# Quasi-Noether Systems and Quasi-Lagrangians

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Symmetries and Conservation Laws of Variational Systems

## 3. Symmetries and Conservation Laws of Quasi-Noether Systems

#### 3.1. Approach Using the Noether Operator Identity

#### 3.2. Approach Using Lagrange Identity

#### 3.3. Quasi-Noether Systems

**Theorem**

**1**

**Proof.**

#### 3.4. Classes of Quasi-Noether Equations

#### 3.4.1. Evolution Equations

- $$\begin{array}{c}\hfill \begin{array}{cc}\hfill A(u,{u}_{x})& =\alpha {H}_{{u}_{x}}/{S}^{\prime}\left(u\right),\hfill \\ \hfill B(u,{u}_{x})& =\alpha ({u}_{x}{H}_{u}-S\left(u\right))/{S}^{\prime}\left(u\right),\hfill \\ \hfill \beta (t,u)& ={e}^{\alpha t}{S}^{\prime}\left(u\right),\hfill \end{array}\end{array}$$In this case the equation $\Delta ={u}_{t}-A{u}_{xx}-B$ is obviously, quasi-Noether since the left hand side turns into total divergence upon multiplication by $\beta $:$$\begin{array}{c}\hfill \beta \Delta ={D}_{t}\left[{e}^{\alpha t}S\left(u\right)\right]-\alpha {D}_{x}\left[{e}^{\alpha t}H(u,{u}_{x})\right].\end{array}$$
- $$\begin{array}{c}\hfill \begin{array}{cc}\hfill A(u,{u}_{x})& =({u}_{x}{G}_{{u}_{x}}-G)/{u}_{x}^{2},\hfill \\ \hfill B(u,{u}_{x})& ={G}_{u}+(b/{u}_{x}+{c}^{\prime}\left(u\right))G+h+{u}_{x}(a+{h}_{u}+h{c}_{u})/b,\hfill \\ \hfill \beta (t,x,u)& =exp(at+bx+c(u\left)\right),\hfill \end{array}\end{array}$$
- The special case $G(u,{u}_{x})={u}_{x}^{2}$, $c\left(u\right)=u$, $h\left(u\right)=0$, and $a=-{b}^{2}$, leads to the following equation:$$\begin{array}{c}\hfill {u}_{t}-{u}_{xx}-{u}_{x}^{2}=0.\end{array}$$Multiplication by $\beta =exp(bx-{b}^{2}t+u)$ turns the LFS of Equation (52) into a total divergence:$$\begin{array}{c}\hfill {D}_{t}\left(\beta u\right)-{D}_{x}\left[\beta ({u}_{x}-b)\right]=0.\end{array}$$
- Choosing $a=-{b}^{2},\phantom{\rule{0.222222em}{0ex}}{c}^{\prime}\left(u\right)=0,\phantom{\rule{0.222222em}{0ex}}G(u,{u}_{x})={{u}_{x}}^{2}+p{u}_{x},\phantom{\rule{0.222222em}{0ex}}p=const,\phantom{\rule{0.222222em}{0ex}}h\left(u\right)=1-bp$ we obtain heat equation, $A=1,\phantom{\rule{0.222222em}{0ex}}B=1$$$\begin{array}{c}\hfill {u}_{t}={u}_{xx}\end{array}$$
- Choosing $a=-{b}^{2},\phantom{\rule{0.222222em}{0ex}}{c}^{\prime}\left(u\right)=0,\phantom{\rule{0.222222em}{0ex}}G(u,{u}_{x})={{u}_{x}}^{2}+{u}^{2}{u}_{x}/2,\phantom{\rule{0.222222em}{0ex}}h\left(u\right)=-b{u}^{2}/2$ we obtain Burgers equation, $A=1,\phantom{\rule{0.222222em}{0ex}}B=-u{u}_{x}$$$\begin{array}{c}\hfill {u}_{t}={u}_{xx}-u{u}_{x}.\end{array}$$
- Choosing $a=-{b}^{2}=-1,\phantom{\rule{0.222222em}{0ex}}{c}^{\prime}\left(u\right)=0,\phantom{\rule{0.222222em}{0ex}}G(u,{u}_{x})={{u}_{x}}^{2}+(u-{u}^{2}){u}_{x},\phantom{\rule{0.222222em}{0ex}}h\left(u\right)=0$ we obtain Fisher equation, $A=1,\phantom{\rule{0.222222em}{0ex}}B=u-{u}^{2}$$$\begin{array}{c}\hfill {u}_{t}={u}_{xx}+u(1-u).\end{array}$$

