SLIDE 1
Lagrangian Multiform Theory
Frank Nijhoff, University of Leeds (joint work with James Atkinson, Steven King, Sarah Lobb, Pavlos Xenitidis, Sikarin Yoo-Kong) RAQIS 2016, Geneva, 26 August 2016
SLIDE 2 The problem
Multidimensional consistency: We know that many ”integrable” equations, discrete and continuous, possess the property of “multidimensional consistency” (MDC).
◮ continuous: commuting flows, higher symmetries & master symmetries,
hierarchies;
◮ discrete: consistency-around-the-cube, B¨
acklund transforms, higher continuous symmetries, commuting discrete flows In all these cases we can think of the dependent variable a (possibly vector-valued) function of many (discrete and continuous) variables u = u(n, m, h, . . . ; x, t1, t2, . . . )
- n which we can impose many equations simultaneously, and it is the compatibility of
those equations that makes the integrability manifest. Key question: How to capture the property of multidimensional consistency within a Lagrange formalism? Main problem: We note that the conventional variational principle, through the EL equations, only produces one equation per component of the dependent variables, but not an entire system of compatible equations on one and the same dependent variable!
SLIDE 3 The answer
Answer: A new variational principle (which we call Lagrangian multiform theory) based on a key observation made about the structure of integrable Lagrangians1: Well-chosen Lagrangians embedded in higher-dimensional space of independent variables obey a special relation, the so-called closure relation, when evaluated on solutions of the equations of motion (i.e., “off-shell”). This allows the interpretation
- f these Lagrangians as differential - or difference forms in a higher-dimensional space
- f independent variables.
This property was proven first for special examples, namely quadrilateral lattice equations, and associated continuous equations, but subsequently extended to other classes of equations (higher-rank, higher-dimensional, finite-dimensional, etc.). Thus, it seems this property is quite universal. This has led to the formulation of a novel variational principle which involves not only the field variables, but also the geometries in the space of independent variables. The principles of the corresponding variational calculus were proposed and elaborated at Leeds, while further refinements and extensions were made by the Berlin group (Boll, Petrera, Suris, Vermeeren). Remark: Suris et al. in recent papers adopted the new name ”Pluri-Lagrangian Systems” deviating from the original name ”Lagrangian multiforms”. This is essentially based on a difference of point of view.
- 1S. Lobb & FWN: Lagrangian multiforms and multidimensional consistency, J. Phys. A:Math Theor. 42 (2009)
454013
SLIDE 4 Outline
- 1. Multidimensional consistency for 2D lattices & closure property;
- 2. Fundamental system of multi-form EL equations;
- 3. Interplay continuous/discrete and continuous Lagrangian multi-forms;
- 4. Multiform structure in 3D: latice KP equation;
- 5. Lagrangian 1-form structure (finite-dimensional multi-time systems);
- 6. Implications for quantum theory.
SLIDE 5 Multidimensional consistency on the lattice
Quadrilateral P∆Es on the 2D lattice: Q(u, T1u, T2u, T1T2u; p1, p2) = 0 notation of shifts on the elementary quadrilateral on a rectangular lattice: u := u(n1, n2), T1u = u(n1 + 1, n2) T2u := u(n1, n2 +1), T1T2u = u(n1 +1, n2 +1)
s s s s ✲ ✲ ❄ ❄ ✲ ✲ ❄ ❄
T2u T1T2u T1u u p1 p1 p2 p2 Consistency-around-the cube:
t
T3u
t
u
t T1u r ❞ T1T2T3u t
T2u
❞
T1T3u
❞
T2T3u
❞ T1T2u
- Verifying consistency: Values at the black disks are initial values, values at open
circles are uniquely determined from them, but there are three different ways to compute T1T2T3u.
SLIDE 6 Conventional variational formalism: discrete Euler-Lagrange equations
Define an action functional: S[u(n1, n2)] =
L (u, T1u, T2u; p1, p2) . Following the usual least-action principle, the lattice equations for u are determined by the demand that S attains a minimum under local variations u(n1, n2) → u(n1, n2) + δu(n1, n2). Thus,
δS =
∂ ∂u L (u, T1u, T2u; p1, p2)δu + ∂ ∂T1u L (u, T1u, T2u; p1, p2)δ(T1u) + ∂ ∂T2u L (u, T1u, T2u; p1, p2)δ(T2u)
Setting δ(Tiu) = Tiδu, and resumming each of the terms we get:
=
∂ ∂u L (u, T1u, T2u; p1, p2) + ∂ ∂u L (T −1
1
u, u, T −1
1
T2u; p1, p2) + ∂ ∂u L (T −1
2
u, T1T −1
2
u, u; p1, p2)
(ignoring boundary terms) and since δu is arbitrary the discrete Euler-Lagrange (EL) equation follow:
∂ ∂u
- L (u, T1u, T2u; p1, p2) + L (T −1
1
u, u, T −1
1
T2u; p1, p2) + L (T −1
2
u, T1T −1
2
u, u; p1, p2)
In principle we can have such Lagrangians in every pair of shifts on a multidimensional
- lattice. However, this doesn’t tell us a priori that the corresponding EL equations are
compatible.
SLIDE 7 Closure relation and Lagrangian multiform structure
Closure property: Multidimensionally consistent systems of lattice equations, possess Lagrangians which obey the following relation 2: ∆1L (u, T2u, T3u; p2, p3) + ∆2L (u, T3u, T1u; p3, p1) + ∆3L (u, T1u, T2u; p1, p2) = 0
- n the solutions of the equations.
Here ∆i = Ti − id denotes the difference operator, i.e.. on functions f of u = u(n1, n2, n3) we have: ∆if (u) = f (Tiu) − f (u).
- This property suggests that the Lagrangians Li,j = L (u, Tiu, Tju; pi, pj) should be
considered as difference forms (i.e., discrete differential forms) for which the closure property means that these forms are closed, but only for functions u which solve the lattice equation.
