# First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach

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## Abstract

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## 1. Introduction

## 2. Basic Definitions and Expressions for Noether-Like Operators and Conservation Laws

**Noether-like symmetry conditions:**

**Noether-like Theorem:**

## 3. Classification of Lane-Emden System w.r.t Noether-Like Operators and Corresponding First Integrals

**Case 1.**Herein $P(x)=1$. If we take $F(f,g)$ and $G(f,g)$ as linear functions of f and g, we obtain a system of linear equations which can further be converted into the simplest two-dimensional system ${f}^{\u2033}={g}^{\u2033}=0$. With these choices we find by employing (5) and (6), nine Noether-like operators and ten first integrals as shown in the following table.

Noether-Like Operators | First Integrals |
---|---|

${\partial}_{x}$ | ${I}_{1,1}={f}^{\prime 2}-{g}^{\prime 2}$ |

${I}_{1,2}={f}^{\prime}{g}^{\prime}$ | |

${\partial}_{f}$, ${\partial}_{g}$ | ${I}_{2,1}={f}^{\prime}$ |

${I}_{2,2}={g}^{\prime}$ | |

$x{\partial}_{f}$, $x{\partial}_{g}$ | ${I}_{3,1}=x{f}^{\prime}-f$ |

${I}_{3,2}=x{g}^{\prime}-g$ | |

$2x{\partial}_{x}+f{\partial}_{f}+g{\partial}_{g}$, $g{\partial}_{f}-f{\partial}_{g}$ | ${I}_{4,1}=-x({f}^{\prime 2}-{g}^{\prime 2})+f{f}^{\prime}-g{g}^{\prime}$ |

${I}_{4,2}=-x{f}^{\prime}{g}^{\prime}+f{g}^{\prime}+{f}^{\prime}g$ | |

${x}^{2}{\partial}_{x}+x\left(f{\partial}_{f}+g{\partial}_{g}\right)$, $x\left(g{\partial}_{f}-f{\partial}_{g}\right)$ | ${I}_{5,1}=x(f{f}^{\prime}-g{g}^{\prime})-\frac{{x}^{2}}{2}({f}^{\prime 2}-{g}^{\prime 2})-\frac{1}{2}({f}^{2}-{g}^{2})$ |

${I}_{5,2}=x(f{g}^{\prime}+{f}^{\prime}g)-{x}^{2}{f}^{\prime}{g}^{\prime}-fg$ |

**Case 2.**Here n and $\alpha $ are related as $n=\frac{1-\alpha}{2}$, and $F(f,g)$, $G(f,g)$ are arbitrary. From Equation (5) with the aid of (7), we obtain ${\xi}_{1}={x}^{\frac{1-\alpha}{2}}$, ${\eta}_{1}={\eta}_{2}=0$ with ${A}_{1},{A}_{2}$ as constants. Only one Noether-like operator exists, viz., $\mathbf{X}={x}^{\frac{1-\alpha}{2}}\frac{\partial}{\partial x}$. Therefore, by utilizing this operator with the related Lagrangians in (6), we deduce the following first integrals.

**Case 3.**$F(f,g)=\frac{{\alpha}_{1}}{2}ln({f}^{2}+{g}^{2})+\gamma f+\delta ,{\alpha}_{1}\ne 0$ and $G(f,g)={\alpha}_{1}arctan\left(g/f\right)+\gamma g,\phantom{\rule{3.33333pt}{0ex}}{\alpha}_{1}\ne 0$. If $n=\frac{1-\alpha}{2}$, we obtain ${\xi}_{1}={x}^{\frac{1-\alpha}{2}},\phantom{\rule{3.33333pt}{0ex}}{\xi}_{2}=0$ and ${\eta}_{1}={\eta}_{2}=0$ while ${A}_{1}={A}_{2}=k$, k a constant. This case is subsumed in

**Case 2.**

**Case 4.**For this case, let us take $F={({f}^{2}+{g}^{2})}^{r/2}cos\theta $, $G={({f}^{2}+{g}^{2})}^{r/2}sin\theta $, where $\theta =arctan\left(g/f\right)$. We discuss the following subcases.

