A modified formal Lagrangian formulation for general differential equations
aa r X i v : . [ m a t h - ph ] S e p A modified formal Lagrangian formulation for generaldifferential equations
Linyu Peng ∗ Department of Mechanical Engineering, Keio University,Yokohama 223-8522, Japan
September 10, 2020
Abstract
In this paper, we propose a modified formal Lagrangian formulation by introducing dummydependent variables and prove the existence of such a formulation for any system of differentialequations. The corresponding Euler–Lagrange equations, consisting of the original system andits adjoint system about the dummy variables, reduce to the original system via a simplesubstitution for the dummy variables. The formulation is applied to study conservation laws ofdifferential equations through Noether’s Theorem and in particular, a nontrivial conservationlaw of the Fornberg–Whitham equation is obtained by using its Lie point symmetries. Finally, acorrespondence between conservation laws of the incompressible Euler equations and variationalsymmetries of the relevant modified formal Lagrangian is shown.
Keywords:
Modified formal Lagrangians; Self-adjointness; Symmetries; Conservation laws;Noether’s Theorem
A lot of differential equations arising from physical phenomena can be derived from variationalcalculus, that studies extrema of functionals, i.e., functions whose arguments are also functions.Variational structure not only allows us to study geometric properties of differential equationssystematically (e.g., [4, 5, 26]), but also serves as an important framework for the developmentof geometric numerical integrators (e.g., [12, 25, 27]). Another great advantage of a variationalstructure is that Noether’s Theorem can immediately be applied to derive conservation laws ofvariational differential equations. Noether’s Theorem, establishing a one-to-one correspondencebetween variational symmetries and conservation laws of the Euler–Lagrange equations, wasproved by Emmy Noether and published in 1918 [28]; see [29] for a modern version and [20] fora history of Noether’s Theorem together with her second theorem.Conservation laws are among the most important properties of differential equations. Toapply Noether’s Theorem, it is necessary to study inverse problems, namely, to distinguishvariational differential equations from the others, and to find the corresponding functionalwhen the system is variational. Unfortunately, a variational structure is not always available forgeneral differential equations. Many methods for deriving conservation laws of general or specialtype of differential equations, nevertheless, have been developed, for instance, Vinogradov’s C -spectral sequence [5, 35, 36], symbolic methods [11, 14], the direct construction method ofAnco & Bluman [2, 3], using partial Lagrangians [19], the formal variational structure andself-adjointness method [10, 16, 17], etc. ∗ Adjunct faculty member at the School of Mathematics and Statistics, Beijing Institute of Technology, Beijing100081, China, and adjunct researcher at the Waseda Institute for Advanced Study, Waseda University, Tokyo 169-8050, Japan. Email: [email protected] n particular, the formal variational structure firstly proposed by Ibragimov defines a formalLagrangian L for a general system of differential equations { F α = 0 , α = 1 , , . . . , l } byintroducing dummy dependent variables v , namely L := X α v α F α . In other words, any system of differential equations can be embedded in a bigger system ofEuler–Lagrange equations consisting of the original system and its adjoint system. Moreover,symmetries of the original system can be extended to variational symmetries of the correspond-ing formal variational problem, and hence conservation laws can be derived using Noether’sTheorem. If, through a proper substitution for the dummy variables, the Euler–Lagrange equa-tions reduce to the original system, namely, the adjoint system is equivalent to the originalsystem, called the self-adjointness of the system, then the so-obtained conservation laws turninto conservation laws of the original system through the same substitution. Although theseconservation laws can sometimes be trivial, this method provides a straightforward algorithmfor computing conservation laws of non-variational differential equations using Noether’s The-orem. Furthermore, it also makes the development of variational integrator for non-variationaldifferential systems possible; see, e.g., [21]. Some studies on extensions to discrete equationsare available, e.g., [32, 33].It was realised, unfortunately, that self-adjointness of many differential equations can notbe expressed using simple/unified substitutions for the dummy variables. More complex sub-stitutions were introduced and successfully applied to some differential equations, for instance,the weak self-adjointness [10] and nonlinear self-adjointness [17]. However, they are often case-by-case depending on the system of interest. In this paper, we propose a systematical methodthat is applicable to study conservation laws of all differential equations by deriving their self-adjointness through the simplest substitution for dummy variables v . This is made possibleby modifying a formal Lagrangian by adding an extra so-called balance function L , which isindependent from the dummy variables: b L := X α v α F α + L This method will be called a modified formal Lagrangian formulation .To make the paper self-contained, we will review relevant fundamental theories on symmetryanalysis in Section 2, for instance, the linearized symmetry condition for determining symmetriesof differential equations, symmetries of variational problems and conservation laws of Euler–Lagrange equations obtained from Noether’s Theorem, and a brief introduction to the formalLagrangian method proposed by Ibragimov. Readers who are familiar with these topics andnotations may move to Section 3 directly. In Section 3, we define the modified formal Lagrangianformulation for a general system of differential equations. Algorithms for extending symmetriesof a system of differential equations to variational symmetries of the corresponding modifiedformal Lagrangian are given too. The viscous Burgers’ equation is used as an illustrative runningexample. Further concrete examples will be studied in Section 4, including the derivation ofa nontrivial conservation law for the Fornberg–Whitham equation using a symmetry extendedfrom its Lie point symmetries and a correspondence between variational symmetries of modifiedformal Lagrangians and conservation laws of fluid equations.
