and ∂ L ( τ i ) ∂ (cid:16) ∂ x ∂τ j (cid:17) = 0 , where i = j , (3.88)since δ ( T k x) and δ (cid:16) ∂ x ∂τ j (cid:17) are nonzero, respectively. These equations are in fact the constraintsin this situation. The first one tells us how the system evolves from discrete k -direction to thecontinuous i -direction while the second one tells us how the system evolves from the continuous i -direction to the continuous j -direction. k τ i x E Γ E Γ Γ Γ x( n k , τ i ) x( n k + 1 , τ i )x( n k + 1 , τ i + δτ i )x( n k , τ i + δτ i ) ( n k + 1 , τ i )( n k , τ i ) ( n k + 1 , τ i + δτ i )( n k , τ i + δτ i )Figure 7: The deformation of the semi-discrete curve. Note that here we use x ( n k , τ i ) ≡ x ( ..., n k , ..., τ i , ... ). The curve Γ is the projection of the curve E Γ onto the space of independent variables and theaction functional is S Γ [x( n, τ )] = Z τ i + δτ i τ i L ( τ i ) (x( n k , τ i ) , x( n k , τ i + δτ i )) d τ i + L ( k ) (x( n k , τ i + δτ i ) , x( n k + 1 , τ i + δτ i )) , (3.89)where the first term represents the evolution along the vertical line and second term representsthe evolution along the horizontal line of the curve Γ . We now introduce another curve Γ which shares the end points with the previous curve Γ . The action functional is given by S Γ [x( n, τ )] = L ( k ) (x( n k , τ i ) , x( n k + 1 , τ i ))+ Z τ i + δτ i τ i L ( τ i ) (x( n k + 1 , τ i )) , x( n k + 1 , τ i + δτ i )) d τ i , (3.90)where the first term represents the evolution along the horizontal line, while, the second termrepresents the evolution along the vertical line of the curve Γ . The variation between twoactions is given by S Γ − S Γ = L ( k ) (x( n k , τ i ) , x( n k + 1 , τ i )) − L ( k ) (x( n k , τ i + δτ i ) , x( n k + 1 , τ i + δτ i ))+ Z τ i + δτ i τ i [ L ( τ i ) (x( n k + 1 , τ i ) , x( n k + 1 , τ i + δτ i )) − L ( τ i ) (x( n k , τ i ) , x( n k , τ i + δτ i ))] d τ i . (3.91) sing the Taylor expansion with respect to δτ i and keeping only for the first two terms in theexpansion, we have now δS ≡ S Γ − S Γ = − δτ i ∂∂τ i L ( k ) (x( n k , τ i ) , x( n k + 1 , τ i )) + Z τ i + δτ i τ i [ L ( τ i ) (x( n k + 1 , τ i )) , x( n k + 1 , τ i + δτ i )) − L ( τ i ) (x( n k , τ i ) , x( n k , τ i + δτ i ))] d τ i = δτ i [ L ( τ i ) (x( n k + 1 , τ i )) , x( n k + 1 , τ i + δτ i )) − L ( τ i ) (x( n k , τ i ) , x( n k , τ i + δτ i ))] − δτ i ∂∂τ i L ( k ) (x( n k , τ i ) , x( n k + 1 , τ i )) . (3.92)By imposing the least action principle, δS = 0, (3.92) holds if ∂∂τ i L ( k ) (x( n k , τ i ) , x( n k + 1 , τ i )) = L ( τ i ) (x( n k + 1 , τ i )) , x( n k + 1 , τ i + δτ i )) − L ( τ i ) (x( n k , τ i ) , x( n k , τ i + δτ i )) , or in short ∂ L ( k ) ∂τ i = T k L ( τ i ) − L ( τ i ) , i = 1 , , ... , (3.93)which is the semi-discrete-time closure relation. Again, this relation tells us that the action isinvariant under the local deformation of the curve on the space of independent variables. We present the semi-discrete time rational Calogero-Moser system [14], which possesses the semi-discrete time closure relation (3.93). The Lagrangian associated with the continuous variable τ is given by L ( τ ) (cid:18) x , T k x , ∂ T k x ∂τ (cid:19) = − N X m,l =1 ∂ T k x l ∂τ m − T k x l − N X m,l =1 m = l (cid:18) ∂ T k x m ∂τ − ∂ T k x l ∂τ (cid:19) T k x m − T k x l + N X m =1 (cid:18) x m − T k x m + ∂ T k x m ∂τ (cid:19) , (3.94)which