# A modified formal Lagrangian formulation for general differential equations

Linyu Peng111Adjunct faculty member at the School of Mathematics and
Statistics, Beijing Institute of Technology, China, and adjunct researcher at
the Waseda Institute for Advanced Study, Waseda University, Japan. Email:
<EMAIL_ADDRESS>
Department of Mechanical Engineering, Keio University,
Yokohama 223-8522, Japan

###### Abstract

In this paper, we propose a modified formal Lagrangian formulation by
introducing dummy dependent variables and prove the existence of such a
formulation for any system of differential equations. The corresponding
Euler–Lagrange equations, consisting of the original system and its adjoint
system about the dummy variables, reduce to the original system via a simple
substitution for the dummy variables. The formulation is applied to study
conservation laws of differential equations through Noether’s Theorem and in
particular, a nontrivial conservation law of the Fornberg–Whitham equation is
obtained by using its Lie point symmetries. Finally, a correspondence between
conservation laws of the incompressible Euler equations and variational
symmetries of the relevant modified formal Lagrangian is shown.

Keywords: Modified formal Lagrangians; Self-adjointness; Symmetries;
Conservation laws; Noether’s Theorem

##  1 Introduction

A lot of differential equations arising from physical phenomena can be derived
from variational calculus, 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 equations
systematically (e.g., [4, 6, 29]), but also serves as an important framework
for the development of geometric numerical integrators (e.g., [13, 28, 30]).
Another great advantage of a variational structure is that Noether’s Theorem
can immediately be applied to derive conservation laws of variational
differential equations. Noether’s Theorem, establishing a one-to-one
correspondence between variational symmetries and conservation laws of the
Euler–Lagrange equations, was proved by Emmy Noether and published in 1918
[31]; see [33] for a modern version and [21] for a history of Noether’s
Theorem together with her second theorem.

Conservation laws are among the most important properties of differential
equations. To apply Noether’s Theorem, it is necessary to study inverse
problems, namely, to distinguish variational differential equations from the
others, and to find the corresponding functional when the system is
variational. Unfortunately, a variational structure is not always available
for general differential equations. Many methods for deriving conservation
laws of general or special type of differential equations, nevertheless, have
been developed, for instance, Vinogradov’s $\mathscr{C}$-spectral sequence [6,
39, 40], symbolic methods [12, 15], the direct construction method of Anco &
Bluman [2, 3], using partial Lagrangians [20], the formal variational
structure and self-adjointness method [5, 11, 17, 18], etc.

In particular, the formal variational structure was firstly proposed by
Atherton & Homsy [5] in 1975 where they called the composite principles of
differential equations; see also [33]. Atherton & Homsy showed that a
composite principle can always be formulated for any general system of
differential equations. Their formulation was then developed by Ibragimov
[17], by studying the self-adjointness of the Euler–Lagrange equations
consisting of the original system and its adjoint system. A formal Lagrangian
$L$ can be defined for a general system of differential equations
$\\{F_{\alpha}=0,\quad\alpha=1,2,\ldots,l\\}$ by introducing dummy dependent
variables $v$, namely

$L:=\sum_{\alpha}v^{\alpha}F_{\alpha}.$

Symmetries of the original system can be extended to variational symmetries of
the corresponding formal variational problem, and hence conservation laws of
the Euler–Lagrange equations, which include both the original system and its
adjoint system, can be derived using Noether’s Theorem. If, through a proper
substitution for the dummy variables, the Euler–Lagrange equations reduce to
the original system, namely, the adjoint system is equivalent to the original
system, called the self-adjointness of the system, then the so-obtained
conservation laws turn into conservation laws of the original system through
the same substitution. Although these conservation laws can sometimes be
trivial, this method provides a straightforward algorithm for computing
conservation laws of non-variational differential equations using Noether’s
Theorem. Furthermore, it also makes the development of variational integrator
for non-variational differential systems possible; see, e.g., [23]. Some
studies on extensions of such techniques to discrete equations are available,
e.g., [36, 37].

Even though every system can be embedded into a bigger variational system,
without the self-adjointneess it is difficult to obtain conservation laws of
the original system by simply using Noether’s Theorem. It was realised,
unfortunately, that self-adjointness of many differential equations can not be
expressed using simple/unified substitutions for the dummy variables. More
complex substitutions were introduced and successfully applied to some
differential equations, for instance, the weak self-adjointness [11] and
nonlinear self-adjointness [18]. However, they are often case-by-case
depending on the system of interest. In this paper, we propose a systematical
method that 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 possible by modifying a formal
Lagrangian by adding an extra so-called balance function $L_{0}$, which is
independent from the dummy variables:

$\widehat{L}:=\sum_{\alpha}v^{\alpha}F_{\alpha}+L_{0}$

This method will be called a modified formal Lagrangian formulation. First of
all, it should be noted that the balance function $L_{0}$ can always be
systematically constructed. Secondly, the corresponding modified
Euler–Lagrange equations reduce to the original system
$\\{F_{\alpha}=0,\quad\alpha=1,2,\ldots,l\\}$ by the simplest substitution for
the dummy variables via $v=u$. By extending symmetries of the original
system—which keep the balance function $L_{0}$ invariant—to variational
symmetries of the modified formal Lagrangian $\widehat{L}$, conservation laws
of the modified Euler–Lagrange equations, which are obtained from Noether’s
Theorem, become that of the original system through the same substitution
$v=u$. This approach of constructing conservation laws works equally well for
other variational symmetries of $\widehat{L}$, which are not necessary
extended from known symmetries of the original system. Worked examples will be
provided.

To make the paper self-contained, we will review relevant fundamental theories
on symmetry analysis in Section 2, for instance, the linearized symmetry
condition for determining symmetries of differential equations, symmetries of
variational problems and conservation laws of Euler–Lagrange equations
obtained from Noether’s Theorem, and a brief introduction to the formal
Lagrangian method proposed by Ibragimov. Readers who are familiar with these
topics and notations may move to Section 3 directly. In Section 3, we define
the modified formal Lagrangian formulation for a general system of
differential equations. Algorithms for extending symmetries of a system of
differential equations to variational symmetries of the corresponding modified
formal Lagrangian are given too. The viscous Burgers’ equation is used as an
illustrative running example. Further concrete examples will be studied in
Section 4, including the derivation of a nontrivial conservation law for the
Fornberg–Whitham equation using a symmetry extended from its Lie point
symmetries and a correspondence between variational symmetries of modified
formal Lagrangians and conservation laws of fluid equations.

##  2 A review of symmetries, conservation laws and Noether’s Theorem

In this section, we briefly review the linearized symmetry condition for
computing symmetries of differential equations and conservation laws obtained
through Noether’s Theorem; details can be found in, e.g., Olver’s book [33].
Ibragimov’s formal Lagrangian approach for computing conservation laws will
also be reviewed.

###  2.1 The linearized symmetry condition

For a system of differential equations, let
$x=(x^{1},x^{2},\ldots,x^{p})\in\mathbb{R}^{p}$ be the independent variables
and let $u=(u^{1},u^{2},\ldots,u^{q})\in\mathbb{R}^{q}$ be the dependent
variables. In the examples though, we will use $t$ and $x$ to denote the time
and space as the independent variables. A system of differential equations is
defined on the jet bundles (cf. [6, 25, 38]) coordinated with

$(x,[u])$

where $[u]$ denotes $u$ and sufficiently many of their derivatives, written in
terms of the multi-index notations

$u^{\alpha}_{\mathbb{J}}=\frac{\partial^{|\mathbb{J}|}u^{\alpha}}{\partial(x^{1})^{j_{1}}\partial(x^{2})^{j_{2}}\ldots\partial(x^{p})^{j_{p}}},$

where $\mathbb{J}=(j_{1},j_{2},\ldots,j_{p})$ and
$|\mathbb{J}|=j_{1}+j_{2}+\cdots+j_{p}$. 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

$\mathcal{A}=\left\\{F_{\alpha}(x,[u])=0,\quad\alpha=1,2,\ldots,q\right\\}.$
(2.1)

Note than we assumed that the number of equations in the system (2.1) is the
same as the dimension of dependent variables $u$. For simplicity, we will
often assume that the system is analytic and totally nondegenerate, the latter
of which means the system itself and its prolongations are of maximal rank and
locally solvable; see, e.g., Definition 2.83 in [33].

