Equivalence Method on Second‐Order Autonomous Odes With Applications in Dynamical System

ABSTRACTThe equivalence problem in differential geometry investigates whether two geometric structures can be transformed into one another through an appropriate change of variables. This issue is central to understanding the invariants associated with dynamical systems, which remain unchanged under such transformations. The Cartan equivalence method provides a powerful and systematic framework for addressing these problems by reducing complex equivalence questions to more tractable ones involving coframes and structure equations. In this study, we apply Cartan's method to second‐order autonomous ordinary differential equations (ODEs), offering insights into the intrinsic geometric structure of these systems under coordinate transformations. Specifically, we examine a class of second‐order ODEs that model nonlinear dynamical behavior, employing the pseudogroup of web transformations to identify differential invariants and symmetry classes. The study introduces a step‐by‐step construction of an invariant coframe, elucidates the structure equations governing the system, and provides a classification of the equations based on their differential invariants. Two representative physical systems are investigated in detail: the motion of a damped spring‐mass system and the van der Pol oscillator. In addition to these applications, the paper explores the interplay between equivalence, symmetry reduction, and integrability. By leveraging Cartan's method, we aim to provide a deeper understanding of the equivalence problem in the context of higher order differential equations, highlighting both the theoretical elegance and practical utility of the approach in uncovering hidden geometric features, symmetries, and conserved quantities.