# Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

## Abstract

**:**

## 1. Introduction

**Definition:**

**Lie–Scheffers Theorem [1]:**

## 2. Vessiot–Guldberg–Lie Algebras with $r\le 3$ for Scalar SODE Systems

#### 2.1. $dim{\mathcal{L}}_{VG}=2$

#### 2.2. $dim{\mathcal{L}}_{VG}=3$

**Proposition 1:**

- ${\mathfrak{r}}_{1,\epsilon}$:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]=\epsilon \phantom{\rule{0.166667em}{0ex}}{X}_{2},& \hfill \left[{X}_{2},{X}_{3}\right]={X}_{1},\phantom{\rule{1.em}{0ex}}\epsilon =-1,1.\end{array}$$
- ${\mathfrak{r}}_{2}$:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]={X}_{1},& \hfill \left[{X}_{2},{X}_{3}\right]=-{X}_{2}.\end{array}$$
- ${\mathfrak{r}}_{3,\lambda}$:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3}& \hfill \left[{X}_{1},{X}_{3}\right]=0,& \hfill \left[{X}_{2},{X}_{3}\right]=\lambda {X}_{1},\phantom{\rule{1.em}{0ex}}\lambda \in \mathbb{R}.\end{array}$$
- ${\mathfrak{r}}_{4}:$$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]=0,& \hfill \left[{X}_{2},{X}_{3}\right]=0.\end{array}$$
- ${\mathfrak{r}}_{5}$:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]=-{X}_{2},& \hfill \left[{X}_{2},{X}_{3}\right]=0.\end{array}$$
- ${\mathfrak{r}}_{6,\lambda}$:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]=-\lambda {X}_{1}+{X}_{3}& \hfill \left[{X}_{1},{X}_{3}\right]=0,& \hfill \left[{X}_{2},{X}_{3}\right]={X}_{1}+\lambda {X}_{3},\phantom{\rule{1.em}{0ex}}\lambda >0.\end{array}$$