In this section, we briefly review the linearized symmetry condition for computing symmetriesof differential equations and conservation laws obtained through Noether’s Theorem; details canbe found in, e.g., Olver’s book [29]. Ibragimov’s formal Lagrangian approach for computingconservation laws will also be reviewed. .1 The linearized symmetry condition For a system of differential equations, let x = ( x , x , . . . , x p ) ∈ R p be the independent variablesand let u = ( u , u , . . . , u q ) ∈ R q be the dependent variables. In the examples though, we willuse t and x to denote the time and space as the independent variables. A system of differentialequations is defined on the jet bundles (cf. [5, 22, 34]) coordinated with( x, [ u ])where [ u ] denotes u and sufficiently many of their derivatives, written in terms of the multi-indexnotations u α J = ∂ | J | u α ∂ ( x ) j ∂ ( x ) j . . . ∂ ( x p ) j p , where J = ( j , j , . . . , j p ) and | J | = j + j + · · · + j p . Each index j i is a non-negative integer,denoting the number of total derivatives with respect to the independent variable x i . Therefore,a system of differential equations can be written locally as A = { F α ( x, [ u ]) = 0 , α = 1 , , . . . , q } . (2.1)Note than we assumed that the number of equations in the system (2.1) is the same as thedimension of dependent variables u . For simplicity, we will often assume that the system isanalytic and totally nondegenerate, the latter of which means the system itself and its prolon-gations are of maximal rank and locally solvable; see, e.g., [29].Consider the following local transformations around ε = 0 x e x ( x, u, ε ) , u e u ( x, u, ε ) , subject to e x ( x, u,
0) = x, e u ( x, u,
0) = u, (2.2)which are then prolonged to the derivatives u α J . They form a local symmetry group of thesystem (2.1) if and only if the group maps one solution u = f ( x ) to another solution e u = e f ( e x ).It is often more convenient to use the corresponding infinitesimal generator X = ξ i ( x, u ) ∂∂x i + φ α ( x, u ) ∂∂u α , (2.3)where ξ i ( x, u ) := dd ε (cid:12)(cid:12)(cid:12) ε =0 e x i , φ α ( x, u ) := dd ε (cid:12)(cid:12)(cid:12) ε =0 e u α . Note that the Einstein summation convention is used from now on. Prolongation of the trans-formations (2.2) to higher jets yields prolongation of the infinitesimal generator given bypr X = ξ i D i + Q α ∂∂u α + · · · + ( D J Q α ) ∂∂u α J + · · · . (2.4)The tuple Q with Q α = φ α − ξ j D i u α is called the characteristic of X and D i is the totalderivative with respect to x i : D i := ∂∂x i + u α i ∂∂u α + · · · + u α J + i ∂∂u α J + · · · , where i is the p -tuple with only one nonzero entry 1 at the i -th place. The multi-index notation D J denotes a multiple number of total derivatives: D J = D j D j · · · D j p p , J = ( j , j , . . . , j p ) . For a system of differential equations (2.1) satisfying the nondegeneracy condition, i.e.,of maximal rank and locally solvable, a vector field X generates a symmetry group of localtransformations if and only if the linearized symmetry condition is satisfied (e.g., [15, 29]),namely pr X ( F α ) = 0 , α = 1 , , . . . , q, whenever the system (2.1) holds . (2.5) he total nondegeneracy and analyticity conditions further allow us to express the linearizedsymmetry condition equivalently to the existence of q × q matrices ( K J ( x, [ u ])) whose entriesare functions of ( x, [ u ]) such thatpr X ( F α ) = X β, J K J αβ ( D J F β ) , α = 1 , , . . . , q. (2.6) Remark 2.1.
Symmetries corresponding to an infinitesimal generator of the form (2.3) arecalled Lie point symmetries. They will be called generalised symmetries when coefficients of theinfinitesimal generator not only depend on x and u but also on derivatives of u , namely whenthe infinitesimal generator is of the form X = ξ i ( x, [ u ]) ∂∂x i + φ α ( x, [ u ]) ∂∂u α . Example 2.2.
The Korteweg–de Vries (KdV) equation u t + uu x + u xxx = 0 (2.7) admits a four-dimensional group of Lie point symmetries generated by X = ∂ t , X = ∂ x , X = t∂ x + ∂ u , X = 3 t∂ t + x∂ x − u∂ u . Consider a variational problem with a functional L [ u ] = Z Ω L ( x, [ u ]) d x (2.8)defined in an open, connected subspace Ω with smooth boundary, where the smooth function L ( x, [ u ]) is called a Lagrangian (density function). Variational calculus leads to the Euler–Lagrange equations E u α ( L ) = 0, α = 1 , , . . . , q , which are written using the Euler operatorsE u α : = X J ( − D ) J ∂∂u α J = ∂∂u α − D i ∂∂u α i + D i D j ∂∂u α i + j − · · · , (2.9)where ( − D ) J is the adjoint of the operator D J : ( − D ) J = ( − | J | D J .Invariance of the variational problem (2.8) with respect to the transformations (2.2) can beexpressed as the infinitesimal invariance criterion [29]pr X ( L ) + LD i ξ i = Div A (2.10)for some p -tuple A ( x, [ u ]), where X is the corresponding infinitesimal generator (2.3). In fact,this can be extended to generalised symmetries equally. Such symmetries generated by X arecalled (divergence) variational symmetries. The divergence of a p -tuple A is defined asDiv A := D i A i . A conservation law of a system (2.1) is a divergence expression of a p -tuple P ( x, [ u ])Div P = 0that vanishes on solutions of the system. Conservation laws can be trivial in two ways: Thefirst kind is that the p -tuple P itself vanishes on solutions of the system, e.g., when eachcomponent P i is a linear combination of the equations { F α } ; The second kind is Div P ≡ u = f ( x ). In particular, for a totally nondegenerate system f differential equations, a conservation law can be equivalently understood as the existence offunctions M J α ( x, [ u ]) such that Div P = X α, J M J α ( D J F α ) , (2.11)which can be integrated by parts to yield its characteristic formDiv b P = Q α F α . (2.12)Here Q ( x, [ u ]) is the characteristic of the equivalent conservation laws P and b P . If furtherthe system is analytic, then a conservation law is trivial if and only if its characteristic Q istrivial; a trivial characteristic is defined as Q = 0 holds on solutions of the system (2.1) . (2.13)When a characteristic is given, the divergence form can be derived by using the homotopyoperator on the variational bicomplex providing the cohomology is trivial (e.g., [4, 5]) or byintuition. A general formula is also available in [29].For a variational problem, Noether’s Theorem establishes a one-to-one correspondence be-tween variational symmetries and conservation laws of the Euler–Lagrange equations. Theorem 2.3 ( Noether’s Theorem).
