Invariant characterization of scalar third-order ODEs that admit the maximal contact symmetry Lie algebra

TL;DR

This paper uses Cartan's method to find invariants characterizing third-order ODEs with maximal contact symmetry, enabling efficient reduction to linear form and illustrating the approach with examples.

Contribution

It provides an invariant characterization and auxiliary functions for third-order ODEs with maximal contact symmetry using Cartan's equivalence method.

Findings

Derived invariant conditions for maximal contact symmetry

Provided auxiliary functions for contact transformation

Illustrated method with multiple examples

Abstract

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits the maximal ten-dimensional contact symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently determine the contact transformation that does the reduction to the simplest linear equation $\bar{u}'''=0$. Furthermore, ample examples are given to illustrate our method.

Equations

$$u^{\prime \prime \prime }+a(x)u=0.$$

$$\begin{array}[]{l}I_{1}=f_{q,q,q,q}\hfill\\ I_{2}={4\,f_{q}^{3}+18\,f_{q}\left({f_{p}-D_{x}f_{q}}\right)+9\,D_{x}^{2}f_{q}-27\,D_{x}f_{p}+54\,f_{u}}\\ \end{array}$$

$$\begin{array}[]{cc}u^{\prime\prime\prime}=f(x,u,u^{\prime},u^{\prime\prime}),&\bar{u}^{\prime\prime\prime}=\bar{f}(\bar{x},\bar{u},\bar{u}^{\prime},\bar{u}^{\prime\prime}),\\ \end{array}$$

$$\bar{x}=\varphi \left(x,u,p\right),\bar{u}=\psi \left(x,u,p\right),\bar{p}=g\left(x,u,p\right),$$

$$\Phi^{*}\left(\begin{array}[]{c}\bar{\omega}^{1}\\ \bar{\omega}^{2}\\ \bar{\omega}^{3}\\ \bar{\omega}^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}a_{1}&0&0&0\\ a_{2}&a_{3}&0&0\\ a_{4}&a_{5}&a_{6}&0\\ a_{7}&a_{8}&0&a_{9}\\ \end{array}\right)\left(\begin{array}[]{c}\omega^{1}\\ \omega^{2}\\ \omega^{3}\\ \omega^{4}\\ \end{array}\right),$$

$$\begin{array}[]{llll}\bar{\omega}^{1}=d\bar{u}-\bar{p}d\bar{x},&\bar{\omega}^{2}=d\bar{p}-\bar{q}d\bar{x},&\bar{\omega}^{3}=d\bar{q}-\bar{f}d\bar{x},&\bar{\omega}^{4}=d\bar{x},\\ \omega^{1}=du-pdx,&\omega^{2}=dp-qdx,&\omega^{3}=dq-fdx,&\omega^{4}=dx.\\ \end{array}$$

$$\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}a_{1}&0&0&0\\ a_{2}&a_{3}&0&0\\ a_{4}&a_{5}&a_{6}&0\\ a_{7}&a_{8}&0&a_{9}\\ \end{array}\right)\left(\begin{array}[]{c}\omega^{1}\\ \omega^{2}\\ \omega^{3}\\ \omega^{4}\\ \end{array}\right).$$

$$d\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&0\\ \alpha_{2}&\alpha_{3}&0&0\\ \alpha_{4}&\alpha_{5}&\alpha_{6}&0\\ \alpha_{7}&\alpha_{8}&0&\alpha_{9}\\ \end{array}\right)\wedge\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)+\left(\begin{array}[]{c}T^{1}_{24}~{}\theta^{2}\wedge\theta^{4}\\ T^{2}_{34}~{}\theta^{3}\wedge\theta^{4}\\ 0\\ 0\\ \end{array}\right)\\$$

$$\left.\begin{array}[]{cccc}d~{}T^{1}_{24}\equiv(\alpha_{1}-\alpha_{3}-\alpha_{9})T^{1}_{24}\\ d~{}T^{2}_{34}\equiv(\alpha_{3}-\alpha_{6}-\alpha_{9})T^{2}_{34}\\ \end{array}\right\}~{}\textrm{mod}~{}(\theta^{1},\theta^{2},\theta^{3},\theta^{4})\\$$