has both discrete- and continuous-time variables. Equation of motion : Substituting the Lagrangian (3.94) into (3.87), we obtain N X l =1 " ∂ T k x l ∂τ m − T k x l ) − ∂ T − k x l ∂τ m − T − k x l ) = 0 , m = 1 , , ...N , (3.95)which are the equations of motion along the continuous variable τ . Constraint equation : Since the Lagrangian (3.94) contains two type of discrete variables,namely T − k x and T k x, we have ∂ L τ i ∂ T − k x = 0 , hich gives − N X l =1 ∂ T k x l ∂τ m − T k x l ) , (3.96)and ∂ L τ i ∂ T k x = 0 , which gives − N X l =1 ∂ T − k x l ∂τ (cid:0) x m − T − k x l (cid:1) . (3.97)Equations (3.96) and (3.97) are the constraints on semi-discrete time level. Combining (3.96)and (3.97) together, we obtain the semi-discrete equations of motion (3.95). Legendre transformation : In the semi-discrete situation, we have both discrete and semi-discrete Lagrangian. The Legendre transformation for discrete Lagrangian has already beengiven in (3.69) and Legendre transformation for semi-discrete Lagrangian is given by H ( τ ) ( T k P ml , T k ρ ml , T k ν ml , T k Ω ml , x) = N X m,l =1 (cid:20) T k ν ml (x m − T k x l ) − T k P ml ∂ T k x l ∂τ (cid:21) + 12 N X m,l =1 m = l " T k Ω ml ( T k x m − T k x l )+ T k ρ ml (cid:18) ∂ T k x m ∂τ − ∂ T k x l ∂τ (cid:19) − L ( τ ) (cid:18) x , T k x , ∂ T k x ∂τ (cid:19) , (3.98)where T k P ml = 1x m − T k x j , (3.99a) T k ρ ml = − T k x m − T k x l , (3.99b) T k ν ml = ∂ T k x l ∂τ m − T k x l ) , (3.99c) T k Ω ml = (cid:18) ∂ T k x m ∂τ − T k x l ∂τ (cid:19) T k x m − T k x l ) . (3.99d)Using (3.98) with (3.94), we obtain H ( τ ) ( T k P ml , T k ρ ml , T k ν ml , T k Ω ml , x) = N X m,l =1 T k ν ml T k P ml − N X m,l =1 m = l T k Ω ml T k ρ ml − N X m =1 " T k P mm + p T k ν mm ( T k P mm ) , (3.100) hich is the semi-discrete time Hamiltonian associated with the continuous time variable τ .The action associated with the continuous curve is given by S ( τ ) = Z τ τ d τ " N X m,l =1 (cid:20) T k ν ml (x m − T k x l ) − T k P ml ∂ T k x l ∂τ (cid:21) + 12 N X m,l =1 m = l " T k Ω ml ( T k x m − T k x l ) + T k ρ ml (cid:18) ∂ T k x m ∂τ − ∂ T k x l ∂τ (cid:19) − H ( τ ) ( T k P ml , T k ρ ml , T k ν ml , T k Ω ml , x) . (3.101)The variation of action (3.101) with T k ν ml → T k ν ml + δ T k ν ml , T k P ml → T k P ml + δ T k P ml , T k ρ ml → T k ρ ml + δ T k ρ ml , T k Ω ml → T k Ω ml + δ T k Ω ml and x m → x m + δ x m results in δS ( τ ) = Z τ τ d τ N X m,l =1 δ T k ν ml (cid:20) (x m − T k x l ) − ∂ H ( τ ) ∂ T k ν ml (cid:21) + N X m,l =1 δ T k P ml (cid:20) − ∂ T k x l ∂τ − ∂ H ( τ ) ∂ T k P ml (cid:21) + N X m,l =1 m = l δ T k Ω ml (cid:20)
12 ( T k x m − T k x l ) − ∂ H ( τ ) ∂ T k Ω ml (cid:21) + N X m,l =1 m = l δ T k ρ ml (cid:20) (cid:18) ∂ T k x m ∂τ − ∂ T k x l ∂τ (cid:19) − ∂ H ( τ ) ∂ T k ρ ml (cid:21) + N X m =1 δ x m " N X l =1 (cid:20) ( T k ν ml − ν ml ) − ∂ P ml ∂τ (cid:21) + 12 N X l =1 m = l (cid:20) (Ω ml − Ω lm ) − (cid:18) ∂ρ ml ∂τ − ∂ρ lm ∂τ (cid:19)(cid:21) − ∂ H ( τ ) ∂ x m . (3.102)According to the condition δS = 0, we obtain ∂ H ( τ ) ∂ T k ν ml = x m − T k x l , (3.103a) ∂ H ( τ ) ∂ T k P ml = − ∂ T k x l ∂τ , (3.103b) ∂ H ( τ ) ∂ T k Ω ml = 12 ( T k x m − T k x l ) , (3.103c) ∂ H ( τ ) ∂ T k ρ ml = 12 (cid:18) ∂ T k x m ∂τ − ∂ T k x l ∂τ (cid:19) , (3.103d) ∂ H ( τ ) ∂ x m = N X l =1 (cid:20) ( T k ν ml − ν ml ) − ∂ P ml ∂τ (cid:21) + 12 N X l =1 