Consider the following invertible local transformations around
$\varepsilon=0$:

$\displaystyle x\mapsto\widetilde{x}(x,u,\varepsilon),\quad
u\mapsto\widetilde{u}(x,u,\varepsilon),$ (2.2) subject to
$\displaystyle\widetilde{x}(x,u,0)=x,\quad\widetilde{u}(x,u,0)=u,$

which can be prolonged to the derivatives $u^{\alpha}_{\mathbb{J}}$. In
practice, it is often more convenient to prolong the corresponding
infinitesimal generator as introduced below. The transformations form a local
symmetry group of the system (2.1) if and only if the group maps one solution
$u=f(x)$ to another solution $\widetilde{u}=\widetilde{f}(\widetilde{x})$. It
is often more convenient to use the corresponding infinitesimal generator

$X=\xi^{i}(x,u)\frac{\partial}{\partial{x^{i}}}+\phi^{\alpha}(x,u)\frac{\partial}{\partial{u^{\alpha}}},$
(2.3)

where

$\xi^{i}(x,u):=\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}\widetilde{x}^{i},\quad\phi^{\alpha}(x,u):=\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}\widetilde{u}^{\alpha}.$

Note that the Einstein summation convention is used from now on. Prolongation
of the transformations (2.2) to higher jets yields prolongation of the
infinitesimal generator given by (cf. Theorem 2.36 in [33])

$\operatorname{pr}\\!X=\xi^{i}D_{i}+Q^{\alpha}\frac{\partial}{\partial
u^{\alpha}}+\cdots+(D_{\mathbb{J}}Q^{\alpha})\frac{\partial}{\partial
u^{\alpha}_{\mathbb{J}}}+\cdots.$ (2.4)

The tuple $Q$ with $Q^{\alpha}=\phi^{\alpha}-\xi^{j}D_{i}u^{\alpha}$ is called
the characteristic of $X$ and $D_{i}$ is the total derivative with respect to
$x^{i}$:

$D_{i}:=\frac{\partial}{\partial
x^{i}}+u^{\alpha}_{\mathbb{1}_{i}}\frac{\partial}{\partial
u^{\alpha}}+\cdots+u^{\alpha}_{\mathbb{J}+\mathbb{1}_{i}}\frac{\partial}{\partial
u^{\alpha}_{\mathbb{J}}}+\cdots,$

where $\mathbb{1}_{i}$ is the $p$-tuple with only one nonzero entry $1$ at the
$i$-th place. The multi-index notation $D_{\mathbb{J}}$ denotes a multiple
number of total derivatives:

$D_{\mathbb{J}}=D_{1}^{j_{1}}D_{2}^{j_{2}}\cdots
D_{p}^{j_{p}},\quad\mathbb{J}=(j_{1},j_{2},\ldots,j_{p}).$

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 local transformations if and only if the
linearized symmetry condition is satisfied (e.g., [16, 33]), namely

$\operatorname{pr}\\!X(F_{\alpha})=0,\quad\alpha=1,2,\ldots,q,\text{ whenever
the system \eqref{eq:de0} holds}.$ (2.5)

The total nondegeneracy and analyticity conditions further allow us to express
the linearized symmetry condition equivalently to the existence of $q\times q$
matrices $(K^{\mathbb{J}}(x,[u]))$ whose entries are functions of $(x,[u])$
such that

$\operatorname{pr}\\!X(F_{\alpha})=\sum_{\beta,\mathbb{J}}K^{\mathbb{J}}_{\alpha\beta}\left(D_{\mathbb{J}}F_{\beta}\right),\quad\alpha=1,2,\ldots,q.$
(2.6)

######  Remark 2.1.

Symmetries corresponding to an infinitesimal generator of the form (2.3) are
called Lie point symmetries. They will be called generalised symmetries when
coefficients of the infinitesimal generator not only depend on $x$ and $u$ but
also on derivatives of $u$, namely when the infinitesimal generator is of the
form

$X=\xi^{i}(x,[u])\frac{\partial}{\partial
x^{i}}+\phi^{\alpha}(x,[u])\frac{\partial}{\partial u^{\alpha}}.$

######  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

$X_{1}=\partial_{t},\quad X_{2}=\partial_{x},\quad
X_{3}=t\partial_{x}+\partial_{u},\quad
X_{4}=3t\partial_{t}+x\partial_{x}-2u\partial_{u}.$

###  2.2 Variational symmetries and Noether’s Theorem

Consider a variational problem with a functional

$\mathscr{L}[u]=\int_{\Omega}L(x,[u])\operatorname{d}\\!x$ (2.8)

defined in an open, connected subspace $\Omega$ with smooth boundary, where
the smooth function $L(x,[u])$ is called a Lagrangian (density function).
Variational calculus leads to the Euler–Lagrange equations
$\mathbb{E}_{u^{\alpha}}(L)=0$, $\alpha=1,2,\ldots,q$, which are written using
the Euler operators (e.g., Section 4.1 in [33])

$\displaystyle\mathbb{E}_{u^{\alpha}}:$
$\displaystyle=\sum_{\mathbb{J}}(-D)_{\mathbb{J}}\frac{\partial}{\partial
u^{\alpha}_{\mathbb{J}}}$ (2.9) $\displaystyle=\frac{\partial}{\partial
u^{\alpha}}-D_{i}\frac{\partial}{\partial
u^{\alpha}_{\mathbb{1}_{i}}}+D_{i}D_{j}\frac{\partial}{\partial
u^{\alpha}_{\mathbb{1}_{i}+\mathbb{1}_{j}}}-\cdots,$

where $(-D)_{\mathbb{J}}$ is the adjoint of the operator $D_{\mathbb{J}}$:
$(-D)_{\mathbb{J}}=(-1)^{|\mathbb{J}|}D_{\mathbb{J}}$.

Invariance of the variational problem (2.8) with respect to the
transformations (2.2) can be expressed as the infinitesimal invariance
criterion (cf. [33], Definition 4.33)

$\operatorname{pr}\\!X(L)+LD_{i}\xi^{i}=\operatorname{Div}A$ (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 $X$ are called (divergence) variational
symmetries. The divergence of a $p$-tuple $A$ is defined as

$\operatorname{Div}A:=D_{i}A^{i}.$

A conservation law of a system (2.1) is a divergence expression of a $p$-tuple
$P(x,[u])$

$\operatorname{Div}P=0$

that vanishes on solutions of the system. Conservation laws can be trivial in
two ways: The first kind is that the $p$-tuple $P$ itself vanishes on
solutions of the system, e.g., when each component $P^{i}$ is a linear
combination of the equations $\\{F_{\alpha}\\}$; The second kind is
$\operatorname{Div}P\equiv 0$ holds identically for all functions $u=f(x)$. In
particular, for a totally nondegenerate system of differential equations, a
conservation law can be equivalently understood as the existence of functions
$M_{\alpha}^{\mathbb{J}}(x,[u])$ such that

$\operatorname{Div}P=\sum_{\alpha,\mathbb{J}}M^{\mathbb{J}}_{\alpha}(D_{\mathbb{J}}F_{\alpha}),$
(2.11)

which can be integrated by parts to yield its characteristic form

$\operatorname{Div}\widehat{P}=Q^{\alpha}F_{\alpha}.$ (2.12)

Here $Q(x,[u])$ is the characteristic of the equivalent conservation laws $P$
and $\widehat{P}$. If further the system is analytic, then a conservation law
is trivial if and only if its characteristic $Q$ is trivial; a trivial
characteristic is defined as

$Q=0\text{ holds on solutions of the system \eqref{eq:de0}}.$ (2.13)

When a characteristic is given, the divergence form can be derived by using
the homotopy operator on the variational bicomplex providing the cohomology is
trivial (e.g., [4, 6]) or by intuition. A general formula is available in [33]
(see Section 5.4 therein).

