# Singular invariant structures for Lie algebras admitted by a system of second-order ODEs

*Muhammad Ayub* (),
*Sadia Sadique* and
*F.M. Mahomed*

*Applied Mathematics and Computation*, 2016, vol. 281, issue C, 137-147

**Abstract:**
Systems of second-order ordinary differential equations (ODEs) arise in mechanics and have several applications. Differential invariants play a key role in the construction of invariant differential equations as well as the classification of a system of ODEs. Like regular invariants, singular invariant structures also possess an important role in the algebraic analysis of a system of ODEs. In this work, singular invariant equations for a system of two second-order ODEs admitting four-dimensional Lie algebras are investigated. Moreover, by using these singular invariant equations, canonical forms for systems of two second-order ODEs are constructed. Furthermore, it is observed that the same Lie algebra admitted by a system of second-order ODEs has different type of realizations some of which are related to regular invariants and some lead to singular invariant equations. Thus realizations of four-dimensional Lie algebras are associated with a regular invariant manifold as well as to a singular invariant manifold defined by a system of second-order ODEs. In addition, a case of a Lie algebra with realization resulting in a conditional singular invariant structure is presented and those cases of singular invariant equations are discussed which do not form a system of second-order ODEs. An integration procedure for the invariant representation of these canonical forms are presented. Illustrative examples are presented with physical applications to mechanics.

**Keywords:** Systems of second-order ODEs; Singular invariant equations; Integrability (search for similar items in EconPapers)

**Date:** 2016

**References:** View complete reference list from CitEc

**Citations:** Track citations by RSS feed

**Downloads:** (external link)

http://www.sciencedirect.com/science/article/pii/S0096300316300042

Full text for ScienceDirect subscribers only

**Related works:**

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:eee:apmaco:v:281:y:2016:i:c:p:137-147

**DOI:** 10.1016/j.amc.2016.01.004

Access Statistics for this article

Applied Mathematics and Computation is currently edited by *Theodore Simos*

More articles in Applied Mathematics and Computation from Elsevier

Bibliographic data for series maintained by Catherine Liu ().