Isomorphism in Epicyclic Gear Trains and Kinematic Chain

Abstract

Isomorphism, a concept prevalent in various fields such as mathematics, chemistry, and computer science, denotes a fundamental relationship between two entities where their structural or functional characteristics remain unchanged despite differences in their appearance or representation. In the realm of mathematics, isomorphism refers to a bijective mapping between two mathematical structures, such as groups, rings, or graphs, preserving their fundamental properties and relationships [1-5]. In graph theory, isomorphism describes the equivalence between two graphs where the nodes and edges can be relabeled or rearranged without altering the fundamental structure of the graph. Formally, two graphs G and H are considered isomorphic if there exists a bijective mapping between their vertices and edges that preserves adjacency and connectivity properties [6]. Isomorphism plays a crucial role in various fields, including chemistry, where it is used to classify molecules with identical structural arrangements of atoms and bonds. In organic chemistry, for instance, isomorphism distinguishes between different chemical compounds that share the same molecular formula but exhibit distinct structural arrangements, such as structural isomers [5-10]. Moreover, isomorphism finds applications in computer science, particularly in the study of data structures and algorithms. In this context, isomorphic data structures have the same underlying organization and