Suppose that a vector field X = ξ i ( x, [ u ]) ∂∂x i + φ α ( x, [ u ]) ∂∂u α satisfies the infinitesimal invariance criterion (2.10) for a variational problem (2.8) . Then itscharacteristic Q α = φ α ( x, [ u ]) − ξ i ( x, [ u ]) u αi is also the characteristic of a conservation lawfor the corresponding Euler–Lagrange equations E u α ( L ) = 0 . Namely, there exists a p -tuple P ( x, [ u ]) such that Div P = Q α E u α ( L ) . (2.14)Various proofs can be found in different contexts. We briefly review Olver’s proof by inte-grating the identity (2.10) by parts:Div A = pr X ( L ) + LD i ξ i = ξ i D i L + X α, J ( D J Q α ) ∂L∂u α J + LD i ξ i = D i ( Lξ i ) + Q α X J ( − D ) J ∂L∂u α J + Div B (2.15)for some p -tuple B ( x, [ u ]). The resulting conservation law isDiv P = Q α E u α ( L ) , where P = A − Lξ − B. (2.16) Example 2.4.
Let us consider the -dimensional linear wave equation u tt − c u xx = 0 asan illustrative example, where c = 0 . The corresponding Lagrangian is L ( u t , u x ) = − u t + c u x and the wave equation can be equivalently written as E u ( L ) = 0 . Since the Lagrangian isexplicitly independently from t and x , it admits the time translational and space translationalsymmetries, namely t t + ε , x x + ε , whose infinitesimal generators are respectively ∂ t , ∂ x . he corresponding conservation laws are written in terms of the characteristics − u t and − u x ,namely D t (cid:18) − u t − c u x (cid:19) + D x (cid:0) c u t u x (cid:1) = − u t ( u tt − c u xx ) ,D t ( − u t u x ) + D x (cid:18) u t + c u x (cid:19) = − u x ( u tt − c u xx ) . Note that in the integration by parts formula (2.15) , A ≡ for both infinitesimal generators. In this subsection, we brief review formal Lagrangians and the self-adjointness approach forcomputing conservation laws.By introducing dummy dependent variables v with the same dimension of the dependentvariables u , the formal Lagrangian for a system of differential equations (2.1) is defined as L ( x, [ u ; v ]) := v α F α ( x, [ u ]) . (2.17)The corresponding Euler–Lagrange equations consist of two parts, namely the original system0 = E v α ( L ) ≡ F α ( x, [ u ]) , α = 1 , , . . . , q, (2.18)and the so-called adjoint system0 = E u α ( L ) := F ∗ α ( x, [ u ; v ]) , α = 1 , , . . . , q. (2.19)The system (2.18) is said to be (quasi) self-adjoint if the adjoint system (2.19) is equivalent toitself via a proper substitution v = h ( u ). In other words, Euler–Lagrange equations governedby the formal Lagrangian reduce to the original system via the substitution v = h ( u ). Remark 2.5.
It was realised that many systems are not self-adjoint through a substitution v = h ( u ) . In recent years, there have been generalisations to, for instance, weak self-adjointness andnonlinear self-adjointness (e.g., [10, 17]), that, however, have been found restricted in derivingnontrivial conservation laws (e.g., [13]); such an example is studied in Section 4.1. Here, weonly introduce the simplest case of self-adjointness. An important observation by Ibragimov [16] is that any symmetry generator X of theoriginal system (2.18) can be extended to a variational symmetry generator X + φ α ∗ ∂ v α forthe formal Lagrangian, yielding a conservation law of the Euler–Lagrange equations (2.18) and(2.19) using Noether’s Theorem. Taking the self-adjointness condition into consideration, thisconservation law becomes a conservation law of the original system (2.18) via the substitution v = h ( u ). Although non triviality and completeness of so-obtained conservation laws arenot promised [1], its simplicity for implementation is a great advantage while on the otherside the formal Lagrangian structure provides necessary foundations for conducting variationalintegrator [21]. We will illustrate the algorithm by considering the KdV equation (2.7) as anexample. Example 2.6. (KdV equation continued.)
