Some Remarks on the Inverse Problem in the Variational Calculus Within the Functional and Antiexact Differential Forms Approach

1. Introduction

Calculus of variations is currently a well-established vast discipline with methods ranging from functional analysis [13,17,44] through geometric formulation in jet spaces in terms of variational bicomplex [6,7,8,20,21,22,23,24,30,39,40,42,43,45,46,7].

One of the main problems in the calculus of variations is the Inverse Problem (IP). In basic formulation: given a system of differential equations, check if they are variational, i.e., if they are Euler-Lagrange equations. In this formulation, the solution of the problem is affirmative usually by modification of original problem, e.g., by adding new equations that correct non-variationality of original equations. One way is to use Hamiltonian structures, see e.g., [36] for further references. In a more restricted problem it reads as follows: given the differential equations ’as they stand’ (without any alteration), check if they are the Euler-Lagrange equations for some Lagrangian. The solution to this classical problem dates back to the works of Helmholtz [18], where the well-known Helmholtz conditions were formulated. A recent summary is presented in [20,29,40,47], the formulation in terms of exterior differential systems in [9,32,33], the summary from the viewpoint of classical mechanics is presented in [38], and the perspective from the functional-analytic viewpoint is given in the classical book by Vainberg [44].

In general we will focus on differential expressions $\mathcal{E}\left[u\right]\in J_u(\Omega ;{\mathbb{R}}^m),$$u\in M,$ over a domain $\Omega \subset {\mathbb{R}}^n,$ understood as a system of differential equations on a mapping u. It can be composed into functional one-form as

$$\alpha \left[u\right]=\langle R\left[u\right]\mathcal{E}\left[u\right]|du\rangle ,$$

(1)

where $R\left[u\right]\in \mathrm{End}\left({\mathbb{R}}^m\right)$ is an a priori chosen operator, mixing the orders of separate equations. The usual way one defines vertical exterior derivative $\widehat{d}:{\Lambda }_{*}^k\left(J(\Omega ;{\mathbb{R}}^m)\right)\to {\Lambda }_{*}^{k+1}\left(J(\Omega ;{\mathbb{R}}^m)\right),k$$\in {\mathbb{Z}}_{+},$ and a related functional form

$$A\left[u\right]:={\int }_{\Omega }dx\alpha \left[u\right],$$

(2)

where, by definition, ${\Lambda }_{*}^k\left(J(\Omega ;{\mathbb{R}}^m)\right):={\Lambda }^k\left(J(\Omega ;{\mathbb{R}}^m)\right)/D{\Lambda }^{k-1}\left(J(\Omega ;{\mathbb{R}}^m)\right),$ where $D:={\sum }_{j=1}^{n}d{x}_j\wedge$${\displaystyle \frac{d}{d{x}_j}}$ is the total differential operator. Naturally, two functional forms (2) are assumed to be equivalent modulo a divergence term ${\sum }_{j=1}^n{\displaystyle \frac{d}{d{x}_j}}{\Lambda }_u^k\left(J(\Omega ;{\mathbb{R}}^m)\right).$

One can define complementary vertical homotopy operator $H:{\Lambda }_{*}^{k+1}\left(J(\Omega ;{\mathbb{R}}^m)\right)\to {\Lambda }_{*}^k\left(J(\Omega ;{\mathbb{R}}^m)\right)$, defined [25,26,30,44] as

$$H\omega ={\int }_0^1{i}_{\mathcal{K}}{\omega |}_{\tau (s,u)}ds,$$

(3)

where $\omega \in {\Lambda }_{*}^{k+1}\left(J(\Omega ;{\mathbb{R}}^m)\right),$$\mathcal{K}={(u-u_0)}^I{\partial }_I$, and $\tau (s,u)=u_0+s(u-u_0),s\in [0,1],$ is the linear homotopy between $u\in J_u(\Omega ;{\mathbb{R}}^m)$ and the center $u_0\in J_{u_0}(\Omega ;{\mathbb{R}}^m)$ of this homotopy. We assume that of the jet-manifold $J(\Omega ;{\mathbb{R}}^m)$ is connected and star-shaped. This homotopy operator proves to be nilpotent $H^2=0$, moreover, there is a homotopy invariance formula

$$\widehat{d}H+H\widehat{d}=I-s_{u_0}^{*},$$

(4)

where $s_{u_0}$ is an injection to a specific solution $u_0\in M$.