- Furthermore, as a consequence of this closedness of the corresponding Lagrangian
2-form on solutions of the equations, the corresponding action will be locally invariant under deformations of the underlying geometry of the lattice, i.e. locally independent
- f the discrete surface in the space of independent variables.
- However, off-shell, i.e. for general field configurations (i.e. values of the dependent
variable u as a function of the lattice) the action is non-trivial functional of those fields, and also of the lattice-surface on which we evaluate the action.
- 2S. Lobb & FWN: Lagrangian multiforms and multidimensional consistency, J. Phys. A:Math Theor. 42 (2009)
454013.
SLIDE 8 Example: H1 (lattice potential KdV eq.)
The equation is Q(u, T1u, T2u, T1T2u; p1, p2) = (u − T1T2u)(T1u − T2u) + p2
1 − p2 2 = 0
The equation in the “3-leg form” is (u + T1u) − (u + T2u) + p2
1 − p2 2
u − T1T2u = 0 The corresponding 3-point Lagrangian is given as3 L (u, T1u, T2u; p1, p2) = u(T1 − T2)u + (p2
1 − p2 2) ln(T1u − T2u)
The discrete Euler-Lagrange equations lead to a slightly weaker equation than H1 itself, but equivalent to a discrete derivative of the equation: T1u − T −1
2
u + p2
1 − p2 2
u − T1T −1
2
u + T −1
1
u − T2u + p2
1 − p2 2
u − T −1
1
T2u = 0
3Capel, H.W., F.W. Nijhoff and V.G. Papageorgiou. Complete Integrability of Lagrangian Mappings and
Lattices of KdV Type. Physics Letters A, 1991: 155, pp.377-387.
SLIDE 9 Closure property for H1:
The lagrangian for H1 obeys the following closure relation: ∆1L (u, T2u, T3u; p2, p3) + ∆2L (u, T3u, T1u; p3, p1) + ∆3L (u, T1u, T2u; p1, p2) = 0
- n the solutions of the quadrilateral equation.
Proof: From the explicit form of the Lagrangians we find ∆1L (u, u2, u3; p2, p3) + ∆2L (u, u3, u1; p3, p1) + ∆3L (u, u1, u2; p1, p2) = (u1,2 − u1,3)u1 + (p2
2 − p2 3) ln(u1,2 − u1,3) − (u2 − u3)u − (p2 2 − p2 3) ln(u2 − u3)
+(u2,3 − u1,2)u2 + (p2
3 − p2 1) ln(u2,3 − u1,2) − (u3 − u1)u − (p2 3 − p2 1) ln(u3 − u1)
+(u1,3 − u2,3)u3 + (p2
1 − p2 2) ln(u1,3 − u2,3) − (u1 − u2)u − (p2 1 − p2 2) ln(u1 − u2)
wehere we have used the abbreviations: ui := Tiu, ui,j := TiTju . Noting that the differences between the double-shifted terms has the form u1,2 − u1,3 = (p2
2 − p2 3)u1 + (p2 3 − p2 1)u2 + (p2 1 − p2 2)u3
(u1 − u2)(u2 − u3)(u3 − u1) (u2 − u3) =: A1,2,3(u2 − u3) where A1,2,3 is invariant under permutations of the indices, the expression reduces to
A1,2,3(u2 − u3)u1 + (p2
2 − p2 3) ln
- A1,2,3(u2 − u3)
- −(u2 − u3)u − (p2
2 − p2 3) ln(u2 − u3)
+A1,2,3(u3 − u1)u2 + (p2
3 − p2 1) ln
- A1,2,3(u3 − u1)
- −(u3 − u1)u − (p2
3 − p2 1) ln(u3 − u1)
+A1,2,3(u1 − u2)u3 + (p2
1 − p2 2) ln
- A1,2,3(u1 − u2)
- −(u1 − u2)u − (p2
1 − p2 2) ln(u1 − u2)
= 0
SLIDE 10 Closure relation for other cases
The closure property was proven for many other lattice equations (often requiring a specific form of the Lagrangian (taking into account that there is the freedom to add total-“derivative” terms):
- Quad-equations in the ABS (Adler-Bobenko-Suris) list of scalar affine-linear
equations;
- Higher-rank equations: lattice Gel’fand-Dikii hierarchy (incl. lattice Boussinesq
systems);
- The higher-dimensional equations: the (bilinear) lattice KP equation.
In many cases these Lagrangians contain the function F(u) = u ln u or F(u) = Li2(u) , the dilogarithm function Li2(z) = − z z−1 ln(1 − z) dz and the closure property relies on the Rogers 5-term relation (pentagon relation):
Li2
1 − y y 1 − x
Li2
1 − y
1 − x
− ln(1 − x) ln(1 − y)
Elliptic lattice systems require an elliptic analogue of the dilogarithm: F(u) ∼ u ln(σ(x)) dx. An example of elliptic case is the so-called Q4 equation,(V. Adler, 1998): pi(u ui + ujui,j) − pj(u uj + uiui,j) − pij(u ui,j + uiuj) + pipjpij(1 + u uiujui,j) = 0 where ui := Tiu, ui,j = TiTju. The parameters are in terms of Jacobi elliptic functions: pi = √ k sn(αi; k) , pj = √ k sn(αj; k) , pij = √ k sn(αi − αj; k) .
SLIDE 11 Surface-dependent actions
The closure relation suggests the introduction of surface-dependent actions. Action functional on a discrete (oriented) surface σ: S[u(n); σ] =
Lij(n) where Lij(n) = −Lji(n) has the interpretation of a discrete Lagrangian 2-form: These are antisymmetric (under exchange of i, j labels) expressions of the form: Lij(n) = L (u(n), u(n + ei), u(n + ej); pi, pj) defined on elementary plaquettes, in a multidimensional lattice, characterized by the
- rdered triplet σij(n) = (n, n + ei, n + ej)
n ei ej σ a discrete surface in the multidimensional lattice consisting of (a connected configuration of) elementary plaquettes σij(n)
SLIDE 12
Surface independence
Independence of the action S under local deformations S → S′ of the surface is equivalent to the closure relation holding. S′ = S − L (u, ui, uj; αi, αj) + L (uk, ui,k, uj,k; αi, αj) + L (ui, ui,j, ui,k; αj, αk) +L (uj, uj,k, ui,j; αk, αi) − L (u, uj, uk; αj, αk) − L (u, uk, ui; αk, αi) taking into account the orientation of the plaquettes.