m = l (cid:20) (Ω ml − Ω lm ) − (cid:18) ∂ρ ml ∂τ − ∂ρ lm ∂τ (cid:19)(cid:21) , (3.103e)which are the Hamilton’s equations for τ -direction in semi-discrete time level and (3.103e)produces the equations of motion in semi-discrete time level (3.95). .3 Continuous-time Lagrangian 1-form structure In this section, we will investigate the variational principle for the continuous-time Lagrangian 1-form structure. Suppose now we have a set of generalised coordinates X ( t ) ≡ ( X ( t ) , X ( t ) , ..., X N ( t ))and a set of time variables t ( s ) ≡ ( t ( s ) , t ( s ) , t ( s ) , ..., t N ( s )) which are parameterised by theparameter s with the boundary: s ≤ s ≤ s . The action functional for this case is written inthe form S Γ [ X ( t )] = Z Γ N X i =1 L ( t i ) d t i ! = Z s s d sL ( s ) , (3.104)where L ( s ) ≡ L ( t i ) [ X ( t ( s )) , { X ( j ) ( t ( s )) } ] d t i / d s is the multi-time Lagrangian and X ( j ) ( t ) ≡ d X/ d t j , where j = 1 , , ..., N . The curve Γ is on the space of independent variables startingfrom point t ( s ) to point t ( s ), see figure 8. t i X ( t ) t j E Γ E Γ ′ Γ Γ ′ X ( t ( s )) X ( t ( s )) t ( s ) t ( s )Figure 8: The curve Γ and E Γ in the X − t configuration. We consider the variation of the curve Γ → Γ ′ on the space of independent variables while theend points are fixed as shown in figure 8. By doing so, it is convenient to write the Lagrangianas a function of times, i.e., L ( t i ) ≡ L ( t i ) ( t ). We then perform the variation t ( s ) → t ( s ) + δt ( s )resulting in a new curve Γ ′ . A new action functional is S Γ ′ [ X ( t + δt )] = Z s s d s N X i =1 L ( t i ) ( t + δt ) d ( t i + δt i ) d s ! . (3.105) erforming the Taylor expansion and keeping only the first two contributions in the series, wefind that the variation of action is δS Γ ≡ S Γ ′ − S Γ = Z s s d s N X i,j =1 δt j ∂ L ( t i ) ∂t j d t i d s + N X i =1 L ( t i ) d δt i d s . (3.106)Using integration by parts on the second term of (3.106), with conditions δt ( s ) = δt ( s ) = 0,we have δS Γ = Z s s d s N X i,j =1 δt j ∂ L ( t i ) ∂t j d t i d s − N X i =1 d L ( t i ) d s δt i . (3.107)Next, using the chain rule relation d L ( t i ) d s = N X j =1 ∂ L ( t i ) ∂t j d t j d s , (3.108)equation (3.107) can be rewritten as δS Γ = Z s s d s N X i,j =1 i = j δt i (cid:18) ∂ L ( t i ) ∂t j − ∂ L ( t j ) ∂t i (cid:19) d t j d s . (3.109)From the least action principle: δS Γ = 0, we obtain the relation ∂ L ( t i ) ∂t j = ∂ L ( t j ) ∂t i , i, j = 1 , , , ..., N and i = j . (3.110)Equations (3.110) are called the continuous-time closure relations which guarantee the invari-ance of action under the local deformation of a curve Γ on the space of independent variables. Remark : We have seen that the variation of the action gives the closure relation. Here, we areapproaching the problem from different perspective, namely from geometric point of view.Suppose that α is a differential (k-1)-form. The generalised Stokes’ theorem states that theintegral of its exterior derivative over the surface of smooth oriented k-dimensional manifold Ω is equal to its integral of along the boundary ∂ Ω of the manifold Ω [32]: Z ∂ Ω α = Z Ω dα . (3.111) We now introduce an object dS given by dS = N X i =1 L ( t i ) dt i , (3.112) as a 1-form on the N -dimensional space of independent variables and, therefore, the action (3.104) becomes S = R Γ dS . Applying an exterior derivative to the smooth function coefficientswhich, in this case, is the Lagrangian, (3.111) becomes I ∂ Ω N X i =1 L ( t i ) dt i = Z Z Ω N X ≤ i In the previous section, a new feature of integrability, called the closure relation, is derivedfor the system with Lagrangian hierarchy from the point of view of the variational principle.We know that basically we can obtain the Hamiltonian from Lagrangian through the Legendretransformation. The action functional is then written in terms of the Hamiltonian and thevariation can be performed with respect to variables on phase space resulting in the Hamilton’sequations. In this section, we set out to construct the Legendre transformation to obtain theHamiltonian hierarchy and of course consider the variational principle on the phase space. To establish the Legendre transformation, we multiply d X/ d s to the Euler-Lagrange equation(3.123) d X d s N X i =1 ∂ L ( t i ) ∂X d t i d s − d X d s N dd s N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s = 0 . (4.133)and introduce the relation d X d s dd s N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s = dd s d X d s N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s − d X d s N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s . (4.134)Using (4.134), (4.133) can be rewritten as0 = d X d s N X i =1 ∂ L ( t i ) ∂X d t i d s + 1 N d X d s N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s − N dd s d X d s N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( i ) d t i / d s d t j / d s . (4.135)Next we consider the relation dd s N X i =1 L ( t i ) d t i d s ! = N X i =1 (cid:18) ∂ L ( t i ) ∂X d X d s d t i d s + ∂ L ( t i ) ∂X ( i ) d X ( i ) d s d t i d s (cid:19) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d X ( j ) d s d t i d s + N X i =1 L ( t i ) d t i d s . (4.136) mposing d t i / d s = 0 and using constraint (3.124), equation (4.136) becomes dd s N X i =1 L ( t i ) d t i d s ! = d X d s N X i =1 ∂ L ( t i ) ∂X d t i d s + 1 N d X d s N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s . (4.137)Substituting (4.137) into (4.135), we obtain dd s N d X d s N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s − N X i =1 L ( t i ) d t i d s = 0 . (4.138)We observe that the term inside the square bracket must be a constant with respect to theparameter s . We then define the momentum variable as P ≡ N N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s , (4.139)and introduce the energy function E ≡ N X i =1 H ( t i ) ( X, P ) d t i d s , (4.140)which is conserved under parametrised time translation, i.e., dd s N X i =1 H ( t i ) ( X, P ) d t i d s ! = 0 . (4.141)Therefore, what we have now are H ( t i ) = X ( i ) P − L ( t i ) , i = 1 , , ..., N , (4.142)which are the Legendre transformations. This means that, with explicit form of the Lagrangianhierarchy, one can obtain the Hamiltonian hierarchy. With the Legendre transformation (4.142), the action functional becomes S E Γ [ X ( t ) , P ( t )] = Z s s d s N X i =1 (cid:0) X ( i ) P − H ( t i ) ( X, P ) (cid:1) d t i d s , (4.143)where the curve E Γ is defined on the phase space, see figure 10. P M N E Γ E ′ Γ X ( t ( s )) X ( t ( s )) P ( t ( s )) P ( t ( s ))Figure 10: The trajectory of the system defined by a curve E Γ on 2 N -dimensional manifold M N embedded in the phase space. In this situation, we have 2 N variables to