The formal Lagrangian for the KdV equation is L = vF where F = u t + uu x + u xxx , and the adjoint equation is F ∗ := − v t − uv x − v xxx = 0 . The adjoint equation turns into the KdV equation through the substitution v = u : F ∗ (cid:12)(cid:12)(cid:12) v = u = − F. Lie point symmetries of the KdV equation were given in Example 2.2. Here we computerthe conservation law corresponding to the scaling symmetry X = 3 t∂ t + x∂ x − u∂ u . he extended variational symmetry for the formal Lagrangian is generated by Y = X + v ∂∂v . By using the characteristics Q u = − u − tu t − xu x and Q v = v − tv t − xv x , the conservationlaw is written in characteristic form D t P t ( t, x, [ u ; v ]) + D x P x ( t, x, [ u ; v ]) = Q u F ∗ + Q v F. Substituting v = u inside, we obtain a conservation law of the KdV equation, the conservationof momentum, as follows D t (cid:18) u (cid:19) + D x (cid:18) u + uu xx − u x (cid:19) = uF. The formal Lagrangian method defined by Ibragimov is limited even when we are restricted toevolutionary equations. Nonlinear improvements have been introduced but they are, in manysituations, case-by-case, in particular to determine the substitution of dummy variables. In thissection, we propose a modification of formal Lagrangians that is applicable to any differentialequations, that we will call the modified formal Lagrangian formulation. Let us start with amotivating example, the viscous Burgers’ equation, which serves as a running example in thissection.
Example 3.1.
The viscous Burgers’ equation reads u t + uu x − au xx = 0; we assume that the viscosity a is nonzero. By using the usual formal Lagrangian method,the adjoint equation for a dummy variable v can be calculated from the formal Lagrangian v ( u t + uu x − au xx ) , namely − v t − uv x − av xx = 0 , which is not equivalent to the viscous Burgers’ equation via any substitution v = h ( u ) .However, if we modify the formal Lagrangian by adding an extra term − au x and define amodified formal Lagrangian as follows b L := v ( u t + uu x − au xx ) − au x , the corresponding (modified) adjoint equation is equivalent to the viscous Burgers’ equation viathe substitution v = u . In fact, the modified Euler–Lagrange equations consist of two parts:variation w.r.t. v gives the viscous Burgers’ equation, while variation w.r.t. u gives the adjointequation, reading E u ( L ) ≡ − v t − uv x − av xx + 2 au xx . Substituting v = u inside gives an equation differing with the Burger’s equation by a minus sign. This example motivates the definition of a modified formal Lagrangian formulation below.
Definition 3.2.
For a system of differential equations (2.1) , namely, A = { F α ( x, [ u ]) = 0 , α = 1 , , . . . , q } , introduce dummy dependent variables v ∈ R q . If there exists a function L ( x, [ u ]) such that theEuler–Lagrange equations governed by the Lagrangian b L ( x, [ u ; v ]) = v α F α ( x, [ u ]) + L ( x, [ u ]) (3.1) reduce to the original system (2.1) via the substitution v = u , then we call the Lagrangian b L ( x, [ u ; v ]) a modified formal Lagrangian and the corresponding function L ( x, [ u ]) a bal-ance function . o distinguish from formal Lagrangians, we will use b L to denote a modified formal La-grangian in the current paper. There are several fundamentally important remarks or factsregarding the modification. Some of them are as follows. • A first remark is that the substitution can be chosen arbitrary as v = h ( x, [ u ]) where h ( x, [ u ]) are arbitrary functions, but v = u is among the simplest ones such that theadjoint system is equivalent to the original system. • Secondly, since the balance function L is independent from v , half of the modified Euler–Lagrange equations, i.e., E v α ( b L ) = 0, is exactly the original system. • A balance function exists for any system of differential equations but not necessarilyuniquely. The existence will be proved in Theorem 3.3. It is not unique due to the existenceof null Lagrangians, namely functions written in a divergence form; see, e.g., [29, 30]. • When the balance function can be written in a divergence form, the corresponding modifiedformal Lagrangian becomes a formal Lagrangian, namely without modification.
Theorem 3.3.
For any system of differential equations A = { F α ( x, [ u ]) = 0 , α = 1 , , . . . , q } , there exists a generic modified formal Lagrangian b L ( x, [ u ; v ]) : = v α F α ( x, [ u ]) − u α F α ( x, [ u ])= ( v α − u α ) F α ( x, [ u ]) . (3.2) The function L ( x, [ u ]) = − u α F α ( x, [ u ]) will be called a generic balance function.Proof. We only need to show that the corresponding Euler–Lagrange equations reduce to theoriginal system A via the substitution v = u . Now the modified Euler–Lagrange equations read0 = E v α ( b L ) ≡ F α ( x, [ u ]) , E u α ( b L ) := b F ∗ α ( x, [ u ; v ]) . (3.3)Direct computation expands the modified adjoint system as follows b F ∗ α = E u α (cid:16) v β F β − u β F β (cid:17) = X β, J ( − D ) J (cid:18)(cid:16) v β − u β (cid:17) ∂F β ∂u α J (cid:19) − F α , (3.4)that obviously reduces to the original system with the substitution v = u , i.e., b F ∗ α (cid:12)(cid:12)(cid:12) v = u = − F α . (3.5)This completes the proof. Remark 3.4.