Similar to [11,12,25,26] one can make use of the homotopy operator (3) to define the set of so called "antiexact" forms

$$\mathcal{A}:=Ker\left(H\right)\subset {\Lambda }_{*}\left(J(\Omega ;{\mathbb{R}}^m)\right),$$

(5)

likewise exact forms $\mathcal{E}:=Ker\left(\widehat{d}\right).$ Suitably, the space of vertical forms can be decomposed into the direct sum as ${\Lambda }_{*}\left(J(\Omega ;{\mathbb{R}}^m)\right)=\mathcal{E}\oplus \mathcal{A}$. In addition, one can define projector operators [11,12,25,26],

$$\begin{array}{ccc}\hfill \widehat{d}H& :& {\Lambda }_{*}\left(J(\Omega ;{\mathbb{R}}^m)\right)\to \mathcal{E}\subset {\Lambda }_{*}\left(J(\Omega ;{\mathbb{R}}^m)\right),\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill H\widehat{d}& :& {\Lambda }_{*}\left(J(\Omega ;{\mathbb{R}}^m)\right)\to \mathcal{A}\subset {\Lambda }_{*}\left(J(\Omega ;{\mathbb{R}}^m)\right),\hfill \end{array}$$

(7)

for which ${\left(\widehat{d}H\right)}^2=\widehat{d}H,{\left(H\widehat{d}\right)}^2=H\widehat{d}$ on ${\Lambda }_{*}\left(J(\Omega ;{\mathbb{R}}^m)\right).$

Proceeding to a variational representation of a priori given set of equations $\mathcal{E}\left[u\right]\subset J_u(\Omega ;{\mathbb{R}}^m),$$u\in M,$ over a domain $\Omega \subset {\mathbb{R}}^n,$ one can redefine its lack of variationality in terms of the related non-zero antiexact part: namely, the associated functional one-form (1) can not be represented as a vertical differential $\widehat{d}\mathcal{L}$ for some Lagrangian $\mathcal{L}:J(\Omega ;{\mathbb{R}}^m)\to \mathbb{R},$ if the related antiexact part is nontrivial, that is $H\widehat{d}\alpha \left[u\right]\ne 0,u\in M.$

The paper is organized as follows: in the next section we present a hybrid varaitional problem that is based on antiexact forms, then next section provides some another ways to treating the inverse variational problem by means of introducing some auxiliary a priori Lagrangian one-forms reduced on suitably constructed submanifolds via the corresponding Dirac type constraints.

2. General Approach to a Hybrid Variation Problem

In general, an arbitrary smooth differential system $\mathcal{E}\subset$$J(\Omega ;{\mathbb{R}}^m)$ can be rewritten as a functional 1-form

$$\alpha \left[u\right]=\langle \mathcal{E}\left[u\right]|du\rangle ,$$

(8)

which is naturally decomposable into the direct sum components as

$$\alpha \left[u\right]=\widehat{d}H\alpha \left[u\right]\oplus H\widehat{d}\alpha \left[u\right],$$

(9)

within which the density

$$\mathcal{L}\left[u\right]:=H\alpha \left[u\right],$$

(10)

is called a quasi-Lagrangian. The splitting (9) can be rewritten as

$$\alpha \left[u\right]=\widehat{d}\mathcal{L}\left[u\right]\oplus \beta \left[u\right],$$

(11)

where, by definition, $\beta \left[u\right]:=H\widehat{d}\alpha \left[u\right],u\in M.$ We can then interpret the identity (11) in the following way: the differential system under regard can be considered as a solution for the following ’optimization’ problem:

$$0=i_X\alpha \left[u\right]=i_X\widehat{d}\mathcal{L}\oplus i_X\beta \left[u\right],$$

(12)

for vector fields $X\in \Gamma \left(J(\Omega ;{\mathbb{R}}^m)\right).$ As the condition (12), in general, is not solvable for all vector fields $X\in \Gamma (J(\Omega ;{\mathbb{R}}^m),$ it is natural to reduce the whole system $\alpha \left[u\right]=0$ on the functional submanifold

$$M_{\beta }:=\{u\in M:\beta \left[u\right]=0\}.$$

(13)

The obtained this way differential system

$$0\sim \alpha \left[u\right]=\widehat{d}{\mathcal{L}|}_{M_{\beta }}=0$$

(14)

becomes a priori Lagrangian on the functional submanifold $M_{\beta }\subset M,$ as

$${\alpha \left[u\right]|}_{M_{\beta }}=\widehat{d}{\mathcal{L}}_{\beta },$$

(15)

where the Lagrangian ${\mathcal{L}}_{\beta }{:=\mathcal{L}|}_{M_{\beta }}\in {\Lambda }^0\left(M_{\beta }\right)$ is well defined on $M_{\beta }.$ The latter partially solves the inverse problem of variational representation for the one-form $\alpha \in {\tilde{\Lambda }}^1\left(J(\Omega ;{\mathbb{R}}^m)\right),$ suitably reduced on the functional submanifold $M_{\beta }\subset \mathcal{S}(\Omega ;\mathbb{R})$. A slight modification of this scheme, based on the hybrid variational analysis, is worked out below.

