aa r X i v : . [ m a t h - ph ] J u l Variational symmetriesand pluri-Lagrangian systems
Yuri B. Suris
Institut für Mathematik, MA 7-2, Technische Universität Berlin,Str. des 17. Juni 136, 10623 Berlin, GermanyE-mail: [email protected]
Abstract
We analyze the relation of the notion of a pluri-Lagrangian system, which recentlyemerged in the theory of integrable systems, to the classical notion of variationalsymmetry, due to E. Noether.
In the last decade, a new understanding of integrability of discrete systems as theirmulti-dimensional consistency has been a major breakthrough [8], [19]. This led to clas-sification of discrete 2-dimensional integrable systems (ABS list) [1], which turned outto be rather influential. According to the concept of multi-dimensional consistency, inte-grable two-dimensional systems can be imposed in a consistent way on all two-dimensionalsublattices of a lattice Z m of arbitrary dimension. This means that the resulting multi-dimensional system possesses solutions whose restrictions to any two-dimensional sub-lattice are generic solutions of the corresponding two-dimensional system. To put thisidea differently, one can impose the two-dimensional equations on any quad-surface in Z m (i.e., a surface composed of elementary squares), and transfer solutions from onesuch surface to another one, if they are related by a sequence of local moves, each oneinvolving one three-dimensional cube, like the moves shown of Fig. 1.Figure 1: Local move of a quad-surface involving one three-dimensional cube1 Introduction
A further fundamental conceptual development was initiated in [16] and deals withvariational (Lagrangian) formulation of discrete multi-dimensionally consistent systems.Solutions of any ABS equation on any quad surface Σ in Z m are critical points of a certainaction functional S Σ = R Σ L obtained by integration of a suitable discrete Lagrangian2-form L . It was observed in [16] that the critical value of the action remains invariantunder local changes of the underlying quad-surface, or, in other words, that the 2-form L is closed on solutions of quad-equations, and it was suggested to consider this as adefining feature of integrability. Results of [16], found on the case-by-case basis for someequations of the ABS list, have been extended to the whole list and were given a moreconceptual proof in [10]. Subsequently, this research was pushed in various directions: formulti-field two-dimensional systems [17, 2], for dKP, the fundamental three-dimensionaldiscrete integrable system [18], and for the discrete time Calogero-Moser system, animportant one-dimensional integrable system [25]. A general theory of multi-time one-dimensional Lagrangian systems has been developed in [24]. In [11] the general structureof multi-time Euler-Lagrange equations for two-dimensional Lagrangian problems wasstudied, and it was shown that the corresponding 2-form L is closed not only on solutionsof (non-variational) quad-equations, but also on general solutions of the correspondingEuler-Lagrange equations.As argued in the latter paper, the original idea of [16] has significant precursors. Theseinclude: • Theory of pluriharmonic functions and, more generally, of pluriharmonic maps[23, 21, 13]. By definition, a pluriharmonic function of several complex variables f : C m → R minimizes the Dirichlet functional E Γ = R Γ | ( f ◦ Γ) z | dz ∧ d ¯ z alongany holomorphic curve in its domain Γ : C → C m . Differential equations governingpluriharmonic functions (and maps) are heavily overdetermined. Therefore it isnot surprising that they belong to the theory of integrable systems. • Baxter’s Z-invariance of solvable models of statistical mechanics [3, 4]. This conceptis based on invariance of the partition functions of solvable models under elementarylocal transformations of the underlying planar graphs. It is well known (see, e.g.,[7]) that one can identify planar graphs underlying these models with quad-surfacesin Z m . On the other hand, the classical mechanical analogue of the