The relation between modified adjoint system and the adjoint system (2.19) (without modification) is b F ∗ α ( x, [ u ; v ]) = F ∗ α ( x, [ u ; v ]) + E u α ( L ) . (3.6)In practice, the balance function may include total derivative terms that we often prefer tomod out since they have no contribution in the Euler–Lagrange equations. For instance, thebalance function − au x for the viscous Burgers’ equation in Example 3.1 is equivalent to thegeneric one − u ( u t + uu x − au xx ) by differing a divergence u ( u t + uu x − au xx ) − au x = D t (cid:18) u (cid:19) + D x (cid:18) u − auu x (cid:19) . (3.7)Since there exists a modified formal Lagrangian for any system of differential equations, itwould be interesting to consider some well-known examples. Evolutionary equations u αt = f α ( x, t, [ u ] x ) , α = 1 , , . . . , q, (3.8)where the short hand notation [ u ] x denotes u and finitely many of their derivatives w.r.t.to x only. Note that x can be multi-dimensional. The generic modified formal Lagrangianreads b L = X α v α ( u αt − f α ) − X α u α ( u αt − f α ) . (3.9)An equivalent modified formal Lagrangian is b L = X α v α ( u αt − f α ) + X α u α f α . (3.10) • A family of Camassa–Holm-type equations u t − ǫu xxt = g ( x, t, [ u ] x ) , ε = 0 . (3.11)The generic modified formal Lagrangian reads b L = v ( u t − ǫu xxt − g ) − u ( u t − ǫu xxt − g ) , (3.12)which is equivalent to b L = v ( u t − ǫu xxt − g ) + ǫuu xxt + ug. (3.13)For concrete examples, further divergence terms can appear and they can also be modded out.Next, we are going to show the connections between symmetries of the original system andvariational symmetries of the modified formal Lagrangian. Such connections allow us to deriveconservation laws of the modified Euler–Lagrange equations using Noether’s Theorem, that canlead to conservation laws of the original system. Theorem 3.5.
Consider a system of differential equations A = { F α ( x, [ u ]) = 0 , α = 1 , , . . . , q } , that is totally nondegenerate and analytic, and that admits a symmetry generated by X = ξ i ( x, [ u ]) ∂∂x i + φ α ( x, [ u ]) ∂∂u α . Then X can be extended to a variational symmetry Y = X + φ α ∗ ( x, [ u ; v ]) ∂∂v α of the generic modified formal functional c L [ u ; v ] := Z Ω b L ( x, [ u ; v ]) d x, where the functions φ ∗ are to be determined and the generic modified formal Lagrangian is b L ( x, [ u ; v ]) := v α F α ( x, [ u ]) − u α F α ( x, [ u ]) . Proof.
First of all, as the system is totally nondegenerate and analytic, the linearized symmetrycondition is replaced by (2.6), namelypr X ( F α ) = X β, J K J αβ ( D J F β ) , α = 1 , , . . . , q, or some functions K J αβ ( x, [ u ]). The extended infinitesimal generator Y satisfies the infinitesimalinvariance criterion for the modified formal functional, that is,pr Y ( b L ) + b LD i ξ i = Div A for some p -tuple A ( x, [ u ; v ]). Its left-hand side can be integrated by parts as followspr Y ( b L )+ b LD i ξ i = pr X ( b L ) + φ α ∗ F α + b LD i ξ i = ( v α − u α ) pr X ( F α ) − φ α F α + φ α ∗ F α + ( v α − u α ) ( D i ξ i ) F α = X α,β, J ( v α − u α ) K J αβ ( D J F β ) − φ α F α + φ α ∗ F α + ( v α − u α ) ( D i ξ i ) F α = X α n ( − D ) J h(cid:16) v β − u β (cid:17) K J βα i − φ α + φ α ∗ + ( v α − u α ) ( D i ξ i ) o F α + Div B (3.14)for some p -tuple B ( x, [ u ; v ]). Clearly, the undetermined functions φ ∗ can be chosen as φ α ∗ = φ α − ( − D ) J h(cid:16) v β − u β (cid:17) K J βα i − ( v α − u α ) ( D i ξ i ) (3.15)and consequently A = B . This finishes the proof.Theorem 3.5 implies that any symmetry of the original system amounts to a conservationlaw of the Euler–Lagrange equations governed by the modified formal Lagrangian. However,be noted that the extension of symmetries may not be unique and the choice in Theorem 3.5,i.e., Equation (3.15), is in fact not the ideal one, because the conservation law correspondingto the so-extended generator Y becomes a trivial conservation law of the original system whenthe substitution v = u is applied: φ α ∗ (cid:12)(cid:12)(cid:12) v = u = φ α and hence Q v α (cid:12)(cid:12)(cid:12) v = u = Q u α , (3.16)and then we have Div P (cid:12)(cid:12)(cid:12) v = u = (cid:16) Q u α b F ∗ α + Q v α F α (cid:17) (cid:12)(cid:12)(cid:12) v = u = (cid:16) − Q u α + Q v α (cid:12)(cid:12)(cid:12) v = u (cid:17) F α = 0 . (3.17)The relation (3.5) is applied here.Fortunately, the extension to a variational symmetry Y may not be unique, particularlywhen − φ α F α can be written in divergence form and hence can be moved into the divergenceDiv B in (3.14). In fact, to derive a nontrivial conservation law for the original system, we mustchoose those extensions such that (3.16) can not happen. Let us consider the running exampleagain. Example 3.6. (The viscous Burgers’ equation continued.)