SLIDE 13
Surface independence
Independence of the action S under local deformations S → S′ of the surface is equivalent to the closure relation holding. S′ = S − L (u, ui, uj; αi, αj) + L (uk, ui,k, uj,k; αi, αj) + L (ui, ui,j, ui,k; αj, αk) +L (uj, uj,k, ui,j; αk, αi) − L (u, uj, uk; αj, αk) − L (u, uk, ui; αk, αi) taking into account the orientation of the plaquettes.
SLIDE 14 Fundamental EL system for quad-equations
We will now describe how to resolve the issue of “weak equations”: that the closure requires stronger equations than the variational one4. Assuming the 3-point form of the Lagrangians: Li,j(u, Tiu, Tju) := L (u, Tiu, Tju; pi, pj) , the lattice EL can be written as: (EL0)
∂ ∂u
i
u, u, T −1
i
Tju) + Li,j(u, Tiu, Tju) + Li,j(T −1
j
u, TiT −1
j
u, u)
This represents the “planar” EL eqs, illustrated by the diagram (embedded in 3D lattice):
Figure : EL in flat 2D lattice.
This is the weak (non-quadrilateral) form of the equations. However, by the extended variational principle of the multiform structure, the quad-lattice equation is recovered.
4S.B.Lobb & F.W. Nijhoff. A variational principle for discrete integrable systems. ArXiv: 1312.1440.
- R. Boll, M. Petrera and Yu. Suris, What is integrability of discrete variational systems?, arXiv:1307.0523.
SLIDE 15
Lattice action for the closed cube surface
To derive elementary configurations we need action over the (decorated) full oriented cube:
TiTju Tiu u TiTku Tku TjTku Tju TiTjTku Figure : Decorated cube.
This gives rise to a lattice action functional: S[u; cube] = Li,j(u, Tiu, Tju) + Lj,k(u, Tju, Tku) + Lk,i(u, Tku, Tiu) −Li,j(Tku, TiTku, TjTku) − Lj,k(Tiu, TiTju, TiTku) − Lk,i(Tju, TjTku, TiTju). The faces joining each vertex involved in the action will give rise to the various elementary surface configurations: the elementary actions that will lead to the fundamental system of EL equations.
SLIDE 16 Elementary configurations for lattice action
Over curved quad-surfaces we need the following types of elementary configurations:
Figure : Elementary lattice configurations in 3D.
The action functionals corresponding to these configurations give rise to the fundamental system of EL equations:
(EL1) ∂ ∂u
- Li,j(u, Tiu, Tju) + Lj,k(u, Tju, Tku) + Lk,i(u, Tku, Tiu)
- = 0,
(EL2) ∂ ∂u
i
u, u, T −1
i
Tju) − Lj,k(u, Tju, Tku) + Lk,i(T −1
i
u, T −1
i
Tku, u)
(EL3) ∂ ∂u
j
(u), u, T −1
j
Tku) + Lk,i(T −1
i
u, T −1
i
Tku, u)
(up to permutations of the lattice indices).
Furthermore, imposing that the action remains invariant under (discrete) deformations
- f the surface (allowing the above equations to hold simultaneously) the system is
supplemented with the closure relation:
(EL4) ∆iL (u, Tju, Tku; pj, pk) + ∆jL (u, Tku, Tiu; pk, pj) + ∆kL (u, Tiu, Tju; pi, pj) = 0 .
SLIDE 17
Main hypothesis: The solutions of above linear system of equations for the Lagrangians Li,j correspond exactly to the Lagrangian components for integrable (in the sense of multidimensional consistency) quad-lattice systems. The unique foundational principle: The Lagrangians themselves are to be viewed as solutions of the extended EL equations. Instead of being put in by hand, (or posed on the basis of external considerations), they are to emerge purely from the extended variational principle itself.
SLIDE 18 Analysis of the EL system
Analysing the fundamental EL system (EL1)-(EL4) under the assumption that u, Tiu, Tju, Tku are independent and can be chosen arbitrarily, and considering the functional dependence on the arbitrary parameters pi, pj, pk we arrive at the following:
Theorem
Suppose u, Tiu, Tju, Tku are independent and can be chosen arbitrarily. Eq (EL1) implies that the anti-symmetric Lagrangian Li,j = L (u, Tiu, Tju; pi, pj) has the form Li,j(u, Tiu, Tju) = Ai(u, Tiu) − Aj(u, Tju) + Bi,j(Tiu, Tju), where Ai(u, Tiu) = A(u, Tiu; pi), and Bi,j(Tiu, Tju) = B(Tiu, Tju; pi, pj) for some functions A, B of the arguments and lattice parameters, where Bi,j = −Bj,i is antisymmetric in the i, j-arguments. Furthermore, from the eqs (EL2), (EL3) one can deduce the following:
Theorem
The Euler-Lagrange equations (EL2),(EL3) determine the following relation on each single quad: (QEL) ∂ ∂u
i
u, u, T −1
i
Tju)
∂u
where h(u) is an arbitrary function, which w.l.o.g. can be absorbed into Aj. The latter equation, in fact, leads directly to the quadrilateral lattice equation Q(u, Tiu, Tju, TiTju; pi, pj) = 0.