For a variational problem, Noether’s Theorem establishes a one-to-one
correspondence between variational symmetries and conservation laws of the
Euler–Lagrange equations.

######  Theorem 2.3 (Noether’s Theorem).

Suppose that a vector field

$X=\xi^{i}(x,[u])\frac{\partial}{\partial{x^{i}}}+\phi^{\alpha}(x,[u])\frac{\partial}{\partial{u^{\alpha}}}$

satisfies the infinitesimal invariance criterion (2.10) for a variational
problem (2.8). Then its characteristic
$Q^{\alpha}=\phi^{\alpha}(x,[u])-\xi^{i}(x,[u])u^{\alpha}_{i}$ is also the
characteristic of a conservation law for the corresponding Euler–Lagrange
equations $\mathbb{E}_{u^{\alpha}}(L)=0$. Namely, there exists a $p$-tuple
$P(x,[u])$ such that

$\operatorname{Div}P=Q^{\alpha}\mathbb{E}_{u^{\alpha}}(L).$ (2.14)

Various proofs can be found in different contexts. We briefly review Olver’s
proof ([33], Theorem 4.29) by integrating the identity (2.10) by parts:

$\displaystyle\operatorname{Div}A$
$\displaystyle=\operatorname{pr}\\!X(L)+LD_{i}\xi^{i}$ (2.15)
$\displaystyle=\xi^{i}D_{i}L+\sum_{\alpha,\mathbb{J}}(D_{\mathbb{J}}Q^{\alpha})\frac{\partial
L}{\partial u^{\alpha}_{\mathbb{J}}}+LD_{i}\xi^{i}$
$\displaystyle=D_{i}(L\xi^{i})+Q^{\alpha}\sum_{\mathbb{J}}(-D)_{\mathbb{J}}\frac{\partial
L}{\partial u^{\alpha}_{\mathbb{J}}}+\operatorname{Div}B$

for some $p$-tuple $B(x,[u])$. The resulting conservation law is

$\operatorname{Div}P=Q^{\alpha}\mathbb{E}_{u^{\alpha}}(L),\text{ where
}P=A-L\xi-B.$ (2.16)

######  Example 2.4.

Let us consider the $1+1$-dimensional linear wave equation
$u_{tt}-c^{2}u_{xx}=0$ as an illustrative example, where $c\neq 0$. The
corresponding Lagrangian is

$L(u_{t},u_{x})=-\frac{1}{2}u_{t}^{2}+\frac{c^{2}}{2}u_{x}^{2}$

and the wave equation can be equivalently written as $\mathbb{E}_{u}(L)=0$.
Since the Lagrangian is explicitly independently from $t$ and $x$, it admits
the time translational and space translational symmetries, namely

$t\mapsto t+\varepsilon_{1},\quad x\mapsto x+\varepsilon_{2},$

whose infinitesimal generators are respectively

$\partial_{t},\quad\partial_{x}.$

The corresponding conservation laws are written in terms of the
characteristics $-u_{t}$ and $-u_{x}$, namely

$\displaystyle
D_{t}\left(-\frac{1}{2}u_{t}^{2}-\frac{c^{2}}{2}u_{x}^{2}\right)+D_{x}\left(c^{2}u_{t}u_{x}\right)$
$\displaystyle=-u_{t}(u_{tt}-c^{2}u_{xx}),$ $\displaystyle
D_{t}\left(-u_{t}u_{x}\right)+D_{x}\left(\frac{1}{2}u_{t}^{2}+\frac{c^{2}}{2}u_{x}^{2}\right)$
$\displaystyle=-u_{x}(u_{tt}-c^{2}u_{xx}).$

Note that in the integration by parts formula (2.15), $A\equiv 0$ for both
infinitesimal generators.

###  2.3 Formal Lagrangians and self-adjointness

In this subsection, we brief review formal Lagrangians and the self-
adjointness approach for computing conservation laws.

By introducing dummy dependent variables $v$ with the same dimension of the
dependent variables $u$, the formal Lagrangian for a system of differential
equations (2.1) is defined as

$L(x,[u;v]):=v^{\alpha}F_{\alpha}(x,[u]).$ (2.17)

The corresponding Euler–Lagrange equations consist of two parts, namely the
original system

$0=\mathbb{E}_{v^{\alpha}}(L)\equiv
F_{\alpha}(x,[u]),\quad\alpha=1,2,\ldots,q,$ (2.18)

and the so-called adjoint system

$\displaystyle
0=\mathbb{E}_{u^{\alpha}}(L):=F_{\alpha}^{\ast}(x,[u;v]),\quad\alpha=1,2,\ldots,q.$
(2.19)

The system (2.18) is said to be (quasi) self-adjoint if the adjoint system
(2.19) is equivalent to itself via a proper substitution $v=h(u)$. In other
words, Euler–Lagrange equations governed by 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 and nonlinear self-adjointness (e.g., [18, 11]), that,
however, have been found restricted in deriving nontrivial conservation laws
(e.g., [14]); such an example is studied in Section 4.1. Here, we only
introduce the simplest case of self-adjointness.

An important observation by Ibragimov [17] is that any symmetry generator $X$
of the original system (2.18) can be extended to a variational symmetry
generator $X+\phi_{*}^{\alpha}\partial_{v^{\alpha}}$ for the 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, this conservation 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 are not promised [1], its
simplicity for implementation is a great advantage while on the other side the
formal Lagrangian structure provides necessary foundations for constructing
variational integrator [23]. We will illustrate the algorithm by considering
the KdV equation (2.7) as an example.

######  Example 2.6.

_(KdV equation continued.)_ The formal Lagrangian for the KdV equation is

$L=vF\text{ where }F=u_{t}+uu_{x}+u_{xxx},$

and the adjoint equation is

$F^{*}:=-v_{t}-uv_{x}-v_{xxx}=0.$

The adjoint equation turns into the KdV equation through the substitution
$v=u$:

$F^{*}\Big{|}_{v=u}=-F.$

Lie point symmetries of the KdV equation were given in Example 2.2. Here we
compute the conservation law corresponding to the scaling symmetry

$X_{4}=3t\partial_{t}+x\partial_{x}-2u\partial_{u}.$

The extended variational symmetry for the formal Lagrangian is generated by

$Y_{4}=X_{4}+v\frac{\partial}{\partial v}.$

By using the characteristics $Q^{u}=-2u-3tu_{t}-xu_{x}$ and
$Q^{v}=v-3tv_{t}-xv_{x}$, the conservation law 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 conservation of momentum, as follows

$D_{t}\left(\frac{1}{2}u^{2}\right)+D_{x}\left(\frac{1}{3}u^{3}+uu_{xx}-\frac{1}{2}u_{x}^{2}\right)=uF.$

##  3 The modified formal Lagrangian formulation

The formal Lagrangian method defined by Ibragimov is limited even when we are
restricted to evolutionary equations. Nonlinear improvements have been
introduced but they are, in many situations, case-by-case, in particular to
determine the substitution of dummy variables. In this section, we propose a
modification of formal Lagrangians that is applicable to any differential
equations, that we will call the modified formal Lagrangian formulation. Let
us start with a motivating example, the viscous Burgers’ equation, which
serves as a running example in this section.