partitionfunction is the action functional. This suggests the relation of Z-invariance to theconcept of closedness of the Lagrangian 2-form, at least at the heuristic level. Thisrelation has been made mathematically precise for a number of models, throughthe quasiclassical limit [5, 6]. • The classical notion of variational symmetry , going back to the seminal work ofE. Noether [20], turns out to be directly related to the idea of the closedness of theLagrangian form in the multi-time.The first of these precursors motivates the following term which was proposed in [11]to describe the general framework of the deveopment we are speaking about: pluri-Lagrangian systems . A d -dimensional pluri-Lagrangian problem can be set as follows:2 Variational symmetries given a d -form L on an m -dimensional space (called multi-time, m > d ), whose coeffi-cients depend on a sought-after function x of m independent variables (called field), findthose fields x which deliver critical points to the action functionals S Σ = R Σ L for any d -dimensional manifold Σ in the multi-time.The intention of the present note is to explain the relation of this notion to the thirdof the above mentioned precursors, namely to the notion of variational symmetries. Forthis aim, we recall the necessary definitions in section 2, illustrating them with one of themost familiar examples, the sine-Gordon equation and its variational symmetry given bythe modified KdV equation. Then, in section 3 we establish the relation of these classicalnotions with the idea of closedness of the Lagrangian form. Finally, in section 4 wepresent the Euler-Lagrange equations for two-dimensional pluri-Lagrangian problems ofsecond order, and establish the pluri-Lagrangian structure of the sine-Gordon equation. Let us start with reminding the notion of a variational symmetry of Lagrangian differ-ential equations, introduced in the seminal paper by E. Noether [20] (see also a detailedhistorical discussion in [15]).We consider the differential algebra of functions u α ( α = 1 , . . . , q ) of independentvariables x i ( i = 1 , . . . , p ). It has generators u αI , I = ( i , . . . , i n ) being a multiindex. Thederivation D j , understood as a full derivative w.r.t. x j , acts on generators according to D j u αI = u αI + e j . Thus, for any differential function f we have D j f = ∂f∂x j + X u αI + e j ∂f∂u αI . We now define more general derivations (generalized vector fields). In what follows, weonly consider “vertical” (or evolutionary) generalized vector fields, i.e., those acting ondependent variables only. This is done, on one hand, for simplicity of notation, and, onthe other hand, because in the discrete case only these are relevant, due to the absenceof changes of independent variable.
Definition 1. (Generalized evolutionary vector field)
A generalized evolutionaryvector field generated by the set of q differential functions ϕ [ u ] = ( ϕ [ u ] , . . . , ϕ q [ u ]) isgiven by D ϕ = X α,I ϕ αI ∂∂u αI , ϕ αI = D I ϕ α = D i . . . D i n n ϕ α . (1)Usually [22, eq. (5.6)], this is called an (infinite) prolongation of the generalizedevolutionary vector field X α ϕ α ∂∂u α . The following is an adaptation of Definition 5.51 from [22].3
Variational symmetries
Definition 2. (Variational symmetry)
A generalized evolutionary vector field (1) iscalled a variational symmetry of an action functional S [ u ] = Z L [ u ] dx . . . dx p , if the action of D ϕ on the Lagrange function L is a complete divergence, that is, if thereexist functions M [ u ] , . . . , M p [ u ] such that D ϕ L = Div M = p X i =1 D i M i . (2)The intention of this definition is clear: the integral of a complete divergence (byfixed boundary values) vanishes, so D ϕ preserves the value of the action functional. Thefollowing statement (see [22, Theorem 5.53]) justifies the previous definition. Theorem 3.
If a generalized evolutionary vector field D ϕ is a variational symmetry ofthe action functional S , then it is a generalized symmetry of the Euler-Lagrange equations δL/δu = 0 . Example.
