Dimensional Homogeneity in Classifying Second-Order Differential Invariant Systems for Four-Dimensional Lie Algebras

This research investigates a specific mathematical structure called a nonsingular differential invariant structure, which plays a crucial role in describing physical phenomena through differential equations. Focusing on a four-dimensional Lie algebra in (1+3)-dimensional space, this study classifies these structures. Here, a method is developed to identify second-order differential invariants and showed how they translate into systems of three second-order ordinary differential equations for the specific case of a four-dimensional Lie algebra, and these invariants are presented in the form of list in Table 1. To further refine the analysis, these systems are classified based on their “dimensional homogeneity structure,” which reveals their behavior when scaling all variables. This sub-classification described in Table 2, goes beyond the initial broad categorization, potentially linking these structures to specific physical applications. Novelty: To best of our knowledge, the classification of nonsingular differential invariant structure in (1+3)-dimensional space of four-dimensional real Lie algebras and association of dimensional homogeneity structure in classification scheme are new contributions in algebraic classification problems in literature and never reported before this. Such type of classification scheme is fruitful in analysis of many physical systems. In addition, the algebraic properties of these peculiar invariant systems including integrability, are analysed in a detailed manner both mathematically and from a mechanics viewpoint.