2.1. The Reduction Scheme and a Related Hybrid Variational Problem

Let us consider a smooth differential system $\mathcal{E}\left[u\right]\subset J_u(\Omega ;{\mathbb{R}}^m),$$u\in M,$ on the jet -manifold $J(\Omega ;{\mathbb{R}}^m)$ and analyze its virtually assumed Lagrangian structure, that is the existence of such a smooth mapping $\mathcal{L}:J(\Omega ;{\mathbb{R}}^m)\to \mathbb{R},$ that this differential system $\mathcal{E}\left[u\right]$ is equivalent to the gradient grad$\mathcal{L}\left[u\right]\subset T_u^{*}\left(M\right),$ which can be written down in the simplest case as

$$\mathrm{grad}\mathcal{L}\left[u\right]=A\left[u\right]\mathcal{E}\left[u\right],$$

(16)

where one assumes that $\mathcal{E}\left[u\right]\in T_u^{*}\left(M\right)$ and $A\left[u\right]\in$$End$$\left(T_u^{*}\left(M\right)\right)$, $u\in M$, is some nondegenerate operator endomorphism of the cotangent space $T^{*}\left(M\right)$. In the general case it is also well known that the problem under regard is not practically resolvable, if the differential system $\mathcal{E}\left[u\right]=F($grad$\mathcal{L}\left[u\right])$ for some nonlinear and nondegenerate mapping $F:T_u^{*}\left(M\right)\to T_u^{*}\left(M\right)$. Yet, if we are interested in representing a given differential system $\mathcal{E}\subset J(\Omega ;{\mathbb{R}}^m)$ within some kind of a hybrid variational formalism, one can try to express the relationship (16) in the differential geometric language on the jet-manifold $J(\Omega ;{\mathbb{R}}^m)$ as a differential form

$$a\left[u\right]:=\langle A\left[u\right]\mathcal{E}\left[u\right]|du\rangle =\widehat{d}\mathcal{L}\left[u\right]\oplus H\widehat{d}\langle A\left[u\right]\mathcal{E}\left[u\right]|du\rangle ,$$

(17)

where $\mathcal{L}\left[u\right]:=H\langle A\left[u\right]\mathcal{E}\left[u\right]|du\rangle \in {\Lambda }_{*}^0\left(J_u(\Omega ;{\mathbb{R}}^m)\right),$ a mapping $H:{\Lambda }_{*}^1\left(J_u(\Omega ;{\mathbb{R}}^m)\right)\to {\Lambda }_{*}^0\left(J_u(\Omega ;{\mathbb{R}}^m)\right)$ at $u\in M$ denotes the usual [1,11,12,25,26,44] Poincare homotopy operator and $\langle ·|·\rangle$ is the natural bi-linear form on $T^{*}\left(M\right)$$\times T\left(M\right).$ The representation (17) makes it possible to write down the following hybrid variational problem

$$u=arg\underset{u\in M_{\beta }}{inf}\left({\int }_{\Omega }{d}^{n}x\mathcal{L}\left[u\right]\right)$$

(18)

on a functional submanifold $M_{\beta }\subset M,$ defined via the functional relationship

$$M_{\beta }:=\{u\in M:\beta \left[u\right]:=H\widehat{d}\langle A\left[u\right]\mathcal{E}\left[u\right]|du\rangle =0\}.$$

(19)

The latter easily gives rise to the gradient relationship

$$\left(\left.\mathrm{grad}{\mathcal{L}}_{\beta }\left[u\right]|\right|X\right)=0$$

(20)

for the Lagrangian ${\mathcal{L}}_{\beta }{\left[u\right]:=\mathcal{L}\left[u\right]|}_{M_{\beta }}$ and all $X\in T\left(M_{\beta }\right),$ completely equivalent to that of (17), reduced on the submanifold $M_{\beta }\subset M.$ Thus, one can formulate the following proposition.

Proposition 1.

Any smooth differential system $\mathcal{E}\left[u\right]\subset T^{*}\left(M\right),$$u\in M,$ on the jet -manifold $J(\Omega ;{\mathbb{R}}^m)$ admits the hybrid variational representation

$$u=arg\underset{u\in M_{\beta }}{inf}\left({\int }_{\Omega }{d}^{n}x\mathcal{L}\left[u\right]\right)$$

(21)

on the functional submanifold $M_{\beta }\subset M$,defined by the relationship(19).