Symmetries of the viscous Burgers’equation (see Example 3.1) can be calculated using the linearized symmetry condition (2.6) andits Lie point symmetries are generated by the following infinitesimal generators X = ∂ t , X = ∂ x , X = t∂ x + ∂ u ,X = 2 t∂ t + x∂ x − u∂ u , X = t ∂ t + tx∂ x + ( x − tu ) ∂ u . The generic modified formal Lagrangian reads b L = vF − uF, where F := u t + uu x − au xx . The modified adjoint equation is b F ∗ = 0 where b F ∗ = − v t − uv x − av xx + 2 au xx . ake X as an example. Direct calculation shows that pr X ( F ) ≡ . Consequently, Equation (3.14) becomes pr Y ( b L ) = ( φ ∗ − φ ) F = ( φ ∗ − F. (3.18) • The extension (3.15) gives φ ∗ = 1 , and hence Y = t∂ x + ∂ u + ∂ v . This leads to a trivial conservation law of the viscous Burgers’ equation. • Equation (3.18) can be rearranged as follows pr Y ( b L ) = ( φ ∗ − F = φ ∗ F − D t u − D x (cid:18) u − au x (cid:19) . Therefore, we may choose φ ∗ as or − instead of . In both cases, we obtain a nontrivialconservation law of the viscous Burgers’ equation: D t u + D x (cid:18) u − au x (cid:19) = F. As we notice in the example above that the observation needed for obtaining nontrivialconservation laws is relatively strong. Moreover, we often prefer to Lagrangians including no nullinformation, namely without terms written in divergence form. The following theorem providesanother approach for extending symmetries of a system of differential equations to variationalsymmetries of its (not necessary generic) modified formal Lagrangian; in fact, this methodis often more convenient and practical, compared with Theorem 3.5, for deriving nontrivialconservation laws.
Theorem 3.7.
Consider a system of differential equations A = { F α ( x, [ u ]) = 0 , α = 1 , , . . . , q } , that is totally nondegenerate and analytic, and that admits a symmetry generated by X = ξ i ( x, [ u ]) ∂∂x i + φ α ( x, [ u ]) ∂∂u α . Assume b L ( x, [ u ; v ]) := v α F α ( x, [ u ]) + L ( x, [ u ]) . is a modified formal Lagrangian of the system, such that the modified adjoint system is equivalentto the original system via the substitution v = u .If the balance variational problem, whose Lagrangian is the balance function L ( x, [ u ]) , isinvariant w.r.t. X , then X can be extended to a variational symmetry Y = X + φ α ∗ ( x, [ u ; v ]) ∂∂v α of the modified formal functional c L [ u ; v ] := Z Ω b L ( x, [ u ; v ]) d x, where the functions φ ∗ are to be determined. roof. Since the balance variational problem Z Ω L ( x, [ u ]) d x is invariant w.r.t. X , there exists a p -tuple P such thatpr X ( L ) + L D i ξ i = Div P . Then, we havepr Y ( b L ) + b LD i ξ i = pr X ( v α F α + L ) + φ α ∗ F α + ( v α F α + L ) D i ξ i = v α pr X ( F α ) + pr X ( L ) + φ α ∗ F α + v α ( D i ξ i ) F α + L D i ξ i = X α,β, J v α K J αβ ( D J F β ) + v α ( D i ξ i ) F α + φ α ∗ F α + Div P = X α n ( − D ) J h v β K J βα i + v α ( D i ξ i ) + φ α ∗ o F α + Div ( B + P )where the p -tuple B ( x, [ u ; v ]) is the consequence of integration by parts. Therefore, the unde-termined functions φ ∗ can be chosen as φ α ∗ = − n ( − D ) J h v β K J βα i + v α ( D i ξ i ) o (3.19)such that the modified formal functional is invariant w.r.t. Y . Example 3.8. (The viscous Burgers’ equation continued.)
All Lie point symmetries of theviscous Burgers’ equation are listed in Example 3.6. Let us consider the modified formal La-grangian given in Example 3.1: b L = v ( u t + uu x − au xx ) − au x . Recall that it is equivalent to the generic modified formal Lagrangian, leading to the same Euler–Lagrange equations.The balance variational problem with Lagrangian L = − au x is invariant w.r.t. X , X and X such that pr X i ( L ) + L (cid:0) D t ξ ti + D x ξ xi (cid:1) = 0 , i = 1 , , . For each of the three infinitesimal generators, we have pr X i ( F ) ≡ , i = 1 , , . From Equation (3.19) , we obtain φ ∗ = 0 for all of the three infinitesimal generators and hence Y i = X i , i = 1 , , . The corresponding characteristics are Q u = − u t , Q v = − v t ,Q u = − u x , Q v = − v x ,Q u = 1 − tu x , Q v = − tv x . Recall that F = u t + uu x − au xx , b F ∗ = − v t − uv x − av xx + 2 au xx . The corresponding conservation laws of the modified Euler–Lagrange equations are respectivelygiven by D t P ti + D x P xi = Q ui b F ∗ + Q vi F, i = 1 , , , where P t = − uu x v + au xx v + au x , P x = uu t v − au t u x − au tx v + au t v x ,P t = − uv x , P x = uv t − au x + au x v x ,P t = − (1 − tu x ) v, P x = − uv + 2 au x − av x − tu t v − atu x + atu x v x . n fact, they can be simply derived from the integration by parts formula (2.15) and in thisspecial case, we obtain them as P ti = − b Lξ ti − Q ui ∂ b L∂u t , P xi = − b Lξ xi − Q ui ∂ b L∂u x − D x ∂ b L∂u xx ! − ( D x Q ui ) ∂ b L∂u xx , where i = 1 , , . A general formula can be found in [29].Setting v = u in the three conservation laws, only the third one contributes to a nontrivialconservation law of the viscous Burgers’ equation, namely D t u + D x (cid:18) u − au x (cid:19) = F, which is the same as we obtained in Example 3.6. According to the analysis of cohomology of C -spectral sequence, this is the only (local) conservation law for the viscous Burgers’ equation;see, e.g., [5]. Beside the extension of known symmetries of a system to variational symmetries of itsmodified formal functionals through either Theorem 3.5 or Theorem 3.7, one may also usetheir own variational symmetries (not necessary extended from known symmetries) to deriveconservation laws using Noether’s Theorem; see Section 4.2 for an illustrative example fromfluid mechanics.