#### 2.3. Second-Order ODEs with Three-Dimensional Vessiot–Guldberg–Lie Algebra

- ${\mathcal{L}}_{VG}\simeq {\mathfrak{r}}_{1,\epsilon}:$Commutators:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]=\epsilon \phantom{\rule{0.166667em}{0ex}}{X}_{2},& \hfill \left[{X}_{2},{X}_{3}\right]={X}_{1}\end{array}$$Realization:$$\begin{array}{cc}{X}_{1}=v\frac{\partial}{\partial x}+\left(\frac{{G}^{\prime}\left(x\right)}{2}-k\left(x\right)\left(G\left(x\right)-{v}^{2}\right)\right)\frac{\partial}{\partial v},\hfill & {X}_{2}=\sqrt{G\left(x\right)-{v}^{2}}\frac{\partial}{\partial v}\hfill \\ {X}_{3}=-\sqrt{G\left(x\right)-{v}^{2}}\left(\frac{\partial}{\partial x}+v\phantom{\rule{0.166667em}{0ex}}k\left(x\right)\frac{\partial}{\partial v}\right)\hfill \end{array}$$Second-order differential equation:$$\ddot{x}=k\left(x\right)\phantom{\rule{0.166667em}{0ex}}{\dot{x}}^{2}+{F}_{2}\left(t\right)\sqrt{G\left(x\right)-{\dot{x}}^{2}}+\frac{1}{2}{G}^{\prime}\left(x\right)-k\left(x\right)G\left(x\right)$$Constraints:$${G}^{\u2033}\left(x\right)-2{k}^{\prime}\left(x\right)G\left(x\right)+2{k}^{2}\left(x\right)G\left(x\right)-3k\left(x\right){G}^{\prime}\left(x\right)-2\phantom{\rule{0.166667em}{0ex}}\epsilon =0$$
- ${\mathcal{L}}_{VG}\simeq {\mathfrak{r}}_{2}:$Commutators:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]={X}_{1},& \hfill \left[{X}_{2},{X}_{3}\right]=-{X}_{2}\end{array}$$Realization:$$\begin{array}{ccc}\hfill {X}_{1}=v\frac{\partial}{\partial x}+\left(k\left(x\right)+\frac{\left(1+2\phantom{\rule{0.166667em}{0ex}}{G}^{\prime}\left(x\right)\right)}{2G\left(x\right)}{v}^{2}\right)\frac{\partial}{\partial v},& \hfill {X}_{2}=G\left(x\right)\frac{\partial}{\partial v},& \hfill {X}_{3}=-G\left(x\right)\frac{\partial}{\partial x}-v\left(\phantom{\rule{0.166667em}{0ex}}{G}^{\prime}\left(x\right)+1\right)\frac{\partial}{\partial v}\end{array}$$Second-order differential equation:$$\ddot{x}=\frac{{\dot{x}}^{2}}{2}\left(\frac{1+2{G}^{\prime}\left(x\right)}{G\left(x\right)}\right)+k\left(x\right)+{F}_{2}\left(t\right)G\left(x\right)$$Constraints:$${k}^{\prime}\left(x\right)G\left(x\right)-k\left(x\right){G}^{\prime}\left(x\right)-2k\left(x\right)=0$$
- ${\mathcal{L}}_{VG}\simeq {\mathfrak{r}}_{3,\lambda}:$Commutators:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]=0,& \hfill \left[{X}_{2},{X}_{3}\right]=\lambda {X}_{1}\end{array}$$Realization:$$\begin{array}{cc}{X}_{1}=v\frac{\partial}{\partial x}+\left(k\left(x\right)\left(G\left(x\right)-\lambda {v}^{2}\right)+\frac{{G}^{\prime}\left(x\right)}{2\lambda}\right)\frac{\partial}{\partial v},\hfill & {X}_{2}=\sqrt{G\left(x\right)-\lambda {v}^{2}}\frac{\partial}{\partial v},\hfill \\ {X}_{3}=-\sqrt{G\left(x\right)-\lambda {v}^{2}}\left(\frac{\partial}{\partial x}-\lambda vk\left(x\right)\right)\frac{\partial}{\partial v}\hfill \end{array}$$Second-order differential equation:$$\ddot{x}=-\lambda k\left(x\right){\dot{x}}^{2}+G\left(x\right)k\left(x\right)+\frac{{G}^{\prime}\left(x\right)}{2\lambda}+{F}_{2}\left(t\right)\sqrt{G\left(x\right)-\lambda {\dot{x}}^{2}}$$Constraints:$$2{\lambda}^{2}{G}^{\prime}\left(x\right){k}^{2}\left(x\right)+2\lambda G\left(x\right){k}^{\prime}\left(x\right)+3\lambda G\left(x\right)k\left(x\right)+{G}^{\u2033}\left(x\right)=0$$
- ${\mathcal{L}}_{VG}\simeq {\mathfrak{r}}_{4}:$Commutators:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]=0,& \hfill \left[{X}_{2},{X}_{3}\right]=0\end{array}$$Realization:$$\begin{array}{ccc}\hfill {X}_{1}=v\frac{\partial}{\partial x}+\frac{{G}^{\prime}\left(x\right){v}^{2}+\mu {G}^{2}\left(x\right)}{G\left(x\right)}\frac{\partial}{\partial v},& \hfill {X}_{2}=G\left(x\right)\frac{\partial}{\partial v},& \hfill {X}_{3}=-G\left(x\right)\frac{\partial}{\partial x}-v{G}^{\prime}\left(x\right)\frac{\partial}{\partial v}\end{array}$$Second-order differential equation:$$\ddot{x}=\frac{{G}^{\prime}\left(x\right)\phantom{\rule{0.166667em}{0ex}}}{G\left(x\right)}{\dot{x}}^{2}+\left({F}_{2}\left(t\right)+\mu \right)G\left(x\right)$$
- ${\mathcal{L}}_{VG}\simeq {\mathfrak{r}}_{5}:$Commutators:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]={X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]=-{X}_{2},& \hfill \left[{X}_{2},{X}_{3}\right]=0\end{array}$$Realization:$$\begin{array}{ccc}\hfill {X}_{1}=v\frac{\partial}{\partial x}+\left(\frac{{G}^{\prime}\left(x\right)}{G\left(x\right)}{v}^{2}+k\left(x\right)\right)\frac{\partial}{\partial v},& \hfill {X}_{2}=G\left(x\right)\frac{\partial}{\partial v},& \hfill {X}_{3}=-G\left(x\right)\frac{\partial}{\partial x}-v{G}^{\prime}\left(x\right)\frac{\partial}{\partial v}\end{array}$$Second-order differential equation:$$\ddot{x}=\frac{{G}^{\prime}\left(x\right)\phantom{\rule{0.166667em}{0ex}}}{G\left(x\right)}{\dot{x}}^{2}+k\left(x\right)+{F}_{2}\left(t\right)G\left(x\right)$$Constraints:$${k}^{\prime}\left(x\right)G\left(x\right)-k\left(x\right){G}^{\prime}\left(x\right)+G\left(x\right)=0$$
- ${\mathcal{L}}_{VG}\simeq {\mathfrak{r}}_{6,\lambda}:$Commutators:$$\begin{array}{ccc}\hfill \left[{X}_{1},{X}_{2}\right]=-\lambda {X}_{1}+{X}_{3},& \hfill \left[{X}_{1},{X}_{3}\right]=0,& \hfill \left[{X}_{2},{X}_{3}\right]={X}_{1}+\lambda {X}_{3}\end{array}$$Realization:$$\begin{array}{ccc}\hfill {X}_{1}=v\frac{\partial}{\partial x}+\frac{{G}^{2}\left(v\right)}{\left(\beta +\left(1+{\lambda}^{2}\right)x\right)}\frac{\partial}{\partial v},& \hfill {X}_{2}=G\left(v\right)\frac{\partial}{\partial v},& \hfill {X}_{3}=\left(\lambda v-G\left(v\right)\right)\frac{\partial}{\partial x}-\frac{{G}^{2}\left(v\right)\left({G}^{\prime}\left(v\right)-\lambda \right)}{\left(\beta +\left(1+{\lambda}^{2}\right)x\right)}\frac{\partial}{\partial v}\end{array}$$Second-order differential equation:$$\ddot{x}=\frac{{G}^{2}\left(\dot{x}\right)}{\left(\beta +\left(1+{\lambda}^{2}\right)x\right)}+{F}_{2}\left(t\right)G\left(\dot{x}\right)$$Constraints:$$2\lambda G\left(v\right)-G\left(v\right){G}^{\prime}\left(v\right)-\left(1+{\lambda}^{2}\right)v=0$$