$$\begin{array}[]{cc}T^{1}_{24}=-1,&T^{2}_{34}=-1.\\ \end{array}$$

$$\begin{array}[]{cc}\alpha_{6}=2\alpha_{3}-\alpha_{1},&\alpha_{9}=\alpha_{1}-\alpha_{3}.\\ \end{array}$$

$$\begin{array}[]{cc}a_{6}=\frac{a_{3}^{2}}{a_{1}},&a_{9}=\frac{a_{1}}{a_{3}}.\\ \end{array}$$

$$\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}a_{1}&0&0&0\\ a_{2}&a_{3}&0&0\\ a_{4}&a_{5}&\frac{a_{3}^{2}}{a_{1}}&0\\ a_{7}&a_{8}&0&\frac{a_{1}}{a_{3}}\\ \end{array}\right)\left(\begin{array}[]{c}\omega^{1}\\ \omega^{2}\\ \omega^{3}\\ \omega^{4}\\ \end{array}\right).$$

$$d\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&0\\ \alpha_{2}&\alpha_{3}&0&0\\ \alpha_{4}&\alpha_{5}&2\alpha_{3}-\alpha_{1}&0\\ \alpha_{7}&\alpha_{8}&0&\alpha_{1}-\alpha_{3}\\ \end{array}\right)\wedge\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)+\left(\begin{array}[]{c}-\theta^{2}\wedge\theta^{4}\\ -\theta^{3}\wedge\theta^{4}\\ T^{3}_{34}~{}\theta^{3}\wedge\theta^{4}\\ 0\\ \end{array}\right)\\$$

$$\begin{array}[]{cccc}d~{}T^{3}_{34}\equiv(\alpha_{3}-\alpha_{1})T^{3}_{34}+3(\alpha_{2}-\alpha_{5})\\ \end{array}~{}\textrm{mod}~{}(\theta^{1},\theta^{2},\theta^{3},\theta^{4})\\$$

$$\begin{array}[]{cc}T^{3}_{34}=0.\\ \end{array}$$

$$\begin{array}[]{cc}\alpha_{5}=\alpha_{2}.\\ \end{array}$$

$$\begin{array}[]{c}a_{5}=\frac{a_{2}a_{3}}{a_{1}}-\frac{a_{3}^{2}}{3a_{1}}s_{1},\\ \end{array}$$

$$\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}a_{1}&0&0&0\\ a_{2}&a_{3}&0&0\\ a_{4}&\frac{a_{2}a_{3}}{a_{1}}-\frac{a_{3}^{2}}{3a_{1}}s_{1}&\frac{a_{3}^{2}}{a_{1}}&0\\ a_{7}&a_{8}&0&\frac{a_{1}}{a_{3}}\\ \end{array}\right)\left(\begin{array}[]{c}\omega^{1}\\ \omega^{2}\\ \omega^{3}\\ \omega^{4}\\ \end{array}\right).$$

$$d\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&0\\ \alpha_{2}&\alpha_{3}&0&0\\ \alpha_{4}&\alpha_{2}&2\alpha_{3}-\alpha_{1}&0\\ \alpha_{7}&\alpha_{8}&0&\alpha_{1}-\alpha_{3}\\ \end{array}\right)\wedge\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)+\left(\begin{array}[]{c}-\theta^{2}\wedge\theta^{4}\\ -\theta^{3}\wedge\theta^{4}\\ T^{3}_{24}~{}\theta^{2}\wedge\theta^{4}\\ 0\\ \end{array}\right)\\$$

$$\left.\begin{array}[]{ll}d~{}T^{3}_{24}\equiv 2(\alpha_{3}-\alpha_{1})T^{3}_{24}-2\alpha_{4}\\ \end{array}\right\}~{}\textrm{mod}~{}(\theta^{1},\theta^{2},\theta^{3},\theta^{4})\\$$

