Advances in Mathematical Physics

Volume 2017, Article ID 2580968, 12 pages

https://doi.org/10.1155/2017/2580968

## Canonical Forms and Their Integrability for Systems of Three 2nd-Order ODEs

^{1}Department of Mathematics, Azad Jammu and Kashmir University, Muzaffarabad, Pakistan^{2}Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan

Correspondence should be addressed to Muhammad Ayub; moc.liamtoh@5buya_dammahum

Received 5 April 2017; Accepted 25 May 2017; Published 16 July 2017

Academic Editor: Mariano Torrisi

Copyright © 2017 S. Zahida et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Differential invariants and their corresponding canonical forms for systems of three 2nd-order ODEs possessing three-dimensional Lie algebras are constructed. Their extension up to th-order system of three 2nd-order ODEs is presented. Furthermore singularity in invariant structure for the canonical forms is investigated. In addition integrability of these canonical forms is discussed. Illustrative physical examples from mechanics of system of particles are provided.

#### 1. Introduction

There are several physical phenomena whose mathematical modeling is associated with system of 2nd-order ODEs and hence analysis of various aspects of these systems of ordinary differential equations plays a vital role in the applied sciences. Due to this contributing role and importance, these systems have been extensively studied over the years. Different methods and approaches have been introduced and constructed for various aspects of the analysis of these systems.

One of the prominent approaches is Lie symmetry method, initiated by Sophus Lie 1842–1899. This is a general method in which exact solutions and various other aspects of linear and nonlinear differential equations are analysed. Lie theory is based on Lie groups and the corresponding Lie algebras. Lie proved that the integrability of a differential equation is contingent on the properties of the Lie algebra it admits; for details see [1–5].

In Lie theory, there are two approaches for the integrability of differential equations: direct and indirect approach. In the direct approach classical reduction is performed via differential invariants and canonical variables and so forth, while in the indirect approach equations are classified via their admitted Lie algebras in prescribed number of underlying variables with corresponding set of structure constants. Then algebraic realizations are used for integration of these classified forms. The indirect approach is also known as Lie algebraic approach. The direct approach is useful for scalar ordinary differential equations only but does not work for system of ordinary differential equations [6]. The indirect approach is equally valid for both scalar ODEs as well as system of ordinary differential equations.

In this paper, we are interested in Lie algebraic approach (Realization approach) for system of 2nd-order ordinary differential equations.

To apply the algebraic approach, equations have to be classified according to their admitted symmetry algebras. Moreover this classification scheme depends not only upon the abstract Lie algebras (and their subalgebras) but also the realizations of the admitted Lie algebras in prescribed number of variables.

The initial work on classification and realizations of Lie algebras was done by Lie [7]. After Lie, several researchers have been involved in analysis of different aspects of this work; see [1, 6, 8–12]. Although the algebraic approach is more involved, it generates new cases and more insight to mathematical properties whose base is algebra (see [8, 9, 13, 14]). Thus the classification of differential equations via algebraic approach is more efficient than other classical approaches. Unluckily, very limited literature is available on this approach for system of 2nd-order ordinary differential equations.

Like systems of two 2nd-order ODEs, systems of three 2nd-order ODEs have vital role in different mathematical modeling of physical situations such as small oscillation problems, wave propagation problems, and problems of mechanics. There is very limited and restricted work on classification of system of three 2nd-order ODEs according to their symmetry algebras (see [15, 16]). Thus in this paper, we are interested in investigating classification of system of three 2nd-order ODEs via the algebraic approach for dimension 3.

The outline of the rest of the paper is as follows. In Section 2 invariant structure of canonical forms and their Lie algebraic properties are constructed. Section 3 is about the integrability of underlying canonical forms, whereas Section 4 presents illustrative examples of the results obtained in this paper. Paper ends with a brief conclusion.

#### 2. Differential Invariants and Canonical Forms

By fixing number of prescribed variables for a given set of structure constants associated with an underlying Lie algebra one can obtain the most general classes of system of differential equations which admit the investigated Lie algebra in prescribed number of variables.

Here group classification of system of three 2nd order ODEs, admitting three-dimensional Lie algebra, has been investigated via algebraic approach. The classification results of [11] are utilized but in the format of [8].

##### 2.1. Invariant Construction

Following result given in [8] is fundamental in construction of invariants and their corresponding canonical forms.

Proposition 1. *If and are invariants of a Lie algebra admitted by any system of ordinary differential equations, then is also its invariant.*

For the construction of canonical forms the approach developed in [8, 9, 13] is employed. By utilizing realizations of underlying 3-dimensional Lie algebras admitted by system of three 2nd-order ordinary differential equations we find differential invariants associated with them. These invariants are then used to construct invariant system of differential equations, called canonical forms of system.

*Notations.* The following notions are used in rest of the paper: denotes the th Lie algebra of dimension whereas superscripts indicate parameters on which the algebra possibly depends; the column in Table 1 gives the algebra realizations; the realization is referred to by a superscript ; as usual . denotes the elements of a basis of a given Lie algebra; here is less than or equal to the dimension of the underlying real Lie algebra. The rank of the associated realization is denoted by and is the realization of the corresponding algebra.