work with. We then consider the variations X → X + δX and P → P + δP with the fixed end-point conditions resulting in a new curve E ′ Γ withthe action functional S E ′ Γ [ X + δX, P + δP ] = Z s s d s N X i =1 (cid:0) ( X ( i ) + δX ( i ) )( P + δP ) − H ( t i ) ( X + δX, P + δP ) (cid:1) d t i d s . (4.144)Performing the Taylor expansion and keeping the first two contributions in the series, thevariation of action is given by δS E Γ ≡ δS E ′ Γ − δS E Γ = Z s s d s N X i =1 (cid:18) δX ( i ) P + X ( i ) δP − δP ∂ H ( t i ) ∂P − δX ( i ) ∂ H ( t i ) ∂X ( i ) (cid:19) d t i d s . (4.145)Integrating by parts the first and fourth terms in the bracket and using the end-point conditions, δX ( t ( s )) = δX ( t ( s )) = 0, we obtain δS E Γ = Z s s d s N X i =1 (cid:20) δP (cid:18) d X d s − ∂ H ( t i ) ∂P d t i d s (cid:19) − δX (cid:18) d P d s + ∂ H ( t i ) ∂P d t i d s (cid:19)(cid:21) . (4.146)Imposing the least action principle: δS E Γ = 0, we obtain d X d s = N X i =1 ∂ H ( t i ) ∂P d t i d s , (4.147a) − d P d s = N X i =1 ∂ H ( t i ) ∂X d t i d s , (4.147b)since δX = 0 and δP = 0 and they are all independent to each other. Equations (4.147) arenothing but the generalised Hamilton’s equations. .2.2 Variation on independent variables In this case, the time variables are embedded on the 2 N -dimensional manifold M N and, then,they cannot be visualised explicitly. However, we still can consider the variation of the timevariables such that t ( s ) → t ( s ) + δt ( s ) with conditions t ( s ) = t ( s ) = 0. The variation of theaction is given by δS = S [ X ( t ( s ) + δt ( s )) , P ( t ( s ) + δt ( s ))] − S [ X ( t ( s ) , P ( t ( s ))]= Z s s d s " N X i =1 (cid:18) P dd s (cid:0) X ( i ) δt i (cid:1) + d X d s (cid:18) δt i ∂P∂t i (cid:19)(cid:19) − N X i =1 (cid:18) δt i ∂ H ( t i ) ∂t i + H ( t i ) d δt i d s (cid:19) − N X i,j =1 i = j δt j ∂ H ( t i ) ∂t j d δt i d s . (4.148)Integrating by parts the first and the fourth terms, we obtain δS = Z s s d s N X i =1 δt i " N X j =1 i = j (cid:18) ∂P∂t i ∂X∂t j − ∂P∂t j ∂X∂t i + ∂ H ( t i ) ∂t j − ∂ H ( t j ) ∂t i (cid:19) d t j d s . (4.149)From the relation ∂P∂t i ∂X∂t j − ∂P∂t j ∂X∂t i = ∂ H ( t i ) ∂t j − ∂ H ( t j ) ∂t i , (4.150)and, imposing the least action principle: δS = 0, we obtain ∂ H ( t i ) ∂t j − ∂ H ( t j ) ∂t i = 0 , i = j = 1 , , ..., N . (4.151)Equations (4.151) give the characteristic feature of the evolution on the phase space called thecommuting Hamiltonian flows. Here we show that we can obtain the involution directly fromthe variational principle instead of using Poisson structure and Lax matrices [34]. In this section, we work out explicitly on the Legendre transformation and variation of theaction on phase space for the rational Calogero-Moser system. Legendre transformation : With the first two Lagrangians (3.127) in the hierarchy, the Leg-endre transformations are H ( t ) = N X i =1 P i ∂X i ∂t − L ( t ) , (4.152a) H ( t ) = N X i =1 P i ∂X i ∂t − L ( t ) , (4.152b) here P i = ∂X i /∂t is the momentum. From the structure of (4.152), we note that the La-grangians (3.127) share the momentum variable. Substituting (3.128a) and (3.128b) into Leg-endre transformations, we obtain H ( t ) = N X i =1 P i − N X i,j =1 i = j X i − X j ) , (4.153a) H ( t ) = N X i =1 P i − N X i,j =1 i = j P i ( X i − X j ) , (4.153b)which are the first two Hamiltonians in the