As a simple example one can consider the Burgers type dissipative evolution equation

$$\mathcal{E}\left[u\right]\sim u_t-u_{xx}+vu{u}_x=0,$$

(22)

on the jet-manifold $J({\mathbb{R}}^2;\mathbb{R})$for a function $u\in M_u,$ where a parametric function $v\in M_v$ satisfies the adjoint evolution equation

$$v_t+v_{xx}+uv{v}_x=0$$

(23)

on the related jet-manifold $J({\mathbb{R}}^2;\mathbb{R}).$ Having taken the nondegenerate operator endomorphism $A[u,v]=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\in$$End$$\left(T^{*}\left(M\right)\right),$ where $M:=M_u\times M_v,$ we can easily check that the submanifold $M_{\beta }=M$is defiend by the differential form $\beta \left[u\right]:=H\widehat{d}\langle A\left[u\right]\mathcal{E}\left[u\right]|du\rangle =0,$ vanishing identically. The latter makes it possible to state that the corresponding hybrid variational interpretation (21) becomes a true variational problem for the combined Burgers type system (22) and (23).

2.2. An optimal control problem aspect and the related Dirac type reduction scheme

As a typical example, let us consider a dynamical system

$$v_t=K\left[v\right]$$

(24)

on a toric functional manifold $M_v\subset C({\mathbb{T}}^p;{\mathbb{R}}^m),$ which a priori is not of variational type, make its smooth functional parametrical extension

$$v_t{=K[v,u],K[v,u]|}_{\beta [u,v]=0}=K\left[v\right]$$

(25)

with respect to a toric functional variable $u\in M_u\subset C({\mathbb{T}}^p;{\mathbb{R}}^n)$for some smooth differential functional relationship $\beta [u,v]=0$on the product $M_v\times M_u,$ and pose the following Bellman-Pontriagin type optimal control problem [3,31] subject to some smooth Lagrangian density $\mathcal{L}:J_{(v,u)}({\mathbb{T}}^p;{\mathbb{R}}^2)$ on a temporal interval $[0,T]\subset {\mathbb{R}}_{+}:$

$$\underset{v\in M_v}{v=arginf}{\int }_0^{T}dt{\int }_{{\mathbb{T}}^p}\mathcal{L}[v,u]d^{p}x,$$

(26)

for a fixed $u\in M_u$ under the condition that the evolution flow (25) possesses a smooth conserved quantity $\gamma ={\int }_{{\mathbb{T}}^p}\gamma [v,u]d^{p}x\in \mathcal{D}(M_v\times M_u),$ that is $d\gamma /dt=0$ on the combined manifold $M_v\times M_u$ for all $t\in [0,T].$ The latter, in particular, means that we need to determine such an additional evolution flow

$$u_t=F[v,u]$$

(27)

on the extended control manifold $M_u,$ which will ensure the existence of the mentioned above smooth conserved quantity $\gamma \in \mathcal{D}(M_v\times M_u).$ The problem above is solved [31] by means of construction of the extended Lagrangian functional

$$\begin{array}{ccc}\hfill \mathrm{L}[v,\psi ]& :& ={\int }_0^{T}dt{\int }_0^{2\pi }(\mathcal{L}[v,u]+\langle \psi |{(v_t-K[v,u],v_t-F[v,u])}^{⊺}\rangle +\hfill \\ & & +\langle \mu (x,t)|\partial \mathrm{grad}\gamma [u,v]/\partial t\rangle +\langle \mathrm{grad}\gamma [u,v]|{(K[v,u],F[v,u])}^{⊺}\rangle )d^{p}x,\hfill \end{array}$$

(28)

supplemented with Lagrangian multipliers $\mu \in C_0^1({\mathbb{T}}^p\times [0,t];{\mathbb{R}}^2)$and $\psi \in C_0^1([0,T];T^{*}(M_v\times M_u))$ almost everywhere with respect to the temporal parameter $t\in [0,T],$ and next determining its critical points:

$$\begin{array}{c}\delta \mathrm{L}[v;\mu ,\psi ]=0\sim \mathrm{grad}\mathcal{L}[v,u]-{\psi }_t-\\ \\ -{(K[v,u],F[v,u])}^{{⊺}^{\prime ,*}}\psi +\\ \\ +\partial \mathrm{grad}\gamma /\partial t+{(K[v,u],F[v,u])}^{{⊺}^{\prime ,*}}\mathrm{grad}\gamma =0\end{array}$$

