Complete Group Classification of Systems of Two Nonlinear Second-Order Ordinary Differential Equations of the Form ${\bf y}^{\prime\prime}={\bf F}({\bf y})$

TL;DR

This paper provides a complete group classification of systems of two autonomous nonlinear second-order ODEs, filling a significant gap in the literature and applicable to many physical models.

Contribution

It introduces two approaches for classifying such systems, advancing the theoretical understanding of their symmetry properties.

Findings

Complete classification achieved for the specified systems

Two novel methods for group classification introduced

Applicable to various physical systems involving nonlinear ODEs

Abstract

Extensive work has been done on the group classification of systems of equations in the literature. This paper identifies the gap in the literature which concerns the group classification of systems of two autonomous nonlinear second-order ordinary differential equations. We provide a complete group classification using two approaches which form the essence of this paper and argue that this can be applied to systems of autonomous nonlinear second-order ordinary differential equations which occur in many physical applications.