######  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 $v\left(u_{t}+uu_{x}-au_{xx}\right)$,
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
$-au_{x}^{2}$ and define a modified formal Lagrangian as follows

$\widehat{L}:=v\left(u_{t}+uu_{x}-au_{xx}\right)+auu_{xx},$

the corresponding (modified) adjoint equation is equivalent to the viscous
Burgers’ equation via the 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 adjoint
equation, reading

$0=\mathbb{E}_{u}(\widehat{L})\equiv-v_{t}-uv_{x}-av_{xx}+2au_{xx}.$

Substituting $v=u$ inside gives an equation differing with the Burgers’
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,

$\mathcal{A}=\left\\{F_{\alpha}(x,[u])=0,\quad\alpha=1,2,\ldots,q\right\\},$

introduce dummy dependent variables $v\in\mathbb{R}^{q}$. If there exists a
function $L_{0}(x,[u])$ such that the Euler–Lagrange equations governed by the
Lagrangian

$\widehat{L}(x,[u;v])=v^{\alpha}F_{\alpha}(x,[u])+L_{0}(x,[u])$ (3.1)

reduce to the original system (2.1) via the substitution $v=u$, then we call
the Lagrangian $\widehat{L}(x,[u;v])$ a modified formal Lagrangian and the
corresponding function $L_{0}(x,[u])$ a balance function.

To distinguish from formal Lagrangians, we will use $\widehat{L}$ to denote a
modified formal Lagrangian in the current paper. There are several
fundamentally important remarks or facts regarding 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 the adjoint system is equivalent to the original
system.

  * •

Secondly, since the balance function $L_{0}$ is independent from $v$, half of
the modified Euler–Lagrange equations, i.e.,
$\mathbb{E}_{v^{\alpha}}(\widehat{L})=0$, is exactly the original system.

  * •

A balance function exists for any system of differential equations but not
necessarily uniquely. The existence will be proved in Theorem 3.3. It is not
unique due to the existence of null Lagrangians, namely functions written in a
divergence form; see, e.g., [33, 34].

  * •

When the balance function can be written in a divergence form, the
corresponding modified formal Lagrangian becomes a formal Lagrangian, namely
without modification.

######  Theorem 3.3.

For any system of differential equations

$\mathcal{A}=\left\\{F_{\alpha}(x,[u])=0,\quad\alpha=1,2,\ldots,q\right\\},$

there exists a generic modified formal Lagrangian

$\displaystyle\widehat{L}(x,[u;v]):$
$\displaystyle=v^{\alpha}F_{\alpha}(x,[u])-u^{\alpha}F_{\alpha}(x,[u])$ (3.2)
$\displaystyle=\left(v^{\alpha}-u^{\alpha}\right)F_{\alpha}(x,[u]).$

The function $L_{0}(x,[u])=-u^{\alpha}F_{\alpha}(x,[u])$ will be called a
generic balance function.

###### Proof.

We only need to show that the corresponding Euler–Lagrange equations reduce to
the original system $\mathcal{A}$ via the substitution $v=u$. Now the modified
Euler–Lagrange equations read

$\displaystyle 0$ $\displaystyle=\mathbb{E}_{v^{\alpha}}(\widehat{L})\equiv
F_{\alpha}(x,[u]),$ (3.3) $\displaystyle 0$
$\displaystyle=\mathbb{E}_{u^{\alpha}}(\widehat{L}):=\widehat{F}^{*}_{\alpha}(x,[u;v]).$

Direct computation expands the modified adjoint system as follows

$\displaystyle\widehat{F}^{*}_{\alpha}$
$\displaystyle=\mathbb{E}_{u^{\alpha}}\\!\left(v^{\beta}F_{\beta}-u^{\beta}F_{\beta}\right)$
(3.4)
$\displaystyle=\sum_{\beta,\mathbb{J}}(-D)_{\mathbb{J}}\left(\left(v^{\beta}-u^{\beta}\right)\frac{\partial
F_{\beta}}{\partial u^{\alpha}_{\mathbb{J}}}\right)-F_{\alpha},$

that obviously reduces to the original system with the substitution $v=u$,
i.e.,

$\widehat{F}^{*}_{\alpha}\Big{|}_{v=u}=-F_{\alpha}.$ (3.5)

This completes the proof. ∎

######  Remark 3.4.

The relation between modified adjoint system and the adjoint system (2.19)
(without modification) is

$\widehat{F}^{*}_{\alpha}(x,[u;v])=F^{*}_{\alpha}(x,[u;v])+\mathbb{E}_{u^{\alpha}}(L_{0}).$
(3.6)

In practice, the balance function may include total derivative terms that we
often prefer to mod out since they have no contribution in the Euler–Lagrange
equations. For instance, the balance function $auu_{xx}$ for the viscous
Burgers’ equation in Example 3.1 is equivalent to the generic one
$-u(u_{t}+uu_{x}-au_{xx})$ by differing a divergence

$u(u_{t}+uu_{x}-au_{xx})+auu_{xx}=D_{t}\left(\frac{1}{2}u^{2}\right)+D_{x}\left(\frac{1}{3}u^{3}\right).$
(3.7)

Another convenient and equivalent choice is $-au_{x}^{2}$; see Equation
(3.20).

Since there exists a modified formal Lagrangian for any system of differential
equations, it would be interesting to consider some well-known examples.

  * •

Evolutionary equations

$u_{t}^{\alpha}=f^{\alpha}(x,t,[u]_{x}),\quad\alpha=1,2,\ldots,q,$ (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 Lagrangian reads

$\widehat{L}=\sum_{\alpha}v^{\alpha}\left(u_{t}^{\alpha}-f^{\alpha}\right)-\sum_{\alpha}u^{\alpha}\left(u^{\alpha}_{t}-f^{\alpha}\right).$
(3.9)

An equivalent modified formal Lagrangian is

$\widehat{L}=\sum_{\alpha}v^{\alpha}\left(u_{t}^{\alpha}-f^{\alpha}\right)+\sum_{\alpha}u^{\alpha}f^{\alpha}.$
(3.10)

Note that a special case was considered in [23] (Equations (114-115) therein).

  * •

A family of Camassa–Holm-type equations

$u_{t}-\epsilon u_{xxt}=g(x,t,[u]_{x}),\quad\varepsilon\neq 0.$ (3.11)

The generic modified formal Lagrangian reads

$\widehat{L}=v\left(u_{t}-\epsilon u_{xxt}-g\right)-u\left(u_{t}-\epsilon
u_{xxt}-g\right),$ (3.12)

which is equivalent to

$\widehat{L}=v\left(u_{t}-\epsilon u_{xxt}-g\right)+\epsilon uu_{xxt}+ug.$
(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 and variational symmetries of the modified formal Lagrangian. Such
connections allow us to derive conservation laws of the modified
Euler–Lagrange equations using Noether’s Theorem, that can amount to
conservation laws of the original system.