$$\begin{array}[]{c}T^{3}_{24}=0.\\ \end{array}$$

$$\begin{array}[]{c}\alpha_{4}=0.\\ \end{array}$$

$$\begin{array}[]{c}a_{4}=\frac{a^{2}_{2}}{2a_{1}}-\frac{a^{2}_{3}}{18a_{1}}s_{2},\\ \end{array}$$

$$\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}a_{1}&0&0&0\\ a_{2}&a_{3}&0&0\\ \frac{a^{2}_{2}}{2a_{1}}-\frac{a^{2}_{3}}{18a_{1}}s_{2}&\frac{a_{2}a_{3}}{a_{1}}-\frac{a_{3}^{2}}{3a_{1}}s_{1}&\frac{a_{3}^{2}}{a_{1}}&0\\ a_{7}&a_{8}&0&\frac{a_{1}}{a_{3}}\\ \end{array}\right)\left(\begin{array}[]{c}\omega^{1}\\ \omega^{2}\\ \omega^{3}\\ \omega^{4}\\ \end{array}\right).$$

$$d\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&0\\ \alpha_{2}&\alpha_{3}&0&0\\ 0&\alpha_{2}&2\alpha_{3}-\alpha_{1}&0\\ \alpha_{7}&\alpha_{8}&0&\alpha_{1}-\alpha_{3}\\ \end{array}\right)\wedge\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)+\left(\begin{array}[]{c}-\theta^{2}\wedge\theta^{4}\\ -\theta^{3}\wedge\theta^{4}\\ T^{3}_{14}~{}\theta^{1}\wedge\theta^{4}\\ 0\\ \end{array}\right)\\$$

$$\left.\begin{array}[]{cc}d~{}T^{3}_{14}\equiv&-3(\alpha_{1}-\alpha_{3})T^{3}_{14}\\ \end{array}\right\}~{}\textrm{mod}~{}(\theta^{1},\theta^{2},\theta^{3},\theta^{4})\\$$

$$\begin{array}[]{ll}T^{3}_{14}=&-\frac{a_{3}^{3}~{}I_{1}}{a_{1}^{3}},\\ \end{array}$$

$$\begin{array}[]{ll}I_{1}&=J^{3}=\frac{1}{54}\left(2s_{1}s_{2}-3D_{x}s_{2}+54f_{u}\right)\\ &=\frac{1}{54}\left({4\,f_{q}^{3}+18\,f_{q}\left({f_{p}-D_{x}f_{q}}\right)+9\,D_{x}^{2}f_{q}-27\,D_{x}f_{p}+54\,f_{u}}\right).\hfill\\ \end{array}$$

$$d\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&0\\ \alpha_{2}&\alpha_{3}&0&0\\ 0&\alpha_{2}&2\alpha_{3}-\alpha_{1}&0\\ \alpha_{7}&\alpha_{8}&0&\alpha_{1}-\alpha_{3}\\ \end{array}\right)\wedge\left(\begin{array}[]{c}\theta^{1}\\ \theta^{2}\\ \theta^{3}\\ \theta^{4}\\ \end{array}\right)+\left(\begin{array}[]{c}-\theta^{2}\wedge\theta^{4}\\ -\theta^{3}\wedge\theta^{4}\\ 0\\ 0\\ \end{array}\right)\\$$

Full text

Keywords: Invariant characterization, scalar third-order ordinary differential equation, contact symmetries, Cartan equivalence method.

1 Introduction

The idea of tangential transformations was initiated in the early work of Lie [1] wherein he found a local contact transformation which mapped straight lines into spheres in space. Notwithstanding, the theoretical foundation of the theory of contact transformations is given in Lie and Engel [2, 3, 4, 5]. Lie showed that for scalar $n$th-order ordinary differential equations (ODEs), $n=3$, its contact symmetry algebra is of finite dimension. He presented the complete classification of finite-dimensional irreducible contact Lie algebras in two complex variables [4]. In the more recent paper [6], the authors classified $n$th-order ODEs, $n\geq 3$, that admit nontrivial contact symmetry Lie algebras. Lie and Scheffers [7] have shown that a third-order ODE admits at most a ten-dimensional contact symmetry Lie algebra. They proved that the symmetry algebra is ten-dimensional if and only if the third-order ODE is equivalent, up to a local contact transformation, to the simplest third-order ODE (see also [6]). The reader is also referred to [8, 9, 10, 11] on these and related aspects.