**Case 4.1**If $n=\frac{r+2\alpha +1}{r-1}$, we determine from the Noether-like symmetry conditions (5) ${\xi}_{1}=x,$ ${\xi}_{2}=0$ and ${\eta}_{1}=\frac{\alpha +1}{1-r}f,\phantom{\rule{3.33333pt}{0ex}}{\eta}_{2}=\frac{\alpha +1}{1-r}g$ while ${A}_{1}$ and ${A}_{2}$ appear as constants. Hence the Noether-like operators take the form

**Case 4.2**If $n=\frac{r+\alpha +2}{r+1}$ with $r\ne -1$, Equation (5) provides ${\xi}_{1}={x}^{\frac{r-\alpha}{r+1}},\phantom{\rule{3.33333pt}{0ex}}{\xi}_{2}=0$, ${\eta}_{1}=-\frac{\alpha +1}{r+1}{x}^{-\frac{\alpha +1}{r+1}}f$, ${\eta}_{2}=-\frac{\alpha +1}{r+1}{x}^{-\frac{\alpha +1}{r+1}}g$ and the related gauge functions are ${A}_{1}=\frac{{(\alpha +1)}^{2}}{2{(r+1)}^{2}}({f}^{2}-{g}^{2})+C$, ${A}_{2}=\frac{{(\alpha +1)}^{2}}{{(r+1)}^{2}}fg$, where C is constant. Now the Noether-like therorem (6) along with these operators and corresponding Lagrangians, reveal the first integrals

**Case 4.3**Here for $n=\frac{1-\alpha}{2}$, we find that ${\xi}_{1}={x}^{\frac{1-\alpha}{2}}$ and ${\eta}_{1}={\eta}_{2}=0$ with ${A}_{,}\phantom{\rule{3.33333pt}{0ex}}{A}_{2}$ as constants. This falls in

**Case 2**.

**Case 5.**$F,\phantom{\rule{3.33333pt}{0ex}}G$ being quadratic functions of f and g, i.e., ${F}_{1}(f,g)={\alpha}_{1}({f}^{2}-{g}^{2})+{\alpha}_{2}f+{\alpha}_{3}$ and ${F}_{2}(f,g)=2{\alpha}_{1}fg+{\alpha}_{2}g$, ${\alpha}_{1}\ne 0$, we have the following four subcases.

**Case 5.1**If $n=2\alpha +3$, ${\alpha}_{2}=0$ and ${\alpha}_{3}=0$, we find that ${\xi}_{1}=x,\phantom{\rule{3.33333pt}{0ex}}{\xi}_{2}=0$, ${\eta}_{1}=-(\alpha +1)f,{\eta}_{2}=-(\alpha +1)g$ and ${A}_{1},{A}_{2}$ are constants. This case is covered in

**Case 5.1**.

**Case 5.2**By taking $n=2\alpha +3$, $\alpha \ne -1$, ${\alpha}_{2}^{2}=4{\alpha}_{1}{\alpha}_{3}$, Equation (5) with (7) yields ${\xi}_{1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}x,\phantom{\rule{3.33333pt}{0ex}}{\xi}_{2}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$, ${\eta}_{1}=-(1+\alpha )(f+\frac{{\alpha}_{2}}{2{\alpha}_{1}}),{\eta}_{2}=-(1+\alpha )g$, and ${A}_{1}=\frac{{\alpha}_{2}{\alpha}_{3}}{6{\alpha}_{1}}{x}^{3\alpha +3},{A}_{2}=0$. So two Noether-like operators are found which are of the form

**Case 5.3**$n=\frac{\alpha +4}{3}$, $n\ne \frac{1-\alpha}{2},-1$, ${\alpha}_{2}=0$ and ${\alpha}_{3}=0$, we get ${\xi}_{1}={x}^{\frac{2-\alpha}{3}}$, ${\eta}_{1}=-\frac{\alpha +1}{3}{x}^{-\frac{\alpha +1}{3}}f$, ${\eta}_{2}=-\frac{\alpha +1}{3}{x}^{-\frac{\alpha +1}{3}}g$ and ${A}_{1}=\frac{{(\alpha +1)}^{2}}{18}({f}^{2}-{g}^{2})+c,{A}_{2}=\frac{{(\alpha +1)}^{2}}{9}fg$, where c is a constant. This case can be absorbed in