(29)

for all $(v,u)\in M_v\times M_u$ jointly with the condition $d\gamma /dt=0$ for $t\in [0,T].$ The obtained functional relationship (29) under the condition $\mu \left(0\right)=0=\mu \left(T\right)$ reduces to the following generalized Noether-Lax condition

$${(K[v,u],F[v,u])}^{{⊺}^{\prime ,*}}\psi =\mathrm{grad}\mathcal{L}[v,u]$$

(30)

on the Lagrangian multiplier $\psi \in C_0^1([0,T];T^{*}(M_v\times M_u)),$ as the following Noether-Lax equality

$$\partial \mathrm{grad}\gamma /\partial t+{(K[v,u],F[v,u])}^{{⊺}^{\prime ,*}}\mathrm{grad}\gamma =0$$

(31)

holds a priori for any smooth conservation law $\gamma \in \mathcal{D}(M_v\times M_u)$ of the joint dynamical system

$$v_t=K[v,u],u_t=F[v,u]$$

(32)

on the combined manifold $M_v\times M_u.$

A solution $\psi \in T^{*}(M_v\times M_u)$ to the condition (30) allows the unique representation as the direct sum $\psi =\overline{\psi }\oplus \phi$ of its skew symmetric $\overline{\psi }\in T^{*}(M_v\times M_u)$ and strictly symmetric $\phi \in T^{*}(M_v\times M_u)$ components, satisfying, respectively, the following differential-functional equations:

$${\overline{\psi }}_t+{(K[v,u],F[v,u])}^{{⊺}^{\prime ,*}}\overline{\psi }=\mathrm{grad}\mathcal{L}[v,u],$$

(33)

where, by definition, ${\overline{\psi }}^{\prime }\ne {\overline{\psi }}^{\prime ,*}$ on $M_v\times M_u,$ and

$${\phi }_t+{(K[v,u],F[v,u])}^{{⊺}^{\prime ,*}}\phi =0,$$

(34)

where, by definition, ${\phi }^{\prime }={\phi }^{\prime ,*}$ on $M_v\times M_u$for all $u\in M_u.$ Under the a priori assumed condition that the evolution flow (32) is a Hamiltonian system on the functional manifold $M_v$ with respect to the related symplectic structure mapping $\Omega :T(M_v\times M_u)\to$$T^{*}(M_v\times M_u),$ the differential-functional equation (33) is always [1,2,4] solvable, giving rise to the known differential-geometric relationship

$$\Omega ={\overline{\psi }}^{\prime }-{\overline{\psi }}^{\prime ,*},$$

(35)

subject to which the following compatible vector field representation

$${(K,F)}^{⊺}=-{\Omega }^{-1}\mathrm{grad}[\left(\overline{\psi }\right|{(K,F)}^{⊺})-\mathcal{L}]$$

(36)

holds on $M_v\times M_u.$ Simultaneously, the differential-functional equation (34) is also always [1,2,4] solvable under the condition that $\phi =\mathrm{grad}$$\gamma \in T^{*}(M_v\times M_u)$ for some conserved quantity $\gamma \in D(M_v\times M_u)$ of the evolution flow (25) regardless of whether the evolution flow (25) on $M_v$ is Hamiltonian or not. The latter means, evidently, that it is also of variational type, which can be suitably reduced via the Dirac scheme on the functional submanifold

$$M_F:=\{(v,u)\in M_v\times M_u:\beta [u,v]=0,{\beta }_u^{\prime }F[v,u]+{\beta }_v^{\prime }K[v,u]=0\},$$

(37)

concerving its variational type on $M_F\subset M_v\times M_u,$ following from the stated above Hamiltonian representation (36).

2.3. Example: Burgers Equation

Concerning the diffusion type Burgers equation example, considered before,

$$v_t=v_{xx}+2v{v}_x:=K\left[v\right]$$

(38)

on the functional manifold $M_v\subset C(\mathbb{R};\mathbb{R})$, treated within the optimal control problem scheme above, there is suggested the following way of embedding the flow (38) into the Dirac type constrained variariational picture:

• we parametrically extend the flow (38) by means of the simple relationship $\beta [u,v]=uv-v=0$ as

  $$v_t=v_{xx}+2\left(uv\right)v_x{:=K[v,u],K[v,u]|}_{\beta [v,u]=0}=K\left[v\right],$$

  (39)

  and pose the optimal control problem for the flow (39): to detect such an evolution flow

  $$u_t\stackrel{?}{=}F[v,u],$$

  (40)

  on the functional parameter $u\in M_u\subset C(\mathbb{R};\mathbb{R}),$ under which the joint dynamical system

  $$\begin{array}{c}{v}_t=v_{xx}+2uv{v}_x,\\ u_t\stackrel{?}{=}F[v,u],\end{array}$$