Cartan [12], inter alia, provided solution to the linearzation problem for ODEs using his now popular approach called the Cartan equivalent method (see the recent contributions [13, 14]).

Lie and Scheffers [7], Yumaguzhin [11, 15] as well as Wafo et al. [6] studied the local classification of third-order linear ODEs, up to contact transformations. There are three canonical forms that occur for scalar linear third-order ODEs. The maximal contact symmetry Lie algebra case for such ODEs is of dimension ten and corresponds to the simplest equation $u^{\prime\prime\prime}=0$.

The Laguerre-Forsyth canonical form for scalar linear third-order ODEs is given by (see [16])

[TABLE]

where $a(x)$ is an arbitrary function of $x$. If $a(x)\not\equiv 0$, then it is known that (1.1) has a five- or four-dimensional contact symmetry algebra and otherwise ten-dimensional contact symmetry algebra.

Chern [17] was the first to invoke the Cartan equivalence method in order to solve the linearization problem for scalar third-order ODEs via contact transformations. He deduced conditions of equivalence to the equation (1.1) in the two cases $a(x)\equiv 0$ and $a(x)\equiv 1$. Then Neut and Petitot [18] also studied equivalence to (1.1) by means of contact transformations but for arbitrary $a(x)$. In the recent work, Ibragimov and Meleshko [19] investigated the linearization problem for third-order ODEs by utilising a direct approach via both point and contact transformations. It is the case that Ibragimov and Meleshko [19] first studied the second part of the linearization problem for scalar third-order ODEs which is that of constructing the relevant transformations to simpler form, via both point and contact transformations. However, it should be mentioned that the solution of the second part was given as a solution of a non-linear system of PDEs in their investigation. In the works [21, 22, 23], the authors very recently applied a new framework of the Cartan equivalence method to solve the two parts of the linearization problem for scalar third-order ODEs of the form $u^{\prime\prime\prime}=f(x,u,u^{\prime},u^{\prime\prime})$ via point transformations. In these essential works, the transformations can be obtained efficiently as a solution of a system of linear or Riccati equations given in terms of the introduced auxiliary functions.

We emphasise here that invariant characterization of a third-order ODE $u^{\prime\prime\prime}=f(x,u,u^{\prime},u^{\prime\prime})$ which admits the maximal ten contact symmetry algebra was obtained in terms of the function $f$ in the following theorem. We denote $u^{\prime},u^{\prime\prime}$ by $p,q$, respectively in the following and in what transpires in the sequel.

Theorem 1.1**.**

[18]** The third-order equation $u^{\prime\prime\prime}=f(x,u,u^{\prime},u^{\prime\prime})$ is equivalent to the simplest form $u^{\prime\prime\prime}=0$ with ten contact symmetries under contact transformations if and only if the relative invariants

[TABLE]

both vanish identically, where $D_{x}=\frac{\partial}{{\partial x}}+p\frac{\partial}{{\partial u}}+q\frac{\partial}{{\partial p}}+f\frac{\partial}{{\partial q}}$ and $I_{2}$ is the well-known W$\ddot{\textrm{u}}$nschmann relative invariant [18, 20].

In the present paper, we provide the solution of the second part of this equivalence problem, which thereby provides a systematic new way to construct the contact transformations that reduces the third-order ODE with ten contact symmetries to its canonical form. It is opportune to remark that the application of the Cartan equivalence method in this framework is both relevant and new. The new framework of Cartan s equivalence method gives invariant coframe explicitly in terms of auxiliary functions. The invariant coframe is utilized to determine the contact transformations to the equivalent canonical form. The contact transformations can be found efficiently as a solution of a system of linear or Riccati equations given in terms of the introduced auxiliary functions.

2 Application of Cartan s equivalence method to third-order ODEs with ten contact symmetries

For the basic definitions, notations and well-known facts that will be needed in this section, the reader is referred to [13, 14].