#### 2.4. Examples

- The well-known Milne–Pinney equation:$$\ddot{x}=F\left(t\right)\phantom{\rule{0.166667em}{0ex}}x+\frac{c}{{x}^{3}},\phantom{\rule{0.277778em}{0ex}}c\in \mathbb{R}$$$$-\frac{x}{2}{k}^{\prime}\left(x\right)-\frac{3}{2}k\left(x\right)=0$$$${X}_{1}=v\frac{\partial}{\partial x}+\frac{c}{{x}^{3}}\frac{\partial}{\partial v},\phantom{\rule{0.277778em}{0ex}}{X}_{2}=-\frac{x}{2}\frac{\partial}{\partial v},\phantom{\rule{0.277778em}{0ex}}{X}_{3}=-\frac{x}{2}\frac{\partial}{\partial x}-\frac{v}{2}\frac{\partial}{\partial v}$$
- The Kummer–Schwarz equation:$$\ddot{x}=\frac{3}{2\phantom{\rule{0.166667em}{0ex}}x}{\dot{x}}^{2}-2c\phantom{\rule{0.166667em}{0ex}}{x}^{3}+2g\left(t\right)x$$$$x\phantom{\rule{0.166667em}{0ex}}{k}^{\prime}\left(x\right)-3\phantom{\rule{0.166667em}{0ex}}k\left(x\right)=0$$$${X}_{1}=v\frac{\partial}{\partial x}+\left(c\phantom{\rule{0.166667em}{0ex}}{x}^{3}+\frac{3}{2\phantom{\rule{0.166667em}{0ex}}x}\right)\frac{\partial}{\partial v},\phantom{\rule{0.277778em}{0ex}}{X}_{2}=x\frac{\partial}{\partial v},\phantom{\rule{0.277778em}{0ex}}{X}_{3}=-x\frac{\partial}{\partial x}-2v\frac{\partial}{\partial v}$$

## 3. SODE Lie Systems in the Plane

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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Campoamor-Stursberg, R.
Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations. *Symmetry* **2016**, *8*, 15.
https://doi.org/10.3390/sym8030015

**AMA Style**

Campoamor-Stursberg R.
Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations. *Symmetry*. 2016; 8(3):15.
https://doi.org/10.3390/sym8030015

**Chicago/Turabian Style**

Campoamor-Stursberg, Rutwig.
2016. "Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations" *Symmetry* 8, no. 3: 15.
https://doi.org/10.3390/sym8030015