hierarchy. Equation of motion : With Hamiltonians in (4.153), we do have the Hamilton’s equationsas follows. For the first Hamiltonian (4.153a), we have d X i d t = ∂ H ( t ) ∂P i , (4.154a) − d P i d t = ∂ H ( t ) ∂X i , (4.154b)and, for the second Hamiltonian (4.153b), we have d X i d t = ∂ H ( t ) ∂P i , (4.154c) − d P i d t = ∂ H ( t ) ∂X i . (4.154d)Substituting the Hamiltonians (4.153a) and (4.153b) into (4.154), we obtain (3.129a) and(3.129b), respectively. Commuting flows : We now would like to show explicit proof of the relation ∂ H ( t ) ∂t = ∂ H ( t ) ∂t . (4.155)With Hamiltonians (4.153a) and (4.153b), we find that ∂ H ( t ) ∂t = N X i =1 P i ∂P i ∂t + N X i,j =1 i = j (cid:20) ∂X i ∂t X i − X j ) (cid:21) , (4.156) ∂ H ( t ) ∂t = N X i =1 P i ∂P i ∂t − N X i,j =1 i = j (cid:20) ∂P i ∂t X i − X j ) − P i ( X i − X j ) + 8 P i P j ( X i − X j ) (cid:21) . (4.157)We use (3.129a), (3.129b) and (3.131) to simplify (4.156) and (3.129a) to simplify (4.157). Thelast term of (4.157) vanishes because of the antisymmetric property. We find that (4.155) isreduced to ∂ H ( t ) ∂t − ∂ H ( t ) ∂t = N X i,j =1 i = j N X k =1 i = k X i − X j ) X i − X k ) (4.158)= N X i,j =1 i = j X i − X j ) + N X i,j,k =1 i = j = k X i − X j ) ( X i − X k ) . (4.159) he first term and the second term vanish according to the antisymmetric property, see also [14].Here we show an alternative derivation of the involution of the system through (4.155) insteadof using the Poisson bracket { H ( t i ) , H ( t j ) } = 0. In section 3.3, the local variation, where the end points of the curve are fixed, of action wasperformed resulting in the generalised Euler-Lagrange equation, the constraint equation and theclosure relation. In this section, we again consider the variation of the action but without fixedboundary conditions. We start by considering the curve E Γ which is the classical trajectory ofthe system and the action associated with it is given by S E Γ [ X ( t )] = Z s s d s N X i =1 L ( t i ) d t i d s ! , (5.160)where L ( t i ) ≡ L ( t i ) ( X ( t ( s )) , { X ( j ) ( t ( s )) } ). Suppose that there is a neighbouring curve called E ′ Γ , which is a deformation of the curve E Γ such that X ( t i ( s ) , t j ( s )) → X ( t i ( s ′ ) , t j ( s ′ )) and X ( t i ( s ) , t j ( s )) → X ( t i ( s ′ ) , t j ( s ′ )), where s ′ ≡ s + d s and s ′ ≡ s + d s , see figure 11. Anew action is given by S E ′ Γ [ X ( t ) + δX ( t )] = Z s ′ s ′ d s N X i =1 L ′ ( t i ) d t i d s ! , (5.161)where L ′ ( t i ) ≡ L ( t i ) ( X ( t ) + δX ( t ) , { X ( j ) ( t ) + δX ( j ) ( t ) } ). The variation of the action betweentwo curves is δS E Γ = S E ′ Γ [ X ( t ) + δX ( t )] − S E Γ [ X ( t )]= Z s ′ s ′ d s N X i =1 L ′ ( t i ) d t i d s ! − Z s s d s N X i =1 L ( t i ) d t i d s ! = Z s s d s N X i =1 (cid:16) L ′ ( t i ) − L ( t i ) (cid:17) d t i d s + Z s ′ s d s N X i =1 L ′ ( t i ) d t i d s − Z s ′ s d s N X i =1 L ( t i ) d t i d s . (5.162)Using Taylor expansion and substituting (3.120) in (5.162), we have δS E Γ = Z s s d s ( " N X i =1 ∂ L ( t i ) ∂X d t i d s δX + 1 N d δX d s " N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s + 1 N N X i,j =1 i = j δY ij " ∂ L ( t i ) ∂X ( j ) − ∂ L ( t j ) ∂X ( j ) − ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s + ∂ L ( t j ) ∂X ( i ) d t j / d s d t i / d s + N X k =1 k = i,j (cid:20) ∂ L ( t k ) ∂X ( i ) d