**Case 5.2**shown below.

**Case 5.4**If $n=\frac{1-\alpha}{2}$ with ${n}_{1}\ne \frac{\alpha +4}{3}$ and ${\alpha}_{2}$, ${\alpha}_{3}$ are arbitrary constants, we find ${\xi}_{1}={x}^{\frac{1-\alpha}{2}},\phantom{\rule{3.33333pt}{0ex}}{\xi}_{2}=0,\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}{\eta}_{1},\phantom{\rule{3.33333pt}{0ex}}{\eta}_{2}=0$. This subcase is contained in

**Case 2**.

**Case 6.**For the choices $F(f,g)={\alpha}_{1}exp({\alpha}_{2}f)cos({\alpha}_{2}g)+{\alpha}_{3}f+\delta $ and $G(f,g)={\alpha}_{1}exp({\alpha}_{2}f)sin({\alpha}_{2}g)+{\alpha}_{3}g$, where ${\alpha}_{1}\ne 0,{\alpha}_{2}\ne 0$, we consider two subcases here.

**Case 6.1**If $n=\frac{1-\alpha}{2}$, Equation (5) gives ${\xi}_{1}={x}^{\frac{1-\alpha}{2}}$ with ${\eta}_{1}={\eta}_{2}=0$ and ${A}_{1}={A}_{2}=k$, k a constant. This reduces to

**Case 2**.

**Case 6.2**If $n=1,\alpha \ne -1,{\alpha}_{3}=0$ and $\delta =0$, from the Noether-like symmetry condition (5) on using (7) we determine that ${\xi}_{1}=x,\phantom{\rule{3.33333pt}{0ex}}{\xi}_{2}=0$ and ${\eta}_{1}=-\frac{\alpha +1}{{\alpha}_{2}},\phantom{\rule{3.33333pt}{0ex}}{\eta}_{2}=0$ with ${A}_{1}={A}_{2}=k$, k a constant. Therefore we have two Noether-like operators, ${\mathbf{X}}_{1}=x\frac{\partial}{\partial x}-\frac{\alpha +1}{{\alpha}_{2}}\frac{\partial}{\partial f},\phantom{\rule{3.33333pt}{0ex}}{\mathbf{X}}_{2}=\frac{\alpha +1}{{\alpha}_{2}}\frac{\partial}{\partial g}$. Utilization of these operators ${\mathbf{X}}_{1}$ and ${\mathbf{X}}_{2}$ along with the particular pair of real Lagrangians (7) in Equation (6) provides the two first integrals

**Case 7**For $n\ne \frac{1-\alpha}{2}$, and F and G are arbitrary but not of the forms presented in the cases, $\mathbf{1},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}$, the pair of equations (5) yields ${\xi}_{1}={\xi}_{2}=0,{\eta}_{1}={\eta}_{2}=0$ with ${A}_{1}={A}_{2}=$ constant, so in this case no Noether-like operator exists and we cannot find first integrals of the underlying system of Euler-Lagrange equations.

**Case 1.**For $F=exp({\beta}_{1}f)cos({\beta}_{1}g)$ and $G=exp({\beta}_{1}f)sin({\beta}_{1}g)$, Equation (5) with the associated Lagrangians provides a single Noether-like operator $\mathbf{X}={x}^{n}\frac{\partial}{\partial x}$ with gauge terms ${A}_{1}={A}_{2}=0$. So with this operator $\mathbf{X}$, and invocation of (6) with the aid of (7) gives rise to the following conserved quantities

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Cases | Noether-Like Operators | First Integrals |
---|---|---|