SLIDE 19 Example: quadratic 3-point Lagrangian 2-forms
Let us consider the general homogeneous quadratic Lagrangian 2-form component, which must be of the form: Li,j(u, Tiu, Tju) = Ai(u, Tiu) − Aj(u, Tju) + Bi,j(Tiu, Tju) , by setting: Ai(u, Tiu) = 1
2 aiu2+a′ i uTiu+ 1 2 a′′ i (Tiu)2 ,
Bi,j = 1
2 bij(Tiu)2− 1 2 bji(Tju)2+b′ ij(Tiu)Tju ,
where b′
ji = −b′
- ij. Applying the eq. (QEL) in the form:
∂ ∂Tiu
∂ ∂Tiu
(which holds for all directions i, j) we obtain the linear quad-equation: a′
i u + (a′′ i + bij − aj)Tiu + b′ ijTju = a′ jTiTju .
Since these hold for arbitrary i, j-labels we obtain the conditions: a′
i 2 = a′ j 2 , and (a′′ i + bij − aj)a′ i = a′ jb′ ji .
Setting a′
i = a′ j =: a′ and implementing the other condition we get the quad equation:
TiTju = u + 1
a′ b′ ij(Tju − Tiu) , with Lagrangian:
Li,j = 1
2 (ai − aj)u2 + a′u(Tiu − Tju) + 1 2 aj(Tiu)2 − 1 2 ai(Tju)2 − 1 2 b′ ij(Tiu − Tju)2 ,
where the terms with ai can be removed w.l.o.g., and we can set a′ = 1. The closure relation (EL4) leads to the functional relation b′
ij(b′ ik − b′ jk) = 1 − b′ ikb′ jk
⇒ b′
ij = (1 − PiPj)/(Pi − Pj) ,
in terms of new lattice parameter Pi := b′
i,k0 (with k0 fixed).
SLIDE 20 Universal Lagrangian for affine-linear quad-lattices
Under the assumption that the quad-equation Q(u, Tiu, Tju, TiTju; pi, pj) = 0 is affine multi-linear and possesses the symmetries of the square (Kleinian symmetry) we can find the general solution using the method of characteristics5 The general formula is given by: L (u, ui, uj) = u
u0
ui
u0
i
dx dy hpi (x, y) − u
u0
uj
u0
j
dx dy hpj (x, y) − ui
u0
i
uj
u0
j
dx dy hpij (x, y) + ui
u0
i
dx Y (u0,x,u0
ij )
u0
j
dy hpij (x, y) + uj
u0
j
dy X(u0,y,u0
ij )
u0
i
dx hpij (x, y) where ui := Tiu, uij := TiTju, etc., and the functions X and Y are solutions of the equations Q(u0, x, Y , u0
ij; pi, pj) = 0
respectively Q(u0, X, y, u0
ij; pi, pj) = 0 ,
and where the quantities hp(x, y) are biquadratic functions associated with the quad-function Q as follows: small ∂Q ∂(Tju) ∂Q ∂(TiTju) − Q ∂2Q ∂(Tju)∂(TiTju) =: Kpi ,pj hpi (u, Tiu) ∂Q ∂u ∂Q ∂(TiTju) − Q ∂2Q ∂u∂(TiTju) =: −Kpi ,pj hpij (Tiu, Tju) and where Kp,q = −Kq,p is a function of the lattice parameters p, q only.
- 5P. Xenitidis, F.W. Nijhoff and S. Lobb, On the Lagrangian formulation of multidimensionally consistent
systems, Proc. Roy. Soc. A467 (2011), 3295–3317.
SLIDE 21 Interplay Discrete ← → Continuous
The lattice systems we consider here admit a role reversal: lattice parameters pi ↔ lattice variables ni For all quadrilateral P∆Es we have fully consistent system of equations comprising three types of equations, all compatible discrete as well as continuous. P∆E ↔ D∆E ↔ PDE These D(∆)Es can be simultaneously imposed on the same dependent variables: u = u(n1, n2, n3, . . . ; p1, p2, p3, . . . ) and possess the property of multidimensional consistency. The consistency can be expressed as the condition that on solutions of these equations the operators (Tiu)(. . . , ni, . . . ) = u(. . . , ni + 1, . . . ) , mutually commute in all lattice directions and with the differential operators ∂pi , ∂pj , . . . , i.e. among these equations hold on the solutions TiTju = TjTiu , ∂ ∂pi ∂u ∂pj
∂ ∂pj ∂u ∂pi
Ti ∂u ∂pj
∂ ∂pj Tiu .
SLIDE 22 Example: H1 (lattice potential KdV)
P∆E Fully discrete Lagrangian: Lij = u(Tiu − Tju) + (p2
i − p2 j ) ln
Linear quadrilateral lattice equation(H1): (u − TiTju)(Tiu − Tju) + p2
i − p2 j = 0 .
The lattice Lagrangian Lij obeys the discrete closure relation: ∆iLjk + ∆jLki + ∆kLij = 0
- n solutions of the lattice equation.
PDE As a function of the lattice parameters pi the same function u obeys a PDE which can be derived from the Lagrangian6 Lij = 1 4 (p2
i − p2 j )
(∂pi ∂pj u)2 (∂pi u)(∂pj u) + 1 p2
i − p2 j
i p2 i
∂pj u ∂pi u + n2
j p2 j
∂pi u ∂pj u
- which is compatible with the H1 equation. Remarkably, this PDE is a generalization of
Ernst equation of General Relativity. Main property here: it possesses the MDC property! This manifests itself by continuous closure property: ∂Lij ∂pk + ∂Ljk ∂pi + ∂Lki ∂pj = 0 .
6FWN,A. Hone & N. Joshi, On a Schwarzian PDE associated with the KdV hierarchy, Phys. Lett.A267 (2000)
147–156.
SLIDE 23 Variational formalism for continuous Lagrangian 2-forms
Choosing a parametrisation of the surface σ : p = p(s, t) = (pi(s, t)) , (s, t) ∈ Ω ⊂ R2 , where Ω is some open domain in the space of parameters s, t, we can write for the action7: S[u(p); σ] =
Li,jdpi ∧ dpj =
∂(pi, pj) ∂(s, t)
We have two types of variations:
- Variations of the surface:
σ → σ + δσ , (i.e., making a infinitesimal variations p → p + δp, in the parametrisation). The closure relation can be obtained by considering the Lagrangian as a function of the independent variables L(p(s, t)) :=
∂(pi, pj) ∂(s, t)
and apply the usual EL equations: δL δp(s, t) = 0 ⇒ ∂pi Lj,k + ∂pj Lk,i + ∂pk Li,j = 0 .