t k / d s d t i / d s − ∂ L ( t k ) ∂X ( j ) d t k / d s d t j / d s (cid:21) + d s N X i =1 L ( t i ) d t i d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s s . (5.163) ntegrating by parts the second term in (5.163), we obtain δS E Γ = Z s s d s ( δX " N X i =1 ∂ L ( t i ) ∂X d t i d s − N dd s N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s ! + 1 N N X i,j =1 i = j δY ij " ∂ L ( t i ) ∂X ( i ) − ∂ L ( t j ) ∂X ( j ) − ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s + ∂ L ( t j ) ∂X ( i ) d t j / d s d t i / d s + N X k =1 k = i,j (cid:20) ∂ L ( t k ) ∂X ( i ) d t k / d s d t i / d s − ∂ L ( t k ) ∂X ( j ) d t k / d s d t j / d s (cid:21) + δX N N X i =1 ∂ L ( t i ) ∂X ( i ) + N X i,j =1 i = j ∂ L ( t i ) ∂X ( j ) d t i / d s d t j / d s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s s + d s N X i =1 L ( t i ) d t i d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s s . (5.164) t i Xt j X ( t i ( s ) , t j ( s )) X ( t i ( s ) , t j ( s )) δX ( t i ( s ) , t j ( s )) δX ( t i ( s ) , t j ( s )) d X ( t i ( s ) , t j ( s )) X ( t i ( s ′ ) , t j ( s ′ )) X ( t i ( s ′ ) , t j ( s ′ )) d X ( t i ( s ) , t j ( s )) E Γ E ′ Γ Figure 11: The variation of the curve E Γ (classical path) to E ′ Γ without end-point conditions. The coefficients of δX are generalised Euler-Lagrange equations for Lagrangian 1-form structure(3.123) and the coefficients of δY ij are the constraint equations (3.124). Thus, (5.164) becomes δS E Γ = " P δX + d s N X i =1 L ( t i ) d t i d s s s , (5.165) here P is the momentum as given in (4.139). In figure 11, we find that d X = δX + N X i =1 X ( i ) d t i d s d s . (5.166)Inserting (5.166) into (5.165), we obtain δS E Γ = " P d X − d s N X i =1 (cid:2) X ( i ) P − L ( t i ) (cid:3) d t i d s s s . (5.167)Since the second term of (5.167) is the Legendre transformation (4.142) , we obtain δS E Γ = " P d X − d s N X i =1 H ( t i ) d t i d s s s . (5.168)The least action principle δS E Γ = 0 implies that the right-hand side of (5.168) is a constantquantity defined as Q ≡ P d X − d s N X i =1 H ( t i ) d t i d s , (5.169)which is called the generalised N¨oether charge.In the case that the system goes under parametised time invariant: s → s + d s , the N¨oethercharge (5.169) becomes Q s ≡ N X i =1 H ( t i ) d t i d s . (5.170)Since any arbitrary curve Γ on the space of independent variables can be deformed such thatΓ = P Nj =1 Γ j , where Γ j is the curve that only t j is active and d t k / d s = 0 with k = j . Then theN¨oether charge (5.170) is just the linear combination of the N¨oether charge Q t i = H ( t i ) in allpossible time directions. What we have here is a set of the N¨oether charges: { Q t , Q t , ..., Q t N } which is nothing but a set of Hamiltonians: { H ( t ) , H ( t ) , ..., H ( t N ) } . Alternative method onderiving N¨oether charges, based on the notion of the variational symmetries [35] for the pluri-Lagrangian structure for one-dimensional systems (a.k.a Lagrangian 1-form) can be found in [23]. We present a recent development for a notion of integrability from the Lagrangian perspective,called Lagrangian 1-form structure which is the simplest case of the Lagrangian multiforms, inboth discrete- and continuous-time cases. The variation of the action with respect to dependentvariables gives the generalised Euler-Lagrange equations as well as the