######  Theorem 3.5.

Consider a system of differential equations

$\mathcal{A}=\left\\{F_{\alpha}(x,[u])=0,\quad\alpha=1,2,\ldots,q\right\\},$

that is totally nondegenerate and analytic, and that admits a symmetry
generated by

$X=\xi^{i}(x,[u])\frac{\partial}{\partial{x^{i}}}+\phi^{\alpha}(x,[u])\frac{\partial}{\partial{u^{\alpha}}}.$

Then $X$ can be extended to a variational symmetry

$Y=X+\phi_{*}^{\alpha}(x,[u;v])\frac{\partial}{\partial v^{\alpha}}$

of the generic modified formal functional

$\widehat{\mathscr{L}}[u;v]:=\int_{\Omega}\widehat{L}(x,[u;v])\operatorname{d}\\!x,$

where the functions $\phi_{*}$ are to be determined and the generic modified
formal Lagrangian is

$\widehat{L}(x,[u;v]):=v^{\alpha}F_{\alpha}(x,[u])-u^{\alpha}F_{\alpha}(x,[u]).$

###### Proof.

First of all, as the system is totally nondegenerate and analytic, the
linearized symmetry condition is replaced by (2.6), namely

$\operatorname{pr}\\!X(F_{\alpha})=\sum_{\beta,\mathbb{J}}K^{\mathbb{J}}_{\alpha\beta}\left(D_{\mathbb{J}}F_{\beta}\right),\quad\alpha=1,2,\ldots,q,$

for some functions $K_{\alpha\beta}^{\mathbb{J}}(x,[u])$. The extended
infinitesimal generator $Y$ satisfies the infinitesimal invariance criterion
for the modified formal functional, that is,

$\displaystyle\operatorname{pr}\\!Y(\widehat{L})+\widehat{L}D_{i}\xi^{i}=\operatorname{Div}A$

for some $p$-tuple $A(x,[u;v])$. Its left-hand side can be integrated by parts
as follows

$\displaystyle\operatorname{pr}\\!Y$
$\displaystyle(\widehat{L})+\widehat{L}D_{i}\xi^{i}=\operatorname{pr}\\!X(\widehat{L})+\phi_{*}^{\alpha}F_{\alpha}+\widehat{L}D_{i}\xi^{i}$
(3.14)
$\displaystyle=\left(v^{\alpha}-u^{\alpha}\right)\operatorname{pr}\\!X(F_{\alpha})-\phi^{\alpha}F_{\alpha}+\phi_{*}^{\alpha}F_{\alpha}+\left(v^{\alpha}-u^{\alpha}\right)(D_{i}\xi^{i})F_{\alpha}$
$\displaystyle=\sum_{\alpha,\beta,\mathbb{J}}\left(v^{\alpha}-u^{\alpha}\right)K^{\mathbb{J}}_{\alpha\beta}\left(D_{\mathbb{J}}F_{\beta}\right)-\phi^{\alpha}F_{\alpha}+\phi_{*}^{\alpha}F_{\alpha}+\left(v^{\alpha}-u^{\alpha}\right)(D_{i}\xi^{i})F_{\alpha}$
$\displaystyle=\sum_{\alpha}\Big{\\{}(-D)_{\mathbb{J}}\left[\left(v^{\beta}-u^{\beta}\right)K^{\mathbb{J}}_{\beta\alpha}\right]-\phi^{\alpha}+\phi_{*}^{\alpha}+\left(v^{\alpha}-u^{\alpha}\right)(D_{i}\xi^{i})\Big{\\}}F_{\alpha}+\operatorname{Div}B$

for some $p$-tuple $B(x,[u;v])$. Clearly, the undetermined functions
$\phi_{*}$ can be chosen as

$\phi_{*}^{\alpha}=\phi^{\alpha}-(-D)_{\mathbb{J}}\left[\left(v^{\beta}-u^{\beta}\right)K^{\mathbb{J}}_{\beta\alpha}\right]-\left(v^{\alpha}-u^{\alpha}\right)(D_{i}\xi^{i})$
(3.15)

and consequently $A=B$. This finishes the proof.

∎

Theorem 3.5 implies that any symmetry of the original system amounts to a
conservation law 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 corresponding to the so-extended
generator $Y$ becomes a trivial conservation law of the original system when
the substitution $v=u$ is applied:

$\phi_{*}^{\alpha}\Big{|}_{v=u}=\phi^{\alpha}\text{ and hence
}Q^{v^{\alpha}}\Big{|}_{v=u}=Q^{u^{\alpha}},$ (3.16)

and then we have

$\displaystyle\operatorname{Div}P\Big{|}_{v=u}$
$\displaystyle=\left(Q^{u^{\alpha}}\widehat{F}^{*}_{\alpha}+Q^{v^{\alpha}}F_{\alpha}\right)\Big{|}_{v=u}$
(3.17)
$\displaystyle=\left(-Q^{u^{\alpha}}+Q^{v^{\alpha}}\Big{|}_{v=u}\right)F_{\alpha}$
$\displaystyle=0.$

The relation (3.5) is applied here.

Fortunately, the extension to a variational symmetry $Y$ may not be unique,
particularly when $-\phi^{\alpha}F_{\alpha}$ can be written in divergence form
and hence can be moved into the divergence $\operatorname{Div}B$ in (3.14). In
fact, to derive a nontrivial conservation law for the original system, we must
choose those extensions such that (3.16) can not happen. Let us consider the
running example again.

######  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) and its Lie point symmetries are generated by the
following infinitesimal generators

$\displaystyle X_{1}=\partial_{t},\quad X_{2}=\partial_{x},\quad
X_{3}=t\partial_{x}+\partial_{u},$ $\displaystyle
X_{4}=2t\partial_{t}+x\partial_{x}-u\partial_{u},\quad
X_{5}=t^{2}\partial_{t}+tx\partial_{x}+(x-tu)\partial_{u}.$

The generic modified formal Lagrangian reads

$\widehat{L}=vF-uF,\text{ where }F:=u_{t}+uu_{x}-au_{xx}.$

The modified adjoint equation is $\widehat{F}^{*}=0$ where

$\widehat{F}^{*}=-v_{t}-uv_{x}-av_{xx}+2au_{xx}.$

Take $X_{3}$ as an example. Direct calculation shows that

$\operatorname{pr}\\!X_{3}(F)\equiv 0.$

Consequently, Equation (3.14) becomes

$\displaystyle\operatorname{pr}\\!Y_{3}(\widehat{L})$
$\displaystyle=\left(\phi_{*}-\phi\right)F$ (3.18)
$\displaystyle=\left(\phi_{*}-1\right)F.$

  * •

The extension (3.15) gives

$\phi_{*}=1,\text{ and hence }Y_{3}=t\partial_{x}+\partial_{u}+\partial_{v}.$

This leads to a trivial conservation law of the viscous Burgers’ equation.

  * •

Equation (3.18) can be rearranged as follows

$\displaystyle\operatorname{pr}\\!Y_{3}(\widehat{L})$
$\displaystyle=\left(\phi_{*}-1\right)F$
$\displaystyle=\phi_{*}F-D_{t}u-D_{x}\left(\frac{1}{2}u^{2}-au_{x}\right).$

Therefore, we may choose $\phi_{*}$ as $0$ or $-1$ instead of $1$. In both
cases, we obtain a nontrivial conservation law of the viscous Burgers’
equation:

$D_{t}u+D_{x}\left(\frac{1}{2}u^{2}-au_{x}\right)=F.$

As we notice in the example above that the observation needed for obtaining
nontrivial conservation laws is relatively strong. Moreover, we often prefer
to Lagrangians including no null information, namely without terms written in
divergence form. The following theorem provides another approach for extending
symmetries of a system of differential equations to variational symmetries of
its (not necessary generic) modified formal Lagrangian; in fact, this method
is often more convenient and practical, compared with Theorem 3.5, for
deriving nontrivial conservation laws.