- $$\begin{array}{c}\hfill \begin{array}{cc}\hfill A(u,{u}_{x})& =a,\hfill \\ \hfill B(u,{u}_{x})& =a{u}_{x}^{2}({\Phi}_{uu}+b{\Phi}_{u}^{2})/{\Phi}_{u}+1/{\Phi}_{u}+\u03f5{u}_{x},\hfill \\ \hfill \beta (t,x,u)& =v(t,x)exp(-bt+b\Phi ),\hfill \end{array}\end{array}$$$$\begin{array}{c}\hfill {v}_{t}-b{v}_{x}+a{v}_{xx}=0.\end{array}$$
- $$\begin{array}{c}\hfill \begin{array}{cc}\hfill A(u,{u}_{x})& =a,\hfill \\ \hfill B(u,{u}_{x})& =a{u}_{x}^{2}{\ell}_{u}+\u03f5{u}_{x},\hfill \\ \hfill \beta (t,x,u)& =v(x,t)\phantom{\rule{0.166667em}{0ex}}{e}^{\ell},\hfill \end{array}\end{array}$$
- $$\begin{array}{c}\hfill \begin{array}{cc}\hfill A(u,{u}_{x})& ={G}_{u}/{\Phi}_{u},\hfill \\ \hfill B(u,{u}_{x})& ={u}_{x}^{2}({G}_{uu}+\tau {G}_{u}^{2}+\u03f5{G}_{u}{\Phi}_{u})/{\Phi}_{u}+{u}_{x}(\delta +2\sigma {G}_{u}/{\Phi}_{u})+1/{\Phi}_{u},\hfill \\ \hfill \beta (t,x,u)& =exp[-\u03f5t+\sigma (x+\delta t)+\tau G+\u03f5\Phi ]cosh\left[(x+\delta t-\mu )\sqrt{{\sigma}^{2}-\tau}\right]{\Phi}_{u},\hfill \end{array}\end{array}$$

#### 3.4.2. Quasi-Linear Equations

- $$\begin{array}{c}\hfill \begin{array}{cc}\hfill L& =\frac{1}{m}[2{M}_{u}+l{M}_{{u}_{x}}+{u}_{x}{M}_{u{u}_{x}}-{K}_{{u}_{x}}]\hfill \\ \hfill R& =\int \left[lL+{u}_{x}{L}_{u}\right]\phantom{\rule{0.222222em}{0ex}}d{u}_{x}={l}^{2}M+2l{u}_{x}{M}_{u}+{{u}_{x}}^{2}{M}_{uu}-lK-{u}_{x}{K}_{u}\phantom{\rule{0.222222em}{0ex}}+\int {K}_{u}d{u}_{x},\hfill \end{array}\end{array}$$$$\begin{array}{c}\hfill -{f}_{u}+m{f}_{{u}_{t}}+{u}_{t}{f}_{u{u}_{t}}=-{m}^{2}.\end{array}$$
- $$\begin{array}{c}\hfill \begin{array}{cc}\hfill R& =L=0\hfill \\ \hfill K& =(l+q{u}_{x})M+{u}_{x}{M}_{u}+\int [qM+{M}_{u}]d{u}_{x}+r\left(u\right),\hfill \end{array}\end{array}$$$$\begin{array}{c}\hfill f(u,{u}_{t})=-q{u}_{t}^{2}+s\left(u\right){u}_{t}+v\left(u\right),\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}s\left(u\right)=-m+\frac{qv\left(u\right)+{v}^{\prime}\left(u\right)}{m}.\end{array}$$Examples:
- If $M(u,{u}_{x})=c,$ we obtain$$\begin{array}{c}\hfill {u}_{tt}=M{u}_{xx}+(lM+r\left(u\right)){u}_{x}+qM{{u}_{x}}^{2}+R\left(u\right)+(-q{{u}_{t}}^{2}+s{u}_{t}+v),\end{array}$$Choosing $q=s=v=0,\phantom{\rule{0.222222em}{0ex}}r=-lM$ we obtain$$\begin{array}{c}\hfill {u}_{tt}=M{u}_{xx}+R\left(u\right).\end{array}$$The class (72) includes Liouville equation ($R\left(u\right)=k{e}^{\lambda u}$)$$\begin{array}{cc}\hfill {u}_{tt}& =M{u}_{xx}+k{e}^{\lambda u},\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{tt}& =M{u}_{xx}+ksin\lambda u.\hfill \end{array}$$
- Choosing $M(u,{u}_{x})=g\left(u\right),q=s=v=0,r\left(u\right)=-lg\left(u\right),$ we obtain$$\begin{array}{cc}\hfill {u}_{tt}& =g\left(u\right){u}_{xx}+{g}^{\prime}\left(u\right){{u}_{x}}^{2}.\hfill \end{array}$$The Equation (75) is a nonlinear wave equation.