constraints. While, thevariation with respect to independent variables gives the closure relations which guarantees theinvariant of the action under the local deformation of curve in the space of independent variables.An important feature for integrability in this context is of course the closure relation which canbe considered to be a Lagrangian analogue of the commuting Hamiltonian flows (involution).Furthermore, a set of Lagrangian equations, e.g. Euler-Lagrange equations, constraints and losure relations, forms a compatible system of equations delivering the multidimensional con-sistency on the level of Lagrangians. One also find that this set of Lagrangian equations can beused to determine the explicit form of the Lagrangians for the integrable one-dimensional many-body systems, e.g. Calogero-Moser and Ruijsenaars-Schneider systems, in discrete-time case [6].Moreover, we demonstrate that, actually, instead of using the variational principle, one can usethe generalised Stokes’ theorem as an alternative way to establish the closure relation for bothcontinuous- and, consequently, discrete-time cases as the Lagrangian multiform is required tobe a closed form on the solutions of equations of motion. The Lagrangian 1-form must be closedresulting in path independent property of the action with the fixed endpoints on the space ofindependent variables. This means that there exists a bunch of homotopic paths which can becontinuously transformed to each other. This implies that the space of independent variablesmust be smooth and simply connected or cannot contain hole-like objects. We also presentthe Legendre transformation for Lagrangian hierarchy. With this Legendre transformation, theHamiltonians (invariances) can be obtained directly from the Lagrangians. The action can berewritten in terms of phase space variables and the variational principle can be considered. Thevariation of the action with respect to dependent variables gives the generalised Hamilton’sequations and the variation with respect to independent variables gives the involution conditionleading to the commuting Hamiltonian flows. This means that one can obtain Liouville integra-bility in the language of variational principle instead of Poisson bracket and Lax pair. We gofurther on relaxing the local deformation of the curve and the variation of the action results inwhat we called the generalised N¨oether charge. Of course all invariants (action variables) canbe obtained from the N¨oether charge under the translation in their associated angle variables.The question of how to capture a new notion of integrability, namely multidimensional con-sistency in quantum level, is natural to be asked. From the Hamiltonian point of view, Liouvilleintegrability, we find that naive transformation from Poisson bracket to quantum bracket is notapplicable since the encounter example was purposed by Weigert [36]. From the Lagrangianpoint of view, a recent work has been done by King and Nijhoff [37] for the discrete harmonicoscillator (quadratic Lagrangian) which is obtained from the periodic reduction of the latticeKdV equation. They found that the propagator for loop time path makes no contribution and,with fixed end points, the propagator is path independent. This, of course, can be consideredas a direct consequence of the closure relation. 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