In this section, we will study some concrete examples from physics and fluid mechanics. In thefirst example, we obtain a nontrivial conservation for the Fornberg–Whitham equation that hasnot been successfully achieved using the previous formal Lagrangian method. For differentialequations from fluid mechanics, we show how to derive conservation laws from a modified formalLagrangian’s variational symmetries, that are not necessary extended from known symmetriesof the original differential equations.
The Fornberg–Whitham (FW) equation is a nonlinear dispersive wave equation, admitting awave of greatest height, e.g., [9]. Symmetry analysis of a bigger family of nonlinear partial differ-ential equations was conducted in [7]. It was shown in [13] (see also [18]) that the FW equationis neither quasi self-adjoint nor weak self-adjoint through the formal Lagrangian approach; al-though it is nonlinearly self-adjoint but only trivial conservation laws could be obtained. Inthis subsection, we will study its modified formal Lagrangian formulation to derive conservationlaws.The FW equation can be written as F = 0 with F = u t − u xxt + u x + uu x − u x u xx − uu xxx . (4.1)It admits a three-dimensional group of Lie point symmetries whose infinitesimal generators are X = ∂ t , X = ∂ x , X = t∂ x + ∂ u . Let us study the conservation law related to X by considering the following modified formalLagrangian b L = vF + L ( x, t, [ u ]) , where v is the dummy dependent variable and the balance function is chosen as L = uu x u xx . t is equivalent to the generic one by modding out all divergence terms. Direct computationgives the modified Euler–Lagrange equations0 = E v ( b L ) ≡ F, E u ( b L ) := b F ∗ , where the modified adjoint equation is b F ∗ = − v t + v xxt − v x − uv x + 3 u x u xx + uv xxx satisfying b F ∗ (cid:12)(cid:12)(cid:12) v = u = − F. First of all, we shall check that the balance variational problem is invariant w.r.t. X .Namely the infinitesimal invariance criterion (2.10) is satisfied for L ; by noting Div ξ ≡
0, wehave pr X ( L ) = D x (cid:18) u x (cid:19) . It can be checked that pr X ( F ) ≡
0, and hence Equation (3.19) gives the extension of X to a variational symmetry Y = X of the modified formal Lagrangian, whose characteristics iswritten in components as Q u = 1 − tu x , Q v = − tv x . The corresponding conservation law for the modified Euler–Lagrange equation is written incharacteristic form as D t P t + D x P x = Q u b F ∗ + Q v F, where P t = − v + v xx − tuv x + tu xx v x ,P x = − v + tuv t + uv xx − u x v x + 32 u x − tu x v xt − tuu x v xx + tu x v x + tuu xx v x − tu x . Substituting v = u inside, it becomes a nontrivial conservation law of the FW equation writtenin characteristic form as follows D t ( u − u xx ) + D x (cid:18) u + 12 u − u x − uu xx (cid:19) = F. (4.2)A remark on the symmetries X and X is that they are also variational symmetries forthe balance variational problem and hence lead to nontrivial conservation laws for the modifiedEuler–Lagrange equations. But no new nontrivial conservation law of the FW equation can beachieved after the substitution v = u is applied. In this subsection, we will show how well-known conservation laws of fluid systems representedin the Eulerian framework can be derived using the modified Lagrangian formulation. In theLagrangian framework, variational formulation for incompressible flow has been known for quitelong time. In this paper, we study the incompressible Euler equations as an example. The samemethodology applies to study other fluid equations, e.g., the compressible Euler equations andthe compressible and incompressible Navier–Stokes equations, and other differential equationsequally. Note that other variational formulations for the incompressible Euler equations exist,for instance, the Clebsch variational principle and a multisymplectic formulation (see [8] andreferences therein for more details).The incompressible Euler equations are the following system of partial differential equationsfor the velocity u ∈ R n ( n = 2 or 3) and the pressure p ∈ R : u t + u · ∇ u + ∇ p = 0 , ∇ · u = 0 . (4.3) imension of the space variable is the same as the velocity, i.e., n = 2 or 3, and time isone-dimensional. This system models the flow of inviscid, incompressible fluid with constantdensity. Note that in this subsection, we use n to denote the dimension of variables rather than p (and q ) used above as it means a different thing in fluid mechanics. Furthermore, the dummydependent variable corresponding to the pressure p will be denoted by q .In Cartesian coordinates, the system can be written in component form as follows ∂u i ∂t + u j ∂u i ∂x j + ∂p∂x i = 0 , i = 1 , , . . . , n,∂u j ∂x j = 0 . (4.4)In this paper, we only consider the three-dimensional case, i.e., n = 3. Introducing dummydependent variables v ∈ R and q ∈ R , we define the modified formal Lagrangian by b L = q (cid:18) ∂u j ∂x j (cid:19) + X i v i (cid:18) ∂u i ∂t + u j ∂u i ∂x j + ∂p∂x