Case A-2 | $\mathbf{X}={x}^{\frac{1-\alpha}{2}}\frac{\partial}{\partial x}$ | ${I}_{1}=\frac{1}{2}{x}^{1-\alpha}({f}^{\prime 2}-{g}^{\prime 2})+\int (Fdf-Gdg)$ |

${I}_{2}={x}^{1-\alpha}({f}^{\prime}{g}^{\prime})+\int (Gdf+Fdg)$ | ||

Case A-4.1 | ${\mathbf{X}}_{1}=x\frac{\partial}{\partial x}+\left(\frac{1+\alpha}{1-r}\right)f\frac{\partial}{\partial f}$+$\left(\frac{1+\alpha}{1-r}\right)g\frac{\partial}{\partial g}$ | ${I}_{1}=\frac{1}{2}{x}^{\frac{2(\alpha +r)}{r-1}}({f}^{\prime 2}-{g}^{\prime 2})+\frac{{\alpha}_{1}}{r+1}{x}^{\frac{(r+1)(\alpha +1)}{r-1}}{({f}^{2}+{g}^{2})}^{\frac{r+1}{2}}cos\theta -\frac{\alpha +1}{1-r}{x}^{\frac{r+2\alpha +1}{r-1}}(f{f}^{\prime}-g{g}^{\prime})$ |

${\mathbf{X}}_{2}=\frac{1+\alpha}{1-r}\left(g\frac{\partial}{\partial f}-f\frac{\partial}{\partial g}\right)$ | ${I}_{2}={x}^{\frac{2(\alpha +r)}{r-1}}{f}^{\prime}{g}^{\prime}+\frac{{\alpha}_{1}}{r+1}{x}^{\frac{(r+1)(\alpha +1)}{r-1}}{({f}^{2}+{g}^{2})}^{\frac{r+1}{2}}sin\theta -\frac{\alpha +1}{1-r}{x}^{\frac{r+2\alpha +1}{r-1}}(f{g}^{\prime}+{f}^{\prime}g)$ | |

Case A-4.2 | ${\mathbf{X}}_{1}={x}^{\frac{r-\alpha}{r+1}}\frac{\partial}{\partial x}-\left(\frac{1+\alpha}{1+r}\right){x}^{-\frac{\alpha +1}{r+1}}f\frac{\partial}{\partial f}-\left(\frac{1+\alpha}{1+r}\right){x}^{-\frac{\alpha +1}{r+1}}g\frac{\partial}{\partial g}$ | ${I}_{1}=\frac{1}{2}{x}^{2}({f}^{\prime 2}-{g}^{\prime 2})+\left(\frac{{\alpha}_{1}}{r+1}\right){x}^{1+\alpha}{({f}^{2}+{g}^{2})}^{\frac{r+1}{2}}cos\theta +\left(\frac{1+\alpha}{1+r}\right){x}^{-\frac{\alpha +1}{r+1}}g\frac{\partial}{\partial g}$ |

${\mathbf{X}}_{2}=-\left(\frac{1+\alpha}{1+r}\right){x}^{-\frac{\alpha +1}{r+1}}g\frac{\partial}{\partial f}+\left(\frac{1+\alpha}{1+r}\right){x}^{-\frac{\alpha +1}{r+1}}f\frac{\partial}{\partial g}$ | ${I}_{2}={x}^{2}{f}^{\prime}{g}^{\prime}+\left(\frac{{\alpha}_{1}}{r+1}\right){x}^{1+\alpha}{({f}^{2}+{g}^{2})}^{\frac{r+1}{2}}sin\theta +\left(\frac{\alpha +1}{r+1}\right)x(f{g}^{\prime}+{f}^{\prime}g)+{\left(\frac{\alpha +1}{r+1}\right)}^{2}fg$ | |