- Infinitesimal variations of the dependent variable u → u + δu, on an arbitrary, but
fixed, surface. This has two contributions: ♦ tangential contributions, i.e. from components (∇δu) along the surface; ♦ orthogonal contributions, i.e. from components (∇δu)⊥ orthogonal to the surface.
- 7S. Lobb & FWN: Lagrangian multiforms and multidimensional consistency, J. Phys. A:Math Theor. 42 (2009)
454013.
SLIDE 24 Lagrange 2-form in 3D space
In the simple case of smooth 2D surfaces σ embedded in R3, and Li,j = L (u, upi , upj ) depending only on the first jet, we get the following set of equations8:
- From the tangential contributions:
- i<j
∂(pi, pj) ∂(s, t) ∂Lij ∂u − ∂ ∂s ∂(pi, pj) ∂(s, t) pt × n ps × pt · ∂Lij ∂∇u
∂t ∂(pi, pj) ∂(s, t) ps × n ps × pt · ∂Lij ∂∇u
where n = ps × pt/ps × pt is the unit normal to the surface; and:
- From the transversal contributions:
- i<j
∂(pi, pj) ∂(s, t) n · ∂Lij ∂∇u = 0 ;
∂Ljk ∂pi + ∂Lki ∂pj + ∂Lij ∂pk = 0 .
8A different approach to the variational system for the general continuous 2-form case was recently obtained in:
M Vermeeren, Masters Thesis, TU Berlin October 2014].
SLIDE 25 3-dimensional case: lattice bilinear KP equation
The lattice bilinear KP (Hirota, 1982) (∆KP) has the following form Ajk(Tiu)TjTku + Aki(Tju)TiTku + Aij(Tku)TiTju = 0 , for the τ-function, where we take antisymmetric coefficients Aij = −Aji. Define9: Lijk := L(Tiu, Tju, Tku, TiTju, TjTku, TiTku; Aij, Ajk, Aki) = ln (Tku)TiTju (Tju)TiTku
AikTju AjkTiu
Aij(Tku)TiTju Aik(Tju)TiTku
- where Li2 as before denotes the standard dilogarithm function.
Consider the action functional S[u(ni, nj, nk)] =
L(Tiu, Tju, Tku, TiTju, TjTku, TiTku; Aij, Ajk, Aki) , where minimizing the action produces the following discrete Euler-Lagrange equation
0 = δS δu = = 1 u
Aik(Tiu)TjT −1
k
u − AijuTiTjT −1
k
u Ajk(Tju)TiT −1
k
u
Aik(T −1
i
u)T −1
j
Tku − AijuT −1
i
T −1
j
Tku Ajk(Tju)T −1
i
Tku
AikuTiT −1
j
Tku − Aij(Tiu)(T −1
j
Tku Ajk(Tku)TiT −1
j
u
AikuT −1
i
TjT −1
k
u − Aij(T −1
i
u)TjT −1
k
u Ajk(T −1
k
u)T −1
i
Tju
- which is a consequence of the ∆KP equation as a combination of 4 copies of the
equation shifted in different lattice directions.
- 9S. Lobb, FWN & R. Quispel, Lagrangian multiform structure for the lattice KP system, J. Phys. A:Math
- Theor. 42 (2009) 472002.
SLIDE 26 Discrete Lagrangian 3-form
It is well-known that the ∆KP is multidimensionally consistent. To incorporate this into a Lagrangian framework we define a Lagrangian 3-form structure for the lattice KP system. We define the Lagrangian 3-form as follows: Lijk := 1 2
- Lijk + Ljki + Lkij − Likj − Ljik − Lkji
- which when written out explicitly and simplified is (using the shorthand notation
Tiu =: ui, T −1
i
u =: u¯
ı):
Lijk = ln (Tku)TiTju (Tju)TkTiu
AjkTiu
Aki(Tju)TkTiu
(Tiu)TjTku (Tku)TiTju
AkiTju
Aij(Tk)uTiTju
(Tju)TkTiu (Tiu)TjTku
AijTku
AjkTiuTjTku
2
2 +
2 +
2 −
2 −
2 −
2 − ln
- TiTju
- ln
- TjTku
- − ln
- TjTku
- ln
- TkTiu
- − ln
- TkTiu
- ln
- TiTju
- + ln
- Tiu
- ln
- Tju
- + ln
- Tju
- ln
- Tku
- + ln
- Tku
- ln
- Tiu
- −A2
ij − A2 jk − A2 ki + AijAjk + AjkAki + AkiAij
where the constant terms arise from the dilogarithm identities.
SLIDE 27
Considered as a scalar Lagrangian defined in Z3 lattice the action would yield an equation combining 12 shifted copies of the ∆KP equation, which is actually a 19-point equation involving the following configuration of points: ni nj nk
Figure : The 19-point equation δLijk/δτ = 0.
It comprises the 12 shifted copies of ∆KP, 6 of which are illustrated as follows:
SLIDE 28
Figure : Copies of the 6-point equation.
Proposition: The Lagrangian components given above satisfy the following closure relation on solutions of the ∆KPequation when embedded in a 4-dimensional lattice ∆lLijk − ∆iLjkl + ∆jLkli − ∆kLlij = 0, where ∆i = Ti − id is the usual difference operator acting on functions f of u.
SLIDE 29 Constitutive set of EL equations in 3D
Each Lagrangian is associated with an oriented elementary cube νijk(n), embedded in a higher-dimensional lattice, and is specified by the position n = (n, n + ei, n + ej, n + ek) and orientation given by a chosen set of base vectors ei, ej, ek. ej ei ek n
Figure : Elementary oriented cube.