i (cid:19) − X i,j u i u j ∂u j ∂x i . (4.5)It is equivalent to the generic one and the modified Euler–Lagrange equations, consisting of theincompressible Euler equations and the adjoint equations, reduce to the incompressible Eulerequations via the substitution v = u and q = p . It is known that when n = 3, the system of incompressible Euler equations admits the followingLie point symmetries (e.g., [29]): • Moving coordinates: f i ∂ x i + f ′ i ∂ u i − f ′′ i x i ∂ p , i = 1 , , • Time translation: ∂ t ; • Scaling: x i ∂ x i + t∂ t ,t∂ t − u i ∂ u i − p∂ p ; • Rotations: x i ∂ x j − x j ∂ x i + u i ∂ u j − u j ∂ u i , i, j = 1 , , i < j ; • Pressure changes: g∂ p . Here the functions f i and g are arbitrary functions of t .According to Theorem 3.7, these symmetries can be extended to variational symmetries ofthe modified formal Lagrangian if they are variational symmetries of the balance variationalproblem, whose Lagrangian is the balance function L = − X i,j u i u j ∂u j ∂x i . (4.6)Using the infinitesimal invariance criterion (2.10), it is immediate to verify that the balancevariational problem is invariant only w.r.t these symmetries: spatial translations (i.e., movingcoordinates with constant functions f i ), time translation, rotations and pressure changes. Theirinfinitesimal generators X can be extended to variational symmetries Y = X + φ i ∗ ∂ v i + φ q ∗ ∂ q of the modified formal Lagrangian according to Theorem 3.7. Extension of spatial translations ∂ x i : Y i = ∂ x i , i = 1 , , • Extension of time translation ∂ t : Y = ∂ t ; • Extension of rotations x i ∂ x j − x j ∂ x i + u i ∂ u j − u j ∂ u i : Y ij = x i ∂ x j − x j ∂ x i + u i ∂ u j − u j ∂ u i + v i ∂ v j − v j ∂ v i , i, j = 1 , , i < j ; • Extension of pressure changes g ( t ) ∂ p : Y = g ( t ) ∂ p . Noether’s Theorem then yields conservation laws of the modified Euler–Lagrange equations,which, by using the substitution v = u and q = p , turn into conservation laws of the incompress-ible Euler equations. We only give the final results without showing intermediate computationaldetails. Only one nontrivial conservation law is obtained after the substitution, that is D x i (cid:0) g ( t ) u i (cid:1) = g ( t ) ∂u i ∂x i , corresponding to the symmetry of pressure changes. This is the conservation of mass . Except those symmetries extended from symmetries of the incompressible Euler equations, themodified formal Lagrangian (4.5) also admits other variational symmetries. They can also beused to compute conservation laws of the incompressible Euler equations.The first kind of infinitesimal generators is b Y i = ∂ v i + u i ∂ q , i = 1 , , , (4.7)which are variational symmetries for the modified formal Lagrangian (4.5), since the infinitesi-mal invariance criterion (2.10) is satisfied for each b Y i , that ispr b Y i ( b L ) = (cid:18) ∂u i ∂t + u j ∂u i ∂x j + ∂p∂x i (cid:19) + u i (cid:18) ∂u j ∂x j (cid:19) = D t u i + D x j (cid:16) δ ji p + u i u j (cid:17) . (4.8)The conservation laws are already written in characteristic form and they correspond to the conservation of momentum .Another variational symmetry is generated by b Y = u i ∂ v i + X i
12 ( u i ) + p ! ∂ q . (4.9)The conservation law is obtained using Noether’s Theorem again, namelypr b Y ( b L ) = X i u i (cid:18) ∂u i ∂t + u j ∂u i ∂x j + ∂p∂x i (cid:19) + X i ( u i ) + p ! (cid:18) ∂u j ∂x j (cid:19) = D t X i ( u i ) ! + D x i X j ( u j ) u i + pu i = D t (cid:18) | u | (cid:19) + ∇ · (cid:18) | u | u + pu (cid:19) , (4.10)which is the conservation of energy . Conclusions and future work
A modified formal Lagrangian formulation for studying conservations laws of differential equa-tions was defined in this paper. It was proved that any system of differential equations admitsat least one modified formal Lagrangian and its self-adjointness can be achieved via the sim-plest substitution for the dummy variables. Practical algorithms were introduced, that allowus to extend symmetries of the original system to symmetries of its modified formal Lagrangianand hence to compute conservation laws directly from Noether’s Theorem. The same substi-tution for dummy variables would yield conservation laws of the original system. We studiedthe viscous Burger’s equation, the Fornberg–Whitham equation and the incompressible Eulerequations as illustrations.Since the modified formal Lagrangian formulation allows us to define formally a variationalstructure for any system of differential equations, methods for studying variational problemscan be, at least formally, applied to study non-variational differential equations, such as, sym-plectic/multisymplectic structures and variational integrator [6,12,21,25–27], invariant calculusfor variational problems [23, 24, 31], etc., beside Noether’s two theorems.
Acknowledgements
This work was partially supported by JSPS KAKENHI Grant Number JP20K14365, JST-CREST, and Keio Gijuku Academic Development Funds.
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