######  Theorem 3.7.

Consider a system of differential equations

$\mathcal{A}=\left\\{F_{\alpha}(x,[u])=0,\quad\alpha=1,2,\ldots,q\right\\},$

that is totally nondegenerate and analytic, and that admits a symmetry
generated by

$X=\xi^{i}(x,[u])\frac{\partial}{\partial{x^{i}}}+\phi^{\alpha}(x,[u])\frac{\partial}{\partial{u^{\alpha}}}.$

Assume

$\widehat{L}(x,[u;v]):=v^{\alpha}F_{\alpha}(x,[u])+L_{0}(x,[u]).$

is a modified formal Lagrangian of the system, such that the modified adjoint
system is equivalent to the original system via the substitution $v=u$.

If the balance variational problem, whose Lagrangian is the balance function
$L_{0}(x,[u])$, is invariant w.r.t. $X$, then $X$ can be extended to a
variational symmetry

$Y=X+\phi_{*}^{\alpha}(x,[u;v])\frac{\partial}{\partial v^{\alpha}}$

of the modified formal functional

$\widehat{\mathscr{L}}[u;v]:=\int_{\Omega}\widehat{L}(x,[u;v])\operatorname{d}\\!x,$

where the functions $\phi_{*}$ are to be determined.

###### Proof.

Since the balance variational problem

$\int_{\Omega}L_{0}(x,[u])\operatorname{d}\\!x$

is invariant w.r.t. $X$, there exists a $p$-tuple $P_{0}$ such that

$\operatorname{pr}\\!X(L_{0})+L_{0}D_{i}\xi^{i}=\operatorname{Div}P_{0}.$

Then, we have

$\displaystyle\operatorname{pr}\\!Y(\widehat{L})+\widehat{L}D_{i}\xi^{i}$
$\displaystyle=\operatorname{pr}\\!X\left(v^{\alpha}F_{\alpha}+L_{0}\right)+\phi_{*}^{\alpha}F_{\alpha}+\left(v^{\alpha}F_{\alpha}+L_{0}\right)D_{i}\xi^{i}$
$\displaystyle=v^{\alpha}\operatorname{pr}\\!X(F_{\alpha})+\operatorname{pr}\\!X(L_{0})+\phi_{*}^{\alpha}F_{\alpha}+v^{\alpha}(D_{i}\xi^{i})F_{\alpha}+L_{0}D_{i}\xi^{i}$
$\displaystyle=\sum_{\alpha,\beta,\mathbb{J}}v^{\alpha}K^{\mathbb{J}}_{\alpha\beta}\left(D_{\mathbb{J}}F_{\beta}\right)+v^{\alpha}(D_{i}\xi^{i})F_{\alpha}+\phi_{*}^{\alpha}F_{\alpha}+\operatorname{Div}P_{0}$
$\displaystyle=\sum_{\alpha}\Big{\\{}(-D)_{\mathbb{J}}\left[v^{\beta}K_{\beta\alpha}^{\mathbb{J}}\right]+v^{\alpha}(D_{i}\xi^{i})+\phi_{*}^{\alpha}\Big{\\}}F_{\alpha}+\operatorname{Div}\left(B+P_{0}\right)$

where the $p$-tuple $B(x,[u;v])$ is the consequence of integration by parts.
Therefore, the undetermined functions $\phi_{*}$ can be chosen as

$\phi_{*}^{\alpha}=-\Big{\\{}(-D)_{\mathbb{J}}\left[v^{\beta}K_{\beta\alpha}^{\mathbb{J}}\right]+v^{\alpha}(D_{i}\xi^{i})\Big{\\}}$
(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 the
viscous Burgers’ equation are listed in Example 3.6. Let us consider the
modified formal Lagrangian given in Example 3.1:

$\widehat{L}=v\left(u_{t}+uu_{x}-au_{xx}\right)+auu_{xx}.$

Recall that it is equivalent to the generic modified formal Lagrangian,
leading to the same Euler–Lagrange equations.

The balance variational problem with Lagrangian $L_{0}=auu_{xx}$ is
(divergence) invariant w.r.t. $X_{1}$, $X_{2}$ and $X_{3}$ such that

$\operatorname{pr}\\!X_{i}(L_{0})+L_{0}\left(D_{t}\xi^{t}_{i}+D_{x}\xi^{x}_{i}\right)=D_{x}A_{i},\quad
i=1,2,3,$

where

$A_{1}=0,\quad A_{2}=0,\quad A_{3}=au_{x}.$

For each of the three infinitesimal generators, we have

$\operatorname{pr}\\!X_{i}(F)\equiv 0,\quad i=1,2,3.$

From Equation (3.19), we obtain $\phi_{*}=0$ for all of the three
infinitesimal generators and hence $Y_{i}=X_{i}$, $i=1,2,3$. The corresponding
characteristics are

$\displaystyle Q_{1}^{u}=-u_{t},\quad Q_{1}^{v}=-v_{t},$ $\displaystyle
Q_{2}^{u}=-u_{x},\quad Q_{2}^{v}=-v_{x},$ $\displaystyle
Q_{3}^{u}=1-tu_{x},\quad Q_{3}^{v}=-tv_{x}.$

The variational symmetries $Y_{i}$ satisfy

$\operatorname{pr}\\!Y_{i}(\widehat{L})+\widehat{L}\left(D_{t}\xi^{t}_{i}+D_{x}\xi^{x}_{i}\right)=D_{x}A_{i},\quad
i=1,2,3.$

Recall that

$\displaystyle F$ $\displaystyle=u_{t}+uu_{x}-au_{xx},$
$\displaystyle\widehat{F}^{*}$ $\displaystyle=-v_{t}-uv_{x}-av_{xx}+2au_{xx}.$

The corresponding conservation laws of the modified Euler–Lagrange equations
are respectively given by

$D_{t}P^{t}_{i}+D_{x}P^{x}_{i}=Q^{u}_{i}\widehat{F}^{*}+Q^{v}_{i}F,\quad
i=1,2,3,$

where

$\displaystyle P_{1}^{t}=-uu_{x}v+au_{xx}v+au_{x}^{2},\quad
P^{x}_{1}=uu_{t}v-2au_{t}u_{x}-au_{tx}v+au_{t}v_{x},$ $\displaystyle
P^{t}_{2}=-uv_{x},\quad P_{2}^{x}=uv_{t}-au_{x}^{2}+au_{x}v_{x},$
$\displaystyle P_{3}^{t}=-(1-tu_{x})v,\quad
P_{3}^{x}=-uv+2au_{x}-av_{x}-tu_{t}v-atu_{x}^{2}+atu_{x}v_{x}.$

In fact, they can be simply derived from the integration by parts formula
(2.15) and in this special case, we obtain them as

$P_{i}^{t}=-\widehat{L}\xi_{i}^{t}-Q_{i}^{u}\frac{\partial\widehat{L}}{\partial
u_{t}},\quad
P_{i}^{x}=A_{i}-\widehat{L}\xi_{i}^{x}-Q^{u}_{i}\left(\frac{\partial\widehat{L}}{\partial
u_{x}}-D_{x}\frac{\partial\widehat{L}}{\partial
u_{xx}}\right)-\left(D_{x}Q^{u}_{i}\right)\frac{\partial\widehat{L}}{\partial
u_{xx}},$

where $i=1,2,3$. A general formula can be found in Section 5.4 of [33].

Setting $v=u$ in the three conservation laws, only the third one contributes
to a nontrivial conservation law of the viscous Burgers’ equation, namely

$D_{t}u+D_{x}\left(\frac{1}{2}u^{2}-au_{x}\right)=F,$

which is the same as we obtained in Example 3.6. According to the analysis of
cohomology of $\mathscr{C}$-spectral sequence, this is the only (local)
conservation law for the viscous Burgers’ equation; see, e.g., [6].