Sine-Gordon equation, p = 2 , q = 1 : u xy = sin u, (3)is the Euler-Lagrange equation for L [ u ] = 12 u x u y − cos u. (4)We show that the (prolongation of the) evolutionary vector field ϕ ∂/∂u with ϕ [ u ] = u xxx + 12 u x (5)is a variational symmetry for the sine-Gordon equation. The corresponding computationis mentioned in [22, p. 336], but is not presented there in detail, being replaced by a lessdirect method. We show that D ϕ L = D x N + D y M (6)with the following differential functions: M [ u ] = 12 ϕu x − u x + 12 u xx , (7) N [ u ] = 12 ϕu y − u x cos u − u xx ( u xy − sin u ) . (8)Indeed, we compute: D ϕ L = 12 ( ϕ y u x + ϕ x u y ) + ϕ sin u, (9)4 Variational symmetries and closedness of multi-time Lagrangian forms and D y M + D x N = 12 ϕ y u x + 12 ϕu xy − u x u xy + u xx u xxy + 12 ϕ x u y + 12 ϕu xy − u x u xx cos u + 12 u x sin u − u xxx ( u xy − sin u ) − u xx ( u xxy − u x cos u )= 12 ( ϕ y u x + ϕ x u y ) + ϕ sin u + (cid:16) ϕ − u x − u xxx (cid:17) ( u xy − sin u ) . (10)Comparing (9) and (10), we see that identity (6) is satisfied under the choice of thedifferential function ϕ as in (5). By the Noether’s theorem, existence of a variationalsymmetry leads to the corresponding conservation law for the sine-Gordon equation: ϕ δLδu = ϕ (cid:18) ∂L∂u − D x ∂L∂u x − D y ∂L∂u y (cid:19) = ϕ ∂L∂u + ϕ x ∂L∂u x + ϕ y ∂L∂u y − D x (cid:18) ϕ ∂L∂u x (cid:19) − D y (cid:18) ϕ ∂L∂u y (cid:19) = D ϕ L − D x (cid:18) ϕ ∂L∂u x (cid:19) − D y (cid:18) ϕ ∂L∂u y (cid:19) = D x (cid:18) N − ϕ ∂L∂u x (cid:19) + D y (cid:18) M − ϕ ∂L∂u y (cid:19) = D x (cid:18) N − ϕu y (cid:19) + D y (cid:18) M − ϕu x (cid:19) = − D x (cid:18) u x cos u + u xx ( u xy − sin u ) (cid:19) + D y (cid:18) − u x + 12 u xx (cid:19) , which can be also found in [22, p. 336]. Now, we would like to promote an alternative point of view. In the standard approach,reproduced in the previous section, equation (2) is a certain (differential-)algebraic prop-erty of the vector field D ϕ . However, this way of thinking about this equation ignoresone of the main interpretations of the notion of “symmetry”, namely the interpretationas a commuting flow. In this interpretation, one introduces a new independent variable z corresponding to the “flow” of the generalized vector field D ϕ , D z u α = ϕ α [ u ] , (11)and considers simultaneous solutions of the Euler-Lagrange equations δL/δu = 0 and ofthe flow (11) as functions of p + 1 independent variables x , . . . , x p , z . Then equation (2)reads D z L − p X i =1 D i M i = 0 . (12)5 Variational symmetries and closedness of multi-time Lagrangian forms
The key observation is that equation (12) is nothing but the closedness condition of thefollowing p -form in the ( p + 1) -dimensional space: L = L [ u ] dx ∧ . . . ∧ dx p − p X i =1 ( − i M i [ u ] dz ∧ dx ∧ . . . ∧ c dx i ∧ . . . ∧ dx p . (13)Thus, we are led to define the extended action functional S Σ = Z Σ L , (14)where Σ is some p -dimensional surface (with boundary) in the ( p + 1) -dimensional spaceof independent variables x , . . . , x p , z . In particular, the action S Σ over the hypersurface Σ ⊂ { z = const } is the original action S . Equation (12) means that the extended actiondoes not depend on local changes of the p -dimensional integration surface Σ preservingboundary. Of course, this statement only holds on simultaneous solutions of the Euler-Lagrange equations δL/δu = 0 and of the flow (11). Example.