The Lagrangian can depend in principle on the fields at all 8 vertices of the elementary cube and the action can evaluated for any given discrete hypersurface defined as a connected configuration ν of these elementary cubes: S[u(n); ν] =
Lijk(n) =
L (u(n), u(n + ei), · · · , u(n + ei + ej + ek)) In the case of the bilinear discrete KP equation the equation can be written either in Z3 or in the root lattice Q(A3).
SLIDE 30 Euler-Lagrange equations and minimal configurations
The Euler-Lagrange equation in the usual 3-dimensional space is = ∂ ∂u
i
Lijk + T −1
j
Lijk + T −1
k
Lijk +T −1
i
T −1
j
Lijk + T −1
j
T −1
k
Lijk + T −1
k
T −1
i
Lijk + T −1
i
T −1
j
T −1
k
Lijk
with the notation Lijk = Lijk(u, Tiu, Tju, Tku, TiTju, TjTku, TiTku, TiTjTku). The EL equation above is the analogue of the “flat” equation for the 2 dimensional case. Once again the weakness of the equation can be overcome by considering the system actions on elementary configurations. Steps are as follows: Step # 1: Embed the system in 4 dimensions. In 4 dimensions, the smallest closed 3-dimensional closed space (made out of quadrilaterals) is a hypercube (tessaract), consisting of 8 cubes. The action on the elementary hypercube will have the form S(u; hypercube) = ∆lLijk − ∆iLjkl + ∆jLkli − ∆kLlij . Step # 2: Derive the equations for elementary configurations by considering the hyper-faces (cubes) of the closed hypersurface joining at each vertex. (Because of the symmetry, this means we need only take derivatives with respect to u, Tiu, TiTju, TiTjTku and TiTjTkTlu; the other equations will follow by cyclic permutations). They define the elementary actions.
SLIDE 31 This leads to the set of elementary equations:
= ∂ ∂u
- −Lijk + Ljkl − Lkli + Llij
- ,
= ∂ ∂Tiu
- −Lijk − TiLjkl − Lkli + Llij
- ,
= ∂ ∂TiTju
- −Lijk − TiLjkl + TjLkli + Llij
- ,
= ∂ ∂TiTjTku
- −Lijk − TiLjkl + TjLkli − TkLlij
- ,
= ∂ ∂TiTjTkTlu
- TlLijk − TiLjkl + TjLkli − TkLlij
- ,
along with the equivalent shifted versions
= ∂ ∂u
- −Lijk + Ljkl − Lkli + Llij
- ,
= ∂ ∂u
i
Lijk − Ljkl − T −1
i
Lkli + T −1
i
Llij
= ∂ ∂u
i
T −1
j
Lijk − T −1
j
Ljkl + T −1
i
Lkli + T −1
i
T −1
j
Llij
= ∂ ∂u
i
T −1
j
T −1
k
Lijk − T −1
j
T −1
k
Ljkl + T −1
i
T −1
k
Lkli − T −1
i
T −1
j
Llij
= ∂ ∂u
i
T −1
j
T −1
k
Lijk − T −1
j
T −1
k
T −1
l
Ljkl + T −1
i
T −1
k
T −1
l
Lkli − T −1
i
T −1
j
T −1
l
Llij
Proposition: The above set of equations, together with the closure relation, define the complete set of elementary EL equations for (admissable/integrable) Lagrangians 3-forms.
SLIDE 32 Variational principle for Lagrange 1-forms and multi-time systems
To propose a system of Lagrangians associated with higher-time variables t = (t1, t2, . . . ) we consider a Lagrangian 1-form: L =
Lk(x(t), xt1(t), xt2(t), . . . ) dtk with components Lk. The action becomes a functional of the type S[x(t); Γ ] =
L(x(t), xt) = s1
s0
- k
- Lk(x(t(s)), xt1(t(s)), xt2(t(s)), . . . ) dtk
ds
where the functions (t(s)) , s0 ≤ s ≤ s1 form a parametrization of the curve Γ. The variational calculus in terms of 1-forms involves two types of variations:
◮ variations t(s) → t(s) + δt(s) of the curve, i.e. Γ → Γ′ ◮ variations x(t) → x + δx(t) of the dependent variables on a fixed curve Γ;
t1 t2 t1(s0), t2(s0) Γ′ t1(s1), t2(s1) t1 x t2 t1(s0), t2(s0) Γ t1(s1), t2(s1) x(t1(s0), t2(s0)) x(t1(s1), t2(s1)) E ′
Γ
SLIDE 33 1-Form Euler-Lagrange equations
The corresponding set of EL equations (in the 2-time case) comprises the following relations:
◮ variations t → t + δt w.r.t. the independent variables leads to a closure relation
in the form: ∂L2 ∂t1 = ∂L1 ∂t2 ,
◮ variations of the dependent variable x → x + δx on an arbitrary curve Γ in
time-space. Again, we have to distinguish between contributions (δxt) from the variations of the derivatives along the curve, leading to: ∂L1 ∂x dt1 ds + ∂L2 ∂x dt2 ds − d ds
dt/ds2 × dt1 ds 2 ∂L1 ∂xt1 + dt1 ds dt2 ds ∂L1 ∂xt2 + ∂L2 ∂xt1
dt2 ds 2 ∂L2 ∂xt2
◮ and contributions (δxt)⊥ from the variations of the derivatives perpendicular to
the curve, leading to the system of constraint equations: ∂L2 ∂xt1 dt2 ds 2 + ∂L1 ∂xt1 − ∂L2 ∂xt2 dt1 ds dt2 ds − ∂L1 ∂xt2 dt1 ds 2 = 0. Remark: Applied to the cases of the Calogero-Moser, the (finite) Toda chain and the Ruijsenaars-Schneider systems, this leads to to the construction of the higher-time Lagrangians which are naturally mixed objects in all the time-derivatives.