Case A-5.2 | ${\mathbf{X}}_{1}=x\frac{\partial}{\partial x}-(1+\alpha )\left((\frac{{\alpha}_{2}}{2{\alpha}_{1}}+f)\frac{\partial}{\partial f}\right)-(1+\alpha )\left(g\frac{\partial}{\partial g}\right)$ | ${I}_{1}=\frac{1}{2}{x}^{2\alpha +4}({f}^{\prime 2}-{g}^{\prime 2})+\frac{1}{3}{\alpha}_{1}{x}^{3\alpha +3}({f}^{3}-3f{g}^{2})+\frac{1}{2}{\alpha}_{2}{x}^{3\alpha +3}({f}^{2}-{g}^{2})+{\alpha}_{3}{x}^{3\alpha +3}f+(\alpha +1){x}^{2\alpha +3}\times (f{f}^{\prime}-g{g}^{\prime})+(\alpha +1)\frac{{\alpha}_{2}}{2{\alpha}_{1}}{x}^{2\alpha +3}{f}^{\prime}+\left(\frac{\alpha +1}{3\alpha +3}\right)\times \left(\frac{{\alpha}_{2}{\alpha}_{3}}{2{\alpha}_{1}}\right){x}^{3\alpha +3}$ |

${\mathbf{X}}_{2}=(1+\alpha )\left(-g\frac{\partial}{\partial f}+(\frac{{\alpha}_{2}}{2{\alpha}_{1}}+f)\frac{\partial}{\partial g}\right)$ | ${I}_{2}={x}^{2\alpha +4}{f}^{\prime}{g}^{\prime}+\frac{1}{3}{\alpha}_{1}{x}^{3\alpha +3}(3{f}^{2}g-{g}^{3})+{\alpha}_{2}{x}^{3\alpha +3}(fg)+{\alpha}_{3}{x}^{3\alpha +3}g+(\alpha +1){x}^{2\alpha +3}(f{g}^{\prime}+{f}^{\prime}g)+(\alpha +1)\frac{{\alpha}_{2}}{2{\alpha}_{1}}{x}^{2\alpha +3}{g}^{\prime}$ | |

Case A-6.2 | ${\mathbf{X}}_{1}=x\frac{\partial}{\partial x}-\frac{\alpha +1}{{\alpha}_{2}}\frac{\partial}{\partial f}$ | ${I}_{1}=\frac{1}{2}{x}^{2}({f}^{\prime 2}-{g}^{\prime 2})+\frac{1}{{\alpha}_{2}}{x}^{1+\alpha}exp({\alpha}_{2}f)cos({\alpha}_{2}g)+\frac{(\alpha +1)}{{\alpha}_{2}}x{f}^{\prime}$ |

${\mathbf{X}}_{2}=\frac{\alpha +1}{{\alpha}_{2}}\frac{\partial}{\partial g}$ | ${I}_{2}={x}^{2}{f}^{\prime}{g}^{\prime}+\frac{1}{{\alpha}_{2}}{x}^{1+\alpha}exp({\alpha}_{2}f)sin({\alpha}_{2}g)+\frac{(\alpha +1)}{{\alpha}_{2}}x{g}^{\prime}$ | |

Case B-1 | $\mathbf{X}={x}^{n}\frac{\partial}{\partial x}$ | ${I}_{1}=\frac{1}{2}{x}^{2n}({f}^{\prime 2}-{g}^{\prime 2})-\frac{\beta}{{\beta}_{1}}exp({\beta}_{1}f)cos({\beta}_{1}g)$ |

${I}_{2}={x}^{2n}{f}^{\prime}{g}^{\prime}-\frac{\beta}{{\beta}_{1}}exp({\beta}_{1}f)sin({\beta}_{1}g)$ |

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**MDPI and ACS Style**

Farooq, M.U.; Khalique, C.M.; Mahomed, F.M.
First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach. *Symmetry* **2019**, *11*, 1244.
https://doi.org/10.3390/sym11101244

**AMA Style**

Farooq MU, Khalique CM, Mahomed FM.
First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach. *Symmetry*. 2019; 11(10):1244.
https://doi.org/10.3390/sym11101244

**Chicago/Turabian Style**

Farooq, Muhammad Umar, Chaudry Masood Khalique, and Fazal M. Mahomed.
2019. "First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach" *Symmetry* 11, no. 10: 1244.
https://doi.org/10.3390/sym11101244