To clearly see this in our above example of the sine-Gordon equation, were-write the previous computations in our new notation, i.e., we replace ϕ by u z . Wehave: L = L [ u ] dx ∧ dy − M [ u ] dz ∧ dx − N [ u ] dy ∧ dz, (15)where L [ u ] = 12 u x u y − cos u, (16) M [ u ] = 12 u x u z − u x + 12 u xx , (17) N [ u ] = 12 u y u z − u x cos u − u xx ( u xy − sin u ) . (18)Then the previous computation tells us that L z − ( M y + N x ) = − (cid:16) u z − u x − u xxx (cid:17) ( u xy − sin u ) . (19)Thus, the form L is closed as soon as u z = u x + u xxx . This shows us once again thatthe modified KdV equation u z = u xxx + 12 u x (20)is a variational symmetry of the sine-Gordon equation (3).The remarkable factorized form of the right-hand side of (19) shows that it also vanishesas soon as u xy = sin u . This suggests that the above relation could be reversed, namely,that the sine-Gordon equation should be a variational symmetry of the modified KdVequation, as well. Two facts apparently stand in the way of this interpretation: first,modified KdV equation is not Lagrangian, and, second, sine-Gordon equation is notevolutionary. Nevertheless, this interpretation is still possible. To show this, we first6 Pluri-Lagrangian structure of the sine-Gordon equation observe that the function M [ u ] from (17) can be considered as a Lagrangian for theaction S Σ over the hypersurface Σ ⊂ { y = const } . The corresponding Euler-Lagrangeequation δM/δu = 0 is u zx − u x u xx − u xxxx = 0 , (21)the differentiated form of modified KdV. It is this equation for which we want to declarethe derivation D y as a variational symmetry. To overcome the difficulty that D y is notan evolutionary vector field (i.e., that u y is not defined by our differential equations), weobserve that we only need to define the action of D y on the Lagrangian M . However,the latter function does not contain u alone, but only its derivatives (of degree 1 w.r.t z and of higher degrees w.r.t. x ). For such functions the formula D y f = u yz ∂f∂u z + X I : i ≥ u Iy ∂f∂u I works perfectly as an evolutionary vector field. Indeed, one can use the equation u yz = u xxxy + 32 u x u xy = (sin u ) xx + 32 u x sin u = u xx cos u + 12 u x sin u, which is obtained from (20) by differentiation upon use of the sine-Gordon equation, aswell as relations u Iy = D I − e sin u for multiindices I with i ≥ . Next, we regard the multi-time Euler-Lagrange equations for the pluri-Lagrangian prob-lem with the Lagrangian 2-form L . We will not give the complete derivation, but restrictourselves to the statement which covers our main example in this note, namely the sine-Gordon equation. Theorem 4.
Consider a pluri-Lagrangian problem with the 2-form L = X i Theorem 5. Multi-time Euler-Lagrange equations for the pluri-Lagrangian problem withthe 2-form (15) with the components (16)–(18) consist of the sine-Gordon equation (34)and the modified KdV equation (38). On simultaneous solutions of these equations, the2-form L is closed. It is remarkable that multi-time Euler-Lagrange equations are capable of producingevolutionary equations. In subsequent publications, we will address the following problems:– To derive multi-time Euler-Lagrange equations for pluri-Lagrangian problems ofarbitrary order, i.e., for forms L depending on jets of arbitrary order.– To extend the classical De Donder-Weyl theory of calculus of variations to thepluri-Lagrangian context.– To elaborate on the pluri-Lagrangian structure of classical integrable hierarchies,like the KdV or, more generally, Gelfand-Dickey hierarchies. Note that in themonograph [14], which is, in my opinion, one of the best sources on the Lagrangianfield theory and whose program, according to the foreword, is “that the book isabout hierarchies of integrable equations rather than about individual equations”,it is the Lagrangian part (chapters 19, 20) that only deals with individual equations.The reason for this is apparently the absence of the concept of pluri-Lagrangiansystems.