Let $(x,u,p=u^{\prime},q=u^{\prime\prime})\in\mathbb{R}^{4}$ as usual be local coordinates of $J^{2}$, the space of the second-order jets. In local coordinates, the equivalence of two third-order ODEs

[TABLE]

under a contact transformation

[TABLE]

with the contact condition $\bar{p}=\frac{d\bar{u}}{d\bar{x}}$ and non-zero Jacobian, can be expressed as the local equivalence problem for the $G$-structure

[TABLE]

where $\Phi^{*}$ is the pull-back of the smooth map $\Phi:(x,u,p,q)\rightarrow(\bar{x},\bar{u},\bar{p},\bar{q})$ defined by the prolongation of the contact transformation (2.4) and

[TABLE]

In particular, if $\phi_{p}=0$ then transformation (2.4) is a point transformation considered in the previous work [21, 22, 23]. Therefore, we assume in what follows that $\phi_{p}\neq 0$.

One can evaluate the functions $a_{i}=a_{i}(x,u,p,q),i=1\dots 9.$ Here we calculate some of them explicitly as follows $a_{1}=\frac{\phi_{x}\psi_{p}-\phi_{p}\psi_{x}}{p~{}\phi_{p}}$, $a_{2}=\frac{D_{x}a_{1}}{D_{x}\phi},a_{3}=\frac{a_{1}}{D_{x}\phi},a_{7}=\phi_{u},a_{8}=\phi_{p}$.

Now, one can define $\theta$ to be the lifted coframe with an nine-dimensional group $G$

[TABLE]

The application of Cartan’s method to this equivalence problem leads to an ${e}$-structure, which is invariantly associated to the given equation.

The first structure equation is

[TABLE]

where the operation $\wedge$ is the wedge product.

The infinitesimal action on the torsion is

[TABLE]

and a parametric calculation gives $T^{1}_{24}=-\frac{a_{1}}{a_{3}a_{9}}\neq 0$ and $T^{2}_{34}=-\frac{a_{3}}{a_{6}a_{9}}\neq 0$. We normalize the torsion by setting

[TABLE]

This leads to the principal components

[TABLE]

The normalizations force relations on the group $G$ in the form

[TABLE]

The first-order normalizations yield an adapted coframe with the seven-dimensional group $G_{1}$

[TABLE]

This leads to the structure equation

[TABLE]

The infinitesimal action on the torsion is

[TABLE]

and we can translate $T^{3}_{34}$ to zero:

[TABLE]

This leads to the principal components

[TABLE]

The normalizations force relations on the group $G_{1}$ are in the form

[TABLE]

where $s_{1}=f_{q}.$

The second-order normalizations yield an adapted coframe with the six-dimensional group $G_{2}$

[TABLE]

This leads to the structure equation

[TABLE]

The infinitesimal action on the torsion is

[TABLE]

and we can translate $T^{3}_{24}$ to zero:

[TABLE]

This leads to the principal components

[TABLE]

The normalizations force relations on the group $G_{2}$ as

[TABLE]

where $s_{2}=2f_{q}^{2}+9\,f_{p}-3\,D_{x}f_{q}$.

The third-order normalizations yield an adapted coframe with the five-dimensional group $G_{3}$

[TABLE]

This gives rise to the structure equation

[TABLE]

The infinitesimal action on the torsion is

[TABLE]

and here we have a bifurcation in the flowchart depending on whether $T^{3}_{14}$ is zero. A parametric calculation gives

[TABLE]

where

[TABLE]

It is well-known that third-order ODEs with ten contact symmetries have the canonical form $u^{\prime\prime\prime}=0$. It should be noted here that the relative invariants $I_{1}=0$ for the canonical form $u^{\prime\prime\prime}=0$. Therefore, we choose the following branch.

Branch 1. $I_{1}=0$.