  (41)

  becomes Hamiltonian on the combined functional manifold $M_v\times M_u;$
• having solved the optimal problem above, we apply to the obtained Hamiltonian system (41) the Dirac type reduction on the functional submanifold $M_F:=\{(v,u)\in M_v\times M_u:uK+vF-K=0\}$, turning back the joint dynamical system (41) to its previous form (38), yet already upon the functional submanifold $M_F\subset M_v\times M_u;$
• as a result, owing to the fact that the reduced on the submanifold $M_F\subset M_v\times M_u,$ dynamical system (38) persists to be Hamiltonian too, it will represent a true Burgers dynamical system as the one a priori representable in the variational Lagrangian form on this submanifold.

Subject to this Burgers dynamical system (38) the analytic scheme above gives rise to the following evolution flow

$$v_t=-v_{xx}+2v{u}_{x}u=F[v,u]$$

(42)

on the parametric functional manifold $M_u,$ which jointly with the flow (38) represents the following Hamiltonian system:

$$\left.\begin{array}{c}{v}_t=v_{xx}+2uv{v}_x,\\ u_t=-u_{xx}+2uv{u}_x\end{array}\right\}=-{\Omega }^{-1}\mathrm{grad}H[v,u],$$

(43)

where $\Omega$$:T(M_v\times M_u)\to T^{*}(M_v\times M_u)$ is the corresponding canonical symplectic structure mapping on $M_v\times M_u$:

$$\Omega =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

(44)

and $H\in \mathcal{D}(M_v\times M_u)$ is the related to it conserved Hamiltonian function:

$$H={\int }_0^{2\pi }dx[u_x{v}_x+(v^{2}u{u}_x-u^{2}v{v}_x)/2].$$

(45)

Having now applied the classical Dirac type reduction scheme to the Hamiltonian system (43) upon the submanifold $M_F:=\{(v,u)\in M_v\times M_u:uK+vF=K\},$ one obtains a true Burgers dynamical system (38) as a Hamiltonian system on this submanifold $M_F\subset M_v,$ a priori possessing, respectively, the related variational Lagrangian representation. To demonstrate this property, we will make use of the fact that the obtained Hamiltonian system possesses [4,5,34,35] a countable hierarchy of functionally independent conserved quantities ${\gamma }_j\in D(M_v\times M_u),j\in {\mathbb{Z}}_{+},$ amongst them the Hamiltonian (45), whose functional gradients are calculated analytically via the recursion scheme:

$$\mathrm{grad}{\gamma }_j[u,v]={\Lambda }^j\mathrm{grad}H[u,v],$$

(46)

where the gradient recursion operator $\Lambda :T^{*}(M_v\times M_u)\to T^{*}(M_v\times M_u)$ is given by the following integro-differential operator expression:

$$\Lambda =\left(\begin{array}{cc}-\partial -u{\partial }^{-1}{v}_x+\partial u{\partial }^{-1}v& u^2-u{\partial }^{-1}{u}_x-\partial u{\partial }^{-1}u\\ v^2-v{\partial }^{-1}{v}_x-\partial v{\partial }^{-1}v& \partial -v{\partial }^{-1}{u}_x+\partial v{\partial }^{-1}u\end{array}\right).$$

(47)

Having calculated the conserved gradient expression $\mathrm{grad}{\gamma }_1[u,v]=\Lambda \mathrm{grad}H[u,v]$∈$T^{*}(M_v\times M_u):$

$$grad{\gamma }_1[v,u]=\left(\begin{array}{c}-u_{3x}+2{\left(u{u}_{x}v\right)}_x+(\partial u{\partial }^{-1}v-u{\partial }^{-1}{v}_x)(u_{xx}-2u{u}_{x}v)+\\ +u^2{v}_{xx}+2u^{3}v{v}_x-u{\partial }^{-1}{u}_x(v_{xx}+2v{v}_{x}u)-\\ -\partial u{\partial }^{-1}vu{v}_{xx}-2\partial u{\partial }^{-1}\left(u^{2}v{v}_x\right)\\ \\ v^2{u}_{xx}-2u{u}_x{v}^3-v{\partial }^{-1}{v}_x{u}_{xx}+2v{\partial }^{-1}{v}_{x}vu{u}_x-\\ -\partial v{\partial }^{-1}v{u}_{xx}+2\partial v{\partial }^{-1}{v}^{2}u{u}_x+v_{3x}+\\ +2{\left(uv{v}_x\right)}_x-v{\partial }^{-1}{u}_x{v}_{xx}-2v{\partial }^{-1}{u}_{x}uv{v}_x+\\ +\partial v{\partial }^{-1}u{v}_{xx}+2\partial v{\partial }^{-1}{u}^{2}v{v}_x\end{array}\right)$$