– To establish a general relation of pluri-Lagrangian structure to more traditionalnotions of integrability.– To study the general relation of pluri-Lagrangian structure to Z-invariance ofstatistical-mechanical problems, via quasi-classical limit, as exemplified in [5, 6].This research is supported by the DFG Collaborative Research Center TRR 109 “Dis-cretization in Geometry and Dynamics”. 9 eferences References [1] V.E. Adler, A.I. Bobenko, Yu.B. Suris. Classification of integrable equations onquad-graphs. The consistency approach , Commun. Math. Phys., 233 (2003), 513–543.[2] J. Atkinson, S.B. Lobb, F.W. Nijhoff. An integrable multicomponent quad-equationand its lagrangian formulation , Theor. Math. Phys., (2012), 1644–1653.[3] R.J. Baxter. Solvable eight-vertex model on an arbitrary planar lattice , Philos.Trans. R. Soc. London, Ser. A (1978) 315–346.[4] R.J. Baxter. Free-fermion, checkerboard and Z-invariant lattice models in statisticalmechanics . Proc. R. Soc. Lond. A (1986) 1–33.[5] V.V. Bazhanov, V.V. Mangazeev, S.M. Sergeev. Faddeev-Volkov solution of theYang-Baxter equation and discrete conformal geometry , Nucl. Phys. B (2007),234–258.[6] V.V. Bazhanov, V.V. Mangazeev, S.M. Sergeev. A master solution of the quantumYang-Baxter equation and classical discrete integrable equations , Adv. Theor. Math.Phys. (2012) 65–95.[7] A.I. Bobenko, Ch. Mercat, Yu.B. Suris. Linear and nonlinear theories of discreteanalytic functions. Integrable structure and isomonodromic Green’s function. J.Reine Angew. Math. (2005), 117–161.[8] A.I. Bobenko, Yu.B. Suris. Integrable systems on quad-graphs , Intern. Math. Re-search Notices, , Nr. 11 (2002), 573–611.[9] A.I. Bobenko, Yu.B. Suris. Discrete Differential Geometry: Integrable Structures ,Graduate Studies in Mathematics, Vol.98, AMS, 2008.[10] A.I. Bobenko, Yu.B. Suris. On the Lagrangian structure of integrable quad-equations , Lett. Math. Phys. (2010), 17–31.[11] R. Boll, M. Petrera, Yu.B. Suris. What is integrability of discrete variational sys-tems? arXiv:1307.0523 [math-ph] .[12] R. Boll, Yu.B. Suris. On the Lagrangian structure of 3D consistent systems ofasymmetric quad-equations , J. Phys. A: Math. Theor. (2012) 115201.[13] F. Burstall, D. Ferus, F. Pedit, and U. Pinkall. Harmonic tori in symmetric spacesand commuting Hamiltonian systems on loop algebras . Ann. Math. (1993) 173–212.[14] L. Dickey. Soliton equations and Hamiltonian systems . 2nd edition. World Scientific,2003. 10 eferences [15] Y. Kosmann-Schwarzbach. The Noether theorems. Invariance and conservationlaws in the 20th century . Springer, 2011.[16] S. Lobb, F.W. Nijhoff. Lagrangian multiforms and multidimensional consistency ,J. Phys. A: Math. Theor. (2009) 454013.[17] S.B. Lobb, F.W. Nijhoff. Lagrangian multiform structure for the lattice Gel’fand-Dikij hierarchy , J. Phys. A: Math. Theor. (2010) 072003.[18] S.B. Lobb, F.W. Nijhoff, G.R.W. Quispel. Lagrangian multiform structure for thelattice KP system , J. Phys. A: Math. Theor. (2009) 472002.[19] F.W. Nijhoff. Lax pair for the Adler (lattice Krichever-Novikov) system , Phys. Lett.A (2002), 49–58.[20] E. Noether. Invariante Variationsprobleme , Nachrichten von der Gesellschaft derWissenschaften zu Göttingen, Math.-Phys. Kl. (1918), 235–257.[21] Y. Ohnita, G. Valli. Pluriharmonic maps into compact Lie groups and factorizationinto unitons , Proc. London Math. Soc. (1990) 546–570.[22] P. Olver. Applications of Lie groups to differential equations . Graduate Texts inMathematics, Vol. 107. 2nd edition, Springer, 1993.[23] W. Rudin. Function theory in polydiscs . Benjamin (1969).[24] Yu.B. Suris. Variational formulation of commuting Hamiltonian flows: multi-timeLagrangian 1-forms , arXiv:1212.3314 [math-ph] .[25] S. Yoo-Kong, S. Lobb, F. Nijhoff. Discrete-time Calogero-Moser system and La-grangian 1-form structure . J. Phys. A: Math. Theor.44