In this branch, the structure equation has the form

[TABLE]

and there is no more unabsorbable torsion left, so the remaining group variables $a_{1},a_{2},a_{3},a_{7}$ and $a_{8}$ cannot be normalized. In addition, $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{7}$ and $\alpha_{8}$ are not uniquely defined where the following transformation

[TABLE]

keeps the structure equation (2.30) invariant for the free variables $\chi_{4,1},\chi_{4,4},\chi_{5,1}$ and $\chi_{5,2}$. So the problem is indeterminant. Moreover, the system is not in involution and we must prolong.

The prolonged coframe consists of the original lifted coframe

[TABLE]

now viewed as a collection of one-forms on the nine-dimensional space $M^{(1)}=J^{2}\times G_{3}$ with coordinates $x,u,p,q,a_{1},a_{2},a_{3},a_{7},a_{8}$ together with the modified Maurer-Cartan forms

[TABLE]

The prolonged structure group is a four-dimensional abelian group having the $9\times 9$ matrix representation

[TABLE]

The new equivalent equivalence problem

In this section, we will apply Cartan’s method to the lifted coframe with a four-dimensional group $G^{(1)}$

[TABLE]

where $A$ is the new prolonged coframe

[TABLE]

as defined in given in (2.32) and (2.33).

The first structure equation is

[TABLE]

where

[TABLE]

The infinitesimal action on the torsion is

[TABLE]

and we can translate $T^{6}_{13},T^{6}_{23},T^{7}_{23}$ to zero:

[TABLE]

This leads to the principal components

[TABLE]

The normalizations force relations on the group $G^{(1)}$ in the form

[TABLE]

The first-order normalizations yield an adapted coframe with the one-dimensional group $G^{(1)}_{1}$

[TABLE]

where $R_{1}$ is the matrix $R$ given in (2.34) after incorporating the values of the parameters $\chi_{4,1},\chi_{5,1},\chi_{5,2}$ obtained in (2.42).

This leads to the structure equation

[TABLE]

where

[TABLE]

The infinitesimal action on the torsion is

[TABLE]

This means that the four invariants do not depend on the group parameter $\chi_{4,4}$, but only on the original group parameters $a_{1},a_{2},a_{3},a_{7},a_{8}$ and the base variables $x,u,p,q$. Therefore, they are invariants of the original equivalence problem. Moreover, it is noted that

[TABLE]

and here we have a bifurcation in the flowchart depending on whether $T^{9}_{23}$ is zero. A parametric calculation gives

[TABLE]

where

[TABLE]

Similarly, the relative invariant $I_{2}=0$ for the canonical form $u^{\prime\prime\prime}=0$. Thus, we choose the following branch.

Branch 2 $I_{2}=0$.

In this branch, the structure equation have the form

[TABLE]

there is no more unabsorbable torsion left, so the remaining group variables $\chi_{4,4}$ cannot be normalized. Moreover, $\alpha_{2}$ is uniquely defined, so the problem is determinant. This results in the following $e$-structure on the ten-dimensional prolonged space $M^{(2)}=M^{(1)}\times G^{(1)}_{1}$ which consists of the original lifted coframe

[TABLE]

[TABLE]

together with the modified Maurer-Cartan forms given $\textrm{mod}~{}(\omega^{1},\omega^{2},\omega^{3})$ as

[TABLE]

where

[TABLE]

This results in the structure equations

[TABLE]

The invariant structure of the prolonged coframe are all constant. We have produced an invariant coframe with rank zero on the ten-dimensional space coordinates $x,u,p,q,a_{1},a_{2},$ $a_{3},a_{7},a_{8},\chi_{4,4}$. Any such differential equation admits a ten-dimensional symmetry group of contact transformations.

Finally, inserting the point transformation (2.4) into the symmetrical version of the Cartan formulation and using $\bar{s}_{1}=\bar{s}_{2}=0$ for $\bar{f}=0$ and $\bar{a_{1}}=1,\bar{a_{2}}=0,\bar{a_{3}}=1,\bar{a_{7}}=0$, $\bar{a_{8}}=0,\bar{\chi}_{4,4}=0$, results in

[TABLE]

This proves the following theorem.