(48)

and taken into account that it is invariant with respect to the vector field (43) and ensuing from the linear Noether-Lax relationship

$${\displaystyle \frac{\partial }{\partial t}}grad{\gamma }_1[v,u]+K^{\prime ,*}[v,u]grad{\gamma }_1[v,u]=0,$$

(49)

one can proceed to the invariant reduction of the Burgers evolution flow (38) upon the functional submanifold $M_F\subset M_v\times M_u.$ Preliminarily, we need to take the invariant Lagrangian function density ${\mathcal{L}}_1:={\gamma }_1+cH+c_{0}uv$ and reduce our Hamiltonian flow (43) on the 6-dimensional invariant submanifold

$$M_1:=\{(u,v)\in M_v\times M_u:\mathrm{grad}{\mathcal{L}}_1[u,v]=0\}\sim J^2(\mathbb{R}/\left\{2\pi \mathbb{Z}\right\};{\mathbb{R}}^2),$$

as a flow on $J^2(\mathbb{R}/\left\{2\pi \mathbb{Z}\right\};{\mathbb{R}}^2)\subset M_v\times M_u,$ taking into account [4,14,15,16] the classical Gelfand-Dickey relationship:

$$d{\mathcal{L}}_1[u,v]=\langle \mathrm{grad}{\mathcal{L}}_1[v,u]|{(dv,du)}^{⊺}\rangle +d{\alpha }^{\left(1\right)}[v,u]/dx,$$

(50)

determining on the submanifold $M_1$ the nondegenerate symplectic structure ${\omega }^{\left(2\right)}=d{\alpha }^{\left(1\right)}[v,u]\in {\Lambda }^2\left(M_1\right).$ From (50) one easily ensues that the Hamiltonian flow (43) on the submanifold $M_1,$ being Hamiltonian with respect to the constructed above symplectic structure ${\omega }^{\left(2\right)}\in {\Lambda }^2\left(M_1\right),$ can be reduced via the Dirac scheme upon the submanifold

$$M_{1,F}:=\{(v,u)\in M_1:uK+vF=K\},$$

(51)

thus reducing it to the initial Burgers flow (42), equivalently representing it as a Lagrangian variational problem on the jet-submanifold $M_{1,F}\subset M_1\sim J^2(\mathbb{R}/\left\{2\pi \mathbb{Z}\right\};{\mathbb{R}}^2).$

3. Schwinger’s Variational Principle as a Constrained Problem

In this section we study Schiwnger’s ’third way’ of formulating variational problem [10,28,41]. In this approach we make independent variations of field, its tangent and cotangent components. At first it seems that one requires space $T\left(M\right)\times T^{*}\left(M\right)$ for this variation, however, one use only $T\left(M\right)$ and a constraint.

Consider a functional manifold M and a Lagrangian

$$\mathcal{L}:\mathbb{R}\times \left(M\times T\left(M\right)\right)\to \mathbb{R},$$

(52)

representable as smooth mapping $\mathcal{L}[t,\varphi ,\dot{\varphi }]$ in local coordinates $(t,\varphi ,\dot{\varphi })\in \mathbb{R}\times \left(M\times T\left(M\right)\right).$ Its Schwinger extension is defined as a mapping $\mathcal{L}\to {\mathcal{L}}_S,$ where ${\mathcal{L}}_S:\mathbb{R}\times \left(M\times T\left(M\right)\right)\times T^{*}\left(M\right)\to \mathbb{R}$ is some analytical expression, whose simplest form looks as

$${\mathcal{L}}_S(t,\varphi ,\dot{\varphi };p))=\langle p|(\dot{\varphi }-v)\rangle +\mathcal{L}[\varphi ,v].$$

(53)

where variables $\varphi \in M,$$v\in T\left(M\right))$ and $p\in T^{*}\left(M\right)$ are assumed to be independent. Then the following proposition holds.

Proposition 2.

The least action variation of the Schwinger’s Lagrangian functional extension (53) is equivalent to that of the Lagrangian functional (52).

Proof. 

In $\mathcal{L}$ replace $\dot{\varphi }$ by an arbitrary element $v\in \Gamma \left(TM\right)$, and introduce a vector of Lagrange multipliers $\pi =\left\{{\pi }^i\right\}$ for the constraint $\dot{\varphi }=v\in T\left(M\right).$ This gives (53).