SLIDE 34 Lagrangian 1-form structure implies integrability
We will now consider a class of Lagrangians with given kinetic terms, but arbitrary potentials. The form of the Lagrangian 1-form components of L = L1 dt1 + L2 dt2, are assumed to be: L1 =
N
1 2 ∂Xi ∂t1 2 +
N
V (Xi − Xj) , L2 =
N
∂Xi ∂t1 ∂Xi ∂t2 + α ∂Xi ∂t1 3 +
N
i=j
∂Xi ∂t1 W (Xi − Xj) . The Euler-Lagrange equations for the t1- resp. t2 components yields ∂2Xi ∂t2
1
= 2
V ′(Xi − Xj) , ∂2Xi ∂t1 ∂t2 =
∂t1 − W ′(Xj − Xi) ∂Xj ∂t1
Furthermore, we have the the constraint: ∂L2 ∂( ∂Xi
∂t1 )
= 3α ∂Xi ∂t1 2 + ∂Xi ∂t2 +
N
W (Xi − Xj) = 0 .
SLIDE 35 The consistency between the constraint and the EL equations lead to the folowing relation between the potentials V and W : W (x) = −6αV (x) + constant . We now consider the closure relation: ∂L1 ∂t2 = ∂L2 ∂t1 . which leads next to the following condition on W :
N
i=j,i=l,j=l
W ′(Xi − Xj)W (Xj − Xl) = 0 . In particular this should hold for the 3-particle case, and hence setting N = 3 we get a functional relation for W of the form: W ′(x) [W (y) − W (x + y)]−W ′(y) [W (x) − W (x + y)] = W ′(x +y) [W (x) − W (y)] which we easily recognise as the addition formula for the Weierstrass ℘-function:
℘(x) ℘′(x) 1 ℘(y) ℘′(y) 1 ℘(x + y) −℘′(x + y)
thus leading to the general solution of the functional equation: W (x) = β℘(x) + γ (up to a (trivial) scaling freedom in the argument x). Thus, we recover precisely the integrable cases of Calogero-Moser potentials from the Lagrangian 1-form structure!
SLIDE 36 Relativistic case
Here the Lagrangian components L1 and L2 of the 1-form action are: L1 =
N
∂Xi ∂t1 ln
∂t1
N
∂Xj ∂t1 V (Xi − Xj) , L2 =
N
∂t2 ln
∂t1
∂Xi ∂t1 2 + β ∂Xi ∂t2
N
∂Xj ∂t2 W (Xi − Xj) + ∂Xi ∂t1 ∂Xj ∂t1 U(Xi − Xj)
The EL equations for the t1-component,
∂L1 ∂Xi − ∂ ∂t1
∂(∂Xi /∂t1)
∂2Xi ∂t2
1
=
∂Xi/∂t1 ∂Xj ∂t1
- V ′(Xi − Xj) − V ′(Xj − Xi)
- ,
while for the t2-component we have
∂L2 ∂Xi − ∂ ∂t2
∂(∂Xi /∂t2)
∂2Xi ∂t1 ∂t2 =
∂Xi ∂t1 ∂Xj ∂t2
- W ′(Xi − Xj) − W ′(Xj − Xi)
- +
- j=i
∂Xi ∂t1 2 ∂Xj ∂t1
- U′(Xi − Xj) − U′(Xj − Xi)
- ,
SLIDE 37 Furthermore, we have the constraint: ∂L2 ∂( ∂Xi
∂t1 )
= 2α ∂Xi ∂t1 + ∂Xi ∂t2 / ∂Xi ∂t1 +
N
∂Xj ∂t1
- U(Xi − Xj) + U(Xj − Xi)
- = 0 .
Taking its derivative with respect to t1 and eliminating the second order derivatives we get : [U(x) + U(−x)]
- V ′(y) − V ′(−y) + V ′(z) − V ′(−z)
- + [U(z) + U(−z)]
- V ′(x) − V ′(−x) − V ′(y) + V ′(−y)
- = [U(y) + U(−y)]
- W ′(x) − W ′(−x) + W ′(z) − W ′(−z)
- ,
in addition to (discarding trivial cases): V (x) + V (−x) = W (x) + W (−x) + const [U(x) + U(−x) − 2α]
- V ′(x) − V ′(−x)
- = U′(x) − U′(−x) ,
the latter leading to the identification: V (x) + V (−x) = ln (U(x) + U(−x) − 2α) + const. Introducing the function Φ(x) := V (x) + V (−x) we obtain the functional relation eΦ(x) Φ′(y) + Φ′(z)
Φ′(x) + Φ′(z)
Φ′(x) − Φ′(y)
with z = x + y. A solution of this functional equation is given by Φ(x) = − ln (a + b℘(x)) , a, b constants . which corresponds to the full elliptic case of the Ruijsenaars (relativistic CM) system. Conclusion: By posing general forms for the potentials in the higher-order Lagrangians, with given forms for the kinetic terms, the Lagrangian 1-form structure selects the integrable cases.
SLIDE 38 ”Beyond quantum mechanics”
Paul Dirac, in his seminal paper10 of 1933, stated: ”The two formulations [namely that of Hamilton and of Lagrange] are, of course, closely related but there are reasons for believing that the Lagrangian one is more fundamental.” Dirac’s paper contains already the key ideas underlying the path integral, later introduced by Feynman. Based on the Lagrangian 1-form structure one could make a first (tentative) proposal for a quantum multi-form Lagrange theory, namely by considering (` a la Feynman) a 1-form quantum propagator: K(xb, tb, sb; xa, ta, sa) = t(sb)=tb
t(sa)=ta
[Dt(s)] x(tb)=xb
x(ta)=xa
[DΓx(t)] exp (iS[x(t); Γ]) . Here:
◮ [DΓx(t)] is some path integral measure along a curve Γ in the space of dependent
variables x(t);
◮ Γ is a curve in the space of independent variables, parametrised by the parameter
s ∈ [sa, sb], bounded by the points t(sa) = ta and t(sb) = tb;
◮ [Dt(s)] is some path integral measure in the space of independent variables; ◮ the action functional S is given by the Lagrangian 1-form action:
S[x(t); Γ] =
L·dt = sb
sa
ds
N
Lj(x(t(s)), xt1(t(s)), . . . , xtN (t(s)); t(s))· dtj ds .