Note that an equivalent modified formal Lagrangian for the viscous Burgers’
equation can be chosen to simplify the calculations above, which reads

$\widehat{L}=v\left(u_{t}+uu_{x}-au_{xx}\right)-au_{x}^{2}.$ (3.20)

The new balance variational problem with Lagrangian $-au_{x}^{2}$ is
invariant—rather than divergence invariant—with respect to all three
symmetries.

Beside the extension of known symmetries of a system to variational symmetries
of its modified formal functionals through either Theorem 3.5 or Theorem 3.7,
one may also use their own variational symmetries (not necessary extended from
known symmetries) to derive conservation laws using Noether’s Theorem; see
Section 4.2 for an illustrative example from fluid mechanics.

##  4 Concrete examples

In this section, we will study some concrete examples from physics and fluid
mechanics. In the first example, we obtain a nontrivial conservation for the
Fornberg–Whitham equation that has not been successfully achieved using the
previous formal Lagrangian method. For differential equations from fluid
mechanics, we show how to derive conservation laws from a modified formal
Lagrangian’s variational symmetries, that are not necessary extended from
known symmetries of the original differential equations.

###  4.1 The Fornberg–Whitham equation

The Fornberg–Whitham (FW) equation is a nonlinear dispersive wave equation,
admitting a wave of greatest height, e.g., [10]. Symmetry analysis of a bigger
family of nonlinear partial differential equations was conducted in [8]. It
was shown in [14] (see also [19]) that the FW equation is neither quasi self-
adjoint nor weak self-adjoint through the formal Lagrangian approach; although
it is nonlinearly self-adjoint but only trivial conservation laws could be
obtained. In this subsection, we will study its modified formal Lagrangian
formulation to derive conservation laws.

The FW equation can be written as $F=0$ with

$F=u_{t}-u_{xxt}+u_{x}+uu_{x}-3u_{x}u_{xx}-uu_{xxx}.$ (4.1)

It admits a three-dimensional group of Lie point symmetries whose
infinitesimal generators are

$X_{1}=\partial_{t},\quad X_{2}=\partial_{x},\quad
X_{3}=t\partial_{x}+\partial_{u}.$

Let us study the conservation law related to $X_{3}$ by considering the
following modified formal Lagrangian

$\widehat{L}=vF+L_{0}(x,t,[u]),$

where $v$ is the dummy dependent variable and the balance function is chosen
as

$L_{0}=uu_{x}u_{xx}.$

It is equivalent to the generic one by modding out all divergence terms.
Direct computation gives the modified Euler–Lagrange equations

$\displaystyle 0$ $\displaystyle=\mathbb{E}_{v}(\widehat{L})\equiv F,$
$\displaystyle 0$
$\displaystyle=\mathbb{E}_{u}(\widehat{L}):=\widehat{F}^{*},$

where the modified adjoint equation is

$\widehat{F}^{*}=-v_{t}+v_{xxt}-v_{x}-uv_{x}+3u_{x}u_{xx}+uv_{xxx}$

satisfying

$\widehat{F}^{*}\Big{|}_{v=u}=-F.$

First of all, we shall check that the balance variational problem is invariant
w.r.t. $X_{3}$. Namely the infinitesimal invariance criterion (2.10) is
satisfied for $L_{0}$; by noting $\operatorname{Div}\xi_{3}\equiv 0$, we have

$\operatorname{pr}\\!X_{3}(L_{0})=D_{x}\left(\frac{1}{2}u_{x}^{2}\right).$

It can be checked that $\operatorname{pr}\\!X_{3}(F)\equiv 0$, and hence
Equation (3.19) gives the extension of $X_{3}$ to a variational symmetry
$Y_{3}=X_{3}$ of the modified formal Lagrangian, whose characteristics is
written in components as

$Q^{u}=1-tu_{x},\quad Q^{v}=-tv_{x}.$

The corresponding conservation law for the modified Euler–Lagrange equation is
written in characteristic form as

$D_{t}P^{t}+D_{x}P^{x}=Q^{u}\widehat{F}^{*}+Q^{v}F,$

where

$\displaystyle P^{t}=-v+v_{xx}-tuv_{x}+tu_{xx}v_{x},$ $\displaystyle
P^{x}=-v+tuv_{t}+uv_{xx}-u_{x}v_{x}+\frac{3}{2}u_{x}^{2}-tu_{x}v_{xt}-tuu_{x}v_{xx}+tu_{x}^{2}v_{x}+tuu_{xx}v_{x}-tu_{x}^{3}.$

Substituting $v=u$ inside, it becomes a nontrivial conservation law of the FW
equation written in characteristic form as follows

$D_{t}\left(u-u_{xx}\right)+D_{x}\left(u+\frac{1}{2}u^{2}-u_{x}^{2}-uu_{xx}\right)=F.$
(4.2)

A remark on the symmetries $X_{1}$ and $X_{2}$ is that they are also
variational symmetries for the balance variational problem and hence lead to
nontrivial conservation laws for the modified Euler–Lagrange equations. But no
new nontrivial conservation law of the FW equation can be achieved after the
substitution $v=u$ is applied.

###  4.2 Fluid mechanics: The incompressible Euler equations as an
illustration

In this subsection, we will show how well-known conservation laws of fluid
systems represented in the Eulerian framework can be derived using the
modified Lagrangian formulation. In the Lagrangian framework, variational
formulation for incompressible flow has been known for quite long time. In
this paper, we study the incompressible Euler equations as an example. The
same methodology applies to study other fluid equations, e.g., the
compressible Euler equations and the compressible and incompressible
Navier–Stokes equations, and other differential equations equally. Note that
other variational formulations for the incompressible Euler equations exist,
for instance, the Clebsch variational principle and a multisymplectic
formulation (see [9] and references therein for more details).

The incompressible Euler equations are the following system of partial
differential equations for the velocity $u\in\mathbb{R}^{n}$ ($n=2$ or $3$)
and the pressure $p\in\mathbb{R}$:

$\displaystyle u_{t}+u\cdot\nabla u+\nabla p=0,$ (4.3)
$\displaystyle\nabla\cdot u=0.$

Dimension of the space variable is the same as the velocity, i.e., $n=2$ or
$3$, and time is one-dimensional. This system models the flow of inviscid,
incompressible fluid with constant density. 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 dummy
dependent variable corresponding to the pressure $p$ will be denoted by $q$.

In Cartesian coordinates, the system can be written in component form as
follows

$\displaystyle\frac{\partial u^{i}}{\partial t}+u^{j}\frac{\partial
u^{i}}{\partial x^{j}}+\frac{\partial p}{\partial x^{i}}=0,\quad
i=1,2,\ldots,n,$ (4.4) $\displaystyle\frac{\partial u^{j}}{\partial x^{j}}=0.$

In this paper, we only consider the three-dimensional case, i.e., $n=3$.
Introducing dummy dependent variables $v\in\mathbb{R}^{3}$ and
$q\in\mathbb{R}$, we define the modified formal Lagrangian by

$\displaystyle\widehat{L}=q\left(\frac{\partial u^{j}}{\partial
x^{j}}\right)+\sum_{i}v^{i}\left(\frac{\partial u^{i}}{\partial
t}+u^{j}\frac{\partial u^{i}}{\partial x^{j}}+\frac{\partial p}{\partial
x^{i}}\right)-\sum_{i,j}u^{i}u^{j}\frac{\partial u^{j}}{\partial x^{i}}.$
(4.5)

It is equivalent to the generic one and the modified Euler–Lagrange equations,
consisting of the incompressible Euler equations and the adjoint equations

$\displaystyle-\left(\frac{\partial v^{i}}{\partial t}+u^{j}\frac{\partial
u^{i}}{\partial x^{j}}+\frac{\partial q}{\partial
x^{i}}+\frac{\partial}{\partial
x^{j}}\left(v^{i}u^{j}-u^{i}u^{j}\right)\right)=0,\quad i=1,2,\ldots,n,$ (4.6)
$\displaystyle-\frac{\partial v^{j}}{\partial x^{j}}=0,$

reduce to the incompressible Euler equations via the substitutions $v=u$ and
$q=p$.