Theorem 2.1**.**

The necessary and sufficient conditions for equivalence of a scalar third-order ODE $u^{\prime\prime\prime}=f(x,u,u^{\prime},u^{\prime\prime})$ to its canonical form $\bar{u}^{\prime\prime\prime}=0$, with ten contact symmetries via contact transformations (2.4), are the identical vanishing of the relative invariants

[TABLE]

Given that the the system of relative invariants (2.57) is zero, the linearizing contact transformation (2.4) is defined by

[TABLE]

where $a_{1}(x,u,p),a_{2}(x,u,p,q),a_{3}(x,u,p,q),a_{7}(x,u,p),a_{8}(x,u,p),\chi_{4,4}(x,u,p,q)$ are auxiliary functions given by

[TABLE]

where

[TABLE]

Remark 2.2**.**

The last two equations of the system (2.59) ensure the compatibility of the system (2.58).

3 Illustration of the theorem

Example 3.1**.**

[19] Consider the nonlinear ODE

[TABLE]

The function

[TABLE]

satisfies the constraints $I_{1}=I_{2}=0$; as a consequence, this equation admits the ten-dimensional contact symmetry group. Moreover, it is equivalent to the canonical form $\bar{u}^{\prime\prime\prime}=0$. We outline the steps.

Step 1

One can verify that for the following auxiliary functions

[TABLE]

are solution for the system (2.59).

Step 2

Using the obtained auxiliary functions (3.63), a solution of the system (2.58) can be given as follows:

[TABLE]

Step 3

Finally, one can verify that the transformation

[TABLE]

transforms the canonical form $\bar{u}^{\prime\prime\prime}=0$ to the nonlinear ODE (3.61).

Example 3.2**.**

[19] Consider the nonlinear ODE

[TABLE]

The function

[TABLE]

satisfies the constraints $I_{1}=I_{2}=0$; as a consequence, this equation admits the ten-dimensional contact symmetry group. Moreover, it is equivalent to the canonical form $\bar{u}^{\prime\prime\prime}=0$. We outline the steps.

Step 1

One can verify that for the following auxiliary functions

[TABLE]

are solution for the system (2.59).

Step 2

Using the obtained auxiliary functions (3.68), a solution of the system (2.58) can be given as follows:

[TABLE]

Step 3

Finally, one can verify that the transformation

[TABLE]

transforms the canonical form $\bar{u}^{\prime\prime\prime}=0$ to the nonlinear ODE (3.66).

The above two examples are paradigms for equations that possess the maximal contact symmetries. The geometrical and contact transformation properties were discussed in [24]. They respectively describe hyperbolas and circles in the plane. One can easily check in Step 3 of each of the examples that $d\bar{u}/d\bar{x}=\bar{p}$. This is a first check that the contact transformations are indeed correct. For the first example, the solution is immediate. For integrating twice $\bar{u}^{\prime\prime\prime}=0$, one has $\bar{u}^{\prime}=c_{2}\bar{x}+c_{1}$ which gives $-x\sqrt{p}=2c_{2}\sqrt{p}+c_{1}$ and (remember that $p=du/dx$) finally the family of hyperbolas $u=k_{1}+k_{2}(x+k_{3})^{-1}$ after renaming of the constants $c_{i}$. Likewise, in the second example, after integrating two times we have $u-x(p+\sqrt{1+{p}^{2}})=c_{2}(p+\sqrt{1+{p}^{2}})+c_{1}$ which eventually yields after integrating $p=du/dx$ for $u$ as a function of $x$ and then renaming of the constants, the family of circles $(k_{1}x+k_{2})^{2}+(k_{1}u+k_{3})^{2}=1$.

4 Conclusion

We have shown in this work how the Cartan equivalence method can be used to find an invariant characterization of scalar third-order ODEs that admit the ten-dimensional contact symmetry algebra. Importantly this approach provides auxiliary functions which can be effectively utilized to construct the contact transformation in order to find the reduction to the simplest third-order ODE. We have demonstrated the utility of the method by significant examples in a constructive manner.

Acknowledgments

Ahmad Y. Al-Dweik and M. T. Mustafa are thankful to Qatar University for its continuous support as well as excellent research facilities. FMM is grateful to the NRF of South Africa for support.