10P.A.M. Dirac, The Lagrangian in Quantum Mechanics, Physikalische Zeitschrift der Sowjetunion, Bd. 3, Heft
1, (1933)
SLIDE 39 Other connections to Physics
These are PDEs in terms of the lattice parameters associated with a given multidimensionally consistent lattice system. Key Examples:
- Generating PDE for the KdV hierarchy11:
L = 1 4 (p2
i − p2 j )
(∂pi ∂pj u)2 (∂pi u)(∂pj u) + 1 p2
i − p2 j
i p2 i
∂pj u ∂pi u + n2
j p2 j
∂pi u ∂pj u
The equations of motion are multidimensionally consistent in a very similar way as the corresponding lattice equation. Furthermore, this Lagrangian is PSL2(C) invariant, and is reduces under similarity reduction to the full-parameter Painlev´ e VI equation. Physically, it represents a generalization of the Ernst equation of General Relativity.
- Generating PDE for the Boussinesq hierarchy given by the Lagrangian12:
Lij = p3
i − p3 j
J2
ij
FijFji + 3nip2
i
Fij Jij − 3njp2
j
Fji Jji , where Jij =
∂pi u(2) ∂pj u(1) ∂pj u(2)
Fij =
∂pi u(2) ∂pi ∂pj u(1) ∂pi ∂pj u(2)
The Lagrangian is PSL3(C) invariant and the EL equations are again multidimensionally consistent. They reduce (under scaling invariant reduction) to a full 6-parameter case of a Garnier type system. Physically, the equations of motion are generalizations of the Einstein-Maxwell-Weyl equations (for gravitational waves in the presence of Maxwell and neutrino fields).
11FWN, A N Hone & N Joshi, On a Schwarzian PDE associated with the KdV hierarchy, Phys. Lett. A267
(2000) 147–156.
12A Tongas & F W Nijhoff, Generalized hyperbolic Ernst equations for an Einstein-Maxwell-Weyl field, J. Phys.
A 38 (2005) 895–906.
SLIDE 40 Concluding Remarks: Main conclusion: Interpreted within the Lagrangian multi-form scheme, ”integrable Lagrangians” can be seen as critical points of a more fundamental system, namely that of the generalized EL system which involves the variations w.r.t. the space of independent variables.
- The proposed extension of Feynman’s path integral for Lagrangian 1-forms touches
ground with some modern points of view developed in the theory of quantum gravity, in particular those related to background-independent and time-scaling invariant quantum theories (cf., e.g., C. Rovelli, 2011)13
- The above extension of Feynman’s path integral suggests that there is a
“democracy” between dependent and independent variables (as well as the parameters) of the system, where all are effectively on the same footing. Connections with recent interpretations in terms finite geometries (J. Atkinson) suggest similar aspects.
- The connection with work mentioned in statistical mechanics, in particular the
remarkable connection between Q4 and a master solution of the star-triangle relation14 is suggestive of an extension to 2D integrable quantum models. This suggest that possible connections with aspects of quantum geometry.
- 13C. Rovelli, On the structure of background independent quantum theory: Hamilton function, transition
amplitudes, classical limit and continuous limit, arXiv:1108.0832.
14V V Bazhanov & S M Sergeev, A master solution of the quantum Yang-Baxter equation and classical discrete
integrable equations, arXiv 1006.0651.
SLIDE 41 Some references:
- 1. S. Lobb & FWN: Lagrangian multiforms and multidimensional consistency, J. Phys. A:Math
- Theor. 42 (2009) 454013
- 2. S. Lobb, FWN & R. Quispel, Lagrangian multiform structure for the lattice KP system, J.
- Phys. A:Math Theor. 42 (2009) 472002
- 3. S. Lobb & FWN, Lagrangian multiform structure for the lattice Gel’fand-Dikii hierarchy, J.
- Phys. A:Math. Theor. 43 (2010) 072003
- 4. A. Bobenko and Yu. Suris, On the Lagrangian Structure of Integrable Quad-Equations, Lett.
- Math. Phys. 92 (2010) 17–31
- 5. P. Xenitidis, FWN & S. Lobb, On the Lagrangian formulation of multidimensionally
consistent systems, Proc. Roy. Soc. A467 # 2135 (2011) 3295-3317
- 6. S. Yoo-Kong, S. Lobb and F.W. Nijhoff, Discrete-time Calogero-Moser system and
Lagrangian 1-form structure, J. Phys. A: 44 (2011) 365203
- 7. J. Atkinson, S.B. Lobb and F.W. Nijhoff, An integrable multicomponent quad equation and
its Lagrangian formalism, Theor. Math. Phys 173 (2012) # 3 pp. 1644-1653
- 8. S. Yoo-Kong and F.W. Nijhoff, Discrete-time Ruijsenaars-Schneider system and Lagrangian
1-form structure, arXiv:1112.4576
- 9. Yu. Suris, Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian
1-forms, arXiv: 1212.3314
- 10. R. Boll, M. Petrera and Yu. Suris, Multi-time Lagrangian 1-forms for families of B¨
acklund
- transformations. Toda-type systems, arXiv:1302.7144
- 11. Yu. Suris, Variational symmetries and pluri-Lagrangan systems, arXiv:1307.2639.
- 12. S. Lobb and F.W. Nijhoff, A variational principle for discrete integrable systems,
arXiv:1312.1440.
- 13. R. Boll, M. Petrera and Yu. Suris, What is integrability of discrete variational systems?,
arXiv:1307.0523.
- 14. Yu.B Suris and M. Vermeeren, On the Lagrangian Structure of Integrable Hierarchies,
arXiv:1510.03724.