From the variation with respect to $\varphi ,$v and the multipliers p independently we obtain

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}p-{\displaystyle \frac{\partial \mathcal{L}}{\partial \varphi }}=0,p={\displaystyle \frac{\partial \mathcal{L}}{\partial v}},\dot{\varphi }=v.\end{array}$$

(54)

This gives Euler-Lagrange equations for the density $\mathcal{L}$ and the definition of momentum $\pi$. □

One can see that $p\in T^{*}\left(M\right),$ by construction, is the canonical momentum playing the role of a Lagrange multiplier subject to the tangent element $v\in T\left(M\right).$ Since the Hamiltonian function is defined as $\mathcal{H}[\varphi ,p]=\langle p|\dot{\varphi }\rangle -\mathcal{L}[\varphi ,v]{\left)\right|}_{p=\delta \mathcal{L}[\varphi ,v]/\delta v}$ modulo the determining relationship

$$\delta {\mathcal{L}}_S[t,\varphi ,v;p]/\delta v=0\sim p=\delta \mathcal{L}[t,\varphi ,v]/\delta v,$$

one gets right away the classical Hamiltonian equations

$$dx/dt=\delta \mathcal{H}[t,\varphi ,p]/\delta p,dp/dt=-\delta \mathcal{H}[t,\varphi ,p]/\delta \varphi .$$

Turn back now to our problem of Lagrangian representation of a given evolution equation

$$K[v,v_t]=0$$

(55)

on a jet-manifold $J(\mathbb{R};M),$ whose Lagrangian form is either not known or not existing on the whole. To suggest a partial solution to this problem, one can consider a close enough to (55) Lagrangian evolution equation

$$\tilde{K}[v,v_t]=0$$

on a jet-manifold $J(\mathbb{R};M),$ whose invariant reduction on the functional submanifold

$$M_{\beta }:=\{v\in M:\beta \left[v\right]=0,\langle {\beta }^{\prime }\left[v\right]|v_t\rangle =0\},$$

(56)

defined by the evolution invariant constraints $\beta \left[v\right]=0\in \Lambda ,$$\langle {\beta }^{\prime }\left[v\right]|w\rangle =0,$ will coincide with the given evolution equation (55). This means that there exists some smooth extended Schwinger type functional

$${\mathrm{L}}_{\lambda ,\mu }[v,w;p]:={\int }_0^{T}dt{\int }_{\Omega }[\langle p|v_t-w\rangle +\mathcal{L}[v,w]d^{n}x+\left(\lambda \right|\beta \left[v\right])+\left(\mu |\langle {\beta }^{\prime }\left[v\right]|w\rangle \right),$$

(57)

on the whole manifold $M,$ for which the least action condition

$$\delta ({\mathrm{L}}_{\lambda ,\mu }[v,w;p]+\left(\lambda \right|\beta \left[v\right])+\left(\mu |\langle {\beta }^{\prime }\left[v\right]|w\rangle \right))=0$$

(58)

with respect to variables $\left(v,p;w\right)\in M\times T^{*}\left(M\right)\times T\left(M\right)$ and the corresponding Lagrangian multipliers $\left(\lambda ,\mu \right)\in {\Lambda }^{*}\times T^{*}(\Lambda )$ reduces on the submanifold $M_{\beta }\subset M$ to the given evolution equation (55). This means that on the submanifold $M_{\beta }\subset M$

$$\begin{array}{c}p={\mathrm{grad}}_w\mathcal{L}[v,w]+{\beta }^{\prime }{\left[v\right]}^{*}\mu ,\\ p_t+{\mathrm{grad}}_v\mathcal{L}[v,w]+{\beta }^{\prime ,*}\left[v\right]\lambda [v,p]+\langle {\beta }^{\prime *,\prime }\left[v\right]|\left(\mu ,w\right)\rangle =0.\end{array}$$

(59)

The latter makes it possible to deduce from (59) the multiplier

$$\mu [v,w;p]={\beta }^{\prime }{\left[v\right]}^{*,-1}(p-{\mathrm{grad}}_w\mathcal{L}[v,w])\in T^{*}(\Lambda )$$

(60)

and, suitably, the next multiplier $\lambda \in {\Lambda }^{*}.$ Thus, one can formulate the following proposition.

Proposition 3.

A given evolution equation (55), reduced on the invariant functional submanifold (56), allows Lagrangian representation (58), specified by means of the functional parameter (60).

4. Conclusions

We proposed some ways of formulation of the variational problem for problems that are not variational. One uses antiexact forms to construct a constraints for space of all possible variations. The other approach extends the number of variables and equations to make the new system variational and then reduce the extended jet space to submanifold that vanish these additional variables. Still, the algorithmic way of such extension is to be found.