## 4. Quasi-Lagrangians

#### 4.1. A Noether Correspondence

**Theorem**

**2.**

- 1.
- ${X}_{\alpha}L=div$, if and only if $\left({T}^{*}\alpha \right)\xb7\Delta =div$.
- 2.
- If ${X}_{\alpha}L=div$, then α is a sub-symmetry of Δ.Let $L={L}_{0}+\beta \xb7\Delta $. Define ${\Delta}_{f}=\Delta +f$ and ${L}_{f}={L}_{0}+\beta \xb7{\Delta}_{f}$ for $f\in ker\left[T\right]$.
- 3.
- If $\beta \in im\left[{T}^{*}\right]$, ${X}_{\alpha}{L}_{f}=div$ if and only if ${X}_{\alpha}L=div$.

**Proof.**

- The Noether theorem states ${X}_{\alpha}L=div$ if and only if $\alpha \xb7E\left(L\right)=div$; thus, $\alpha \xb7(T\xb7\Delta )=div$. If we integrate by parts, we obtain $\left({T}^{*}\alpha \right)\xb7\Delta =div$, as desired.
- It is well known ${X}_{\alpha}L=div$ implies ${X}_{\alpha}$ is a symmetry of $E\left(L\right)$. Since $E\left(L\right)=T\xb7\Delta $ is a sub-system of a prolongation of $\Delta $, this means ${X}_{\alpha}$ is only a sub-symmetry of $\Delta $.
- Since $\beta ={T}^{*}\gamma $ for some smooth $\gamma $, we have $\beta \xb7f=\left({T}^{*}\gamma \right)\xb7f=\gamma \xb7\left(Tf\right)+div=div$, so we conclude ${X}_{\alpha}{L}_{f}={X}_{\alpha}L+div$.

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

#### 4.2. An Example

^{ε}(x,t), and a translation t → t + ε, respectively. These (three) cosymmetries comprise the (order 4) conservation law generators for Equation (80).

_{x}(1) = 0.

**Remark**

**5.**

**Remark**

**6.**

#### 4.3. Critical Points and Symmetries

**Remark**

**7.**

**Proposition**

**1.**

**Remark**

**8.**

**Example**

**1**

**Example**

**2**

**Example**

**3**

**Theorem**

**3.**

**Remark**

**9.**

**Remark**

**10.**

**Remark**

**11.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Rosenhaus, V.; Shankar, R.
Quasi-Noether Systems and Quasi-Lagrangians. *Symmetry* **2019**, *11*, 1008.
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Rosenhaus V, Shankar R.
Quasi-Noether Systems and Quasi-Lagrangians. *Symmetry*. 2019; 11(8):1008.
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