####  4.2.1 Conservation laws related to extended variational symmetries

It is known that when $n=3$, the system of incompressible Euler equations
admits the following Lie point symmetries (e.g., Example 2.45 in [33]):

  * •

Moving coordinates:

$\displaystyle
f_{i}\partial_{x^{i}}+f^{\prime}_{i}\partial_{u^{i}}-f^{\prime\prime}_{i}x^{i}\partial_{p},\quad
i=1,2,3;$

  * •

Time translation:

$\partial_{t};$

  * •

Scaling:

$\displaystyle x^{i}\partial_{x^{i}}+t\partial_{t},$ $\displaystyle
t\partial_{t}-u^{i}\partial_{u^{i}}-2p\partial_{p};$

  * •

Rotations:

$x^{i}\partial_{x^{j}}-x^{j}\partial_{x^{i}}+u^{i}\partial_{u^{j}}-u^{j}\partial_{u^{i}},\quad
i,j=1,2,3\text{ and }i<j;$

  * •

Pressure changes:

$g\partial_{p}.$

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 of the modified formal Lagrangian if they are variational
symmetries of the balance variational problem, whose Lagrangian is the balance
function

$L_{0}=-\sum_{i,j}u^{i}u^{j}\frac{\partial u^{j}}{\partial x^{i}}.$ (4.7)

Using the infinitesimal invariance criterion (2.10), it is immediate to verify
that the balance variational problem is invariant only w.r.t these symmetries:
spatial translations (i.e., moving coordinates with constant functions
$f_{i}$), time translation, rotations and pressure changes. Their
infinitesimal generators $X$ can be extended to variational symmetries

$Y=X+\phi_{\ast}^{i}\partial_{v^{i}}+\phi_{\ast}^{q}\partial_{q}$

of the modified formal Lagrangian according to Theorem 3.7.

  * •

Extension of spatial translations $\partial_{x^{i}}$:

$Y_{i}=\partial_{x^{i}},\quad i=1,2,3;$

  * •

Extension of time translation $\partial_{t}$:

$Y=\partial_{t};$

  * •

Extension of rotations
$x^{i}\partial_{x^{j}}-x^{j}\partial_{x^{i}}+u^{i}\partial_{u^{j}}-u^{j}\partial_{u^{i}}$:

$Y_{ij}=x^{i}\partial_{x^{j}}-x^{j}\partial_{x^{i}}+u^{i}\partial_{u^{j}}-u^{j}\partial_{u^{i}}+v^{i}\partial_{v^{j}}-v^{j}\partial_{v^{i}},\quad
i,j=1,2,3\text{ and }i<j;$

  * •

Extension of pressure changes $g(t)\partial_{p}$:

$Y=g(t)\partial_{p}.$

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 incompressible Euler equations. We only give the
final results without showing intermediate computational details. Only one
nontrivial conservation law is obtained after the substitution, that is

$D_{x^{i}}\left(g(t)u^{i}\right)=g(t)\frac{\partial u^{i}}{\partial x^{i}},$

corresponding to the symmetry of pressure changes. This is the conservation of
mass.

####  4.2.2 Conservation laws related to other variational symmetries

Except those symmetries extended from symmetries of the incompressible Euler
equations, the modified formal Lagrangian (4.5) also admits other variational
symmetries. They can also be used to compute conservation laws of the
incompressible Euler equations.

The first kind of infinitesimal generators is

$\widehat{Y}_{i}=\partial_{v^{i}}+u^{i}\partial_{q},\quad i=1,2,3,$ (4.8)

which are variational symmetries for the modified formal Lagrangian (4.5),
since the infinitesimal invariance criterion (2.10) is satisfied for each
$\widehat{Y}_{i}$, that is

$\displaystyle\operatorname{pr}\\!\widehat{Y}_{i}(\widehat{L})$
$\displaystyle=\left(\frac{\partial u^{i}}{\partial t}+u^{j}\frac{\partial
u^{i}}{\partial x^{j}}+\frac{\partial p}{\partial
x^{i}}\right)+u^{i}\left(\frac{\partial u^{j}}{\partial x_{j}}\right)$ (4.9)
$\displaystyle=D_{t}u^{i}+D_{x^{j}}\left(\delta_{i}^{j}p+u^{i}u^{j}\right).$

The conservation laws are already written in characteristic form and they
correspond to the conservation of momentum.

Another variational symmetry is generated by

$\widehat{Y}_{0}=u^{i}\partial_{v^{i}}+\left(\sum_{i}\frac{1}{2}(u^{i})^{2}+p\right)\partial_{q}.$
(4.10)

The conservation law is obtained using Noether’s Theorem again, namely

$\displaystyle\operatorname{pr}\\!\widehat{Y}_{0}(\widehat{L})$
$\displaystyle=\sum_{i}u^{i}\left(\frac{\partial u^{i}}{\partial
t}+u^{j}\frac{\partial u^{i}}{\partial x^{j}}+\frac{\partial p}{\partial
x^{i}}\right)+\left(\frac{1}{2}\sum_{i}(u^{i})^{2}+p\right)\left(\frac{\partial
u^{j}}{\partial x_{j}}\right)$ (4.11)
$\displaystyle=D_{t}\left(\frac{1}{2}\sum_{i}(u^{i})^{2}\right)+D_{x^{i}}\left(\frac{1}{2}\sum_{j}(u^{j})^{2}u^{i}+pu^{i}\right)$
$\displaystyle=D_{t}\left(\frac{1}{2}|u|^{2}\right)+\nabla\cdot\left(\frac{1}{2}|u|^{2}u+pu\right),$

which is the conservation of energy.

##  5 Conclusions and future work

A modified formal Lagrangian formulation for studying conservations laws of
differential equations was defined in this paper. It was proved that any
system of differential equations admits at least one modified formal
Lagrangian and its self-adjointness can be achieved via the simplest
substitution for the dummy variables. Practical algorithms were introduced,
that allow us to extend symmetries of the original system to symmetries of its
modified formal Lagrangian and hence to compute conservation laws directly
from Noether’s Theorem. The same substitution for dummy variables would yield
conservation laws of the original system. We studied the viscous Burgers’
equation, the Fornberg–Whitham equation and the incompressible Euler equations
as illustrations.

Moreover, since the modified formal Lagrangian formulation allows us to define
formally a variational structure for any system of differential equations,
methods for studying variational problems can be, at least formally, applied
to study non-variational differential equations, such as,
symplectic/multisymplectic structures [7, 29], structure-preserving numerical
methods [13, 28, 30], invariant calculus for variational problems [26, 27,
35], etc., beside Noether’s two theorems. In particular, we mention that
variational integrator for various non-variational differential equations have
been developed using (modified) formal Lagrangians including, for instance,
the advection equation and the vorticity equation [23], magnetohydrodynamics
[22, 24] and nonlinear dissipative wave equations [32].

## Acknowledgements

This work was partially supported by JSPS KAKENHI Grant Number JP20K14365,
JST-CREST Grant Number JPMJCR1914, Keio Gijuku Academic Development Funds, and
Keio Gijuku Fukuzawa Memorial Fund.

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