Identifications of paths and curves under the plane similarity transformations and their applications to mechanics

Abstract

In this paper, global differential G-invariants of paths in the two-dimensional Euclidean space E2 for the similarity group G=Sim(E2) and the orientation-preserving similarity group G=Sim+(E2) are investigated. A general form of a path in terms of its global G-invariants is obtained. For given two paths ξ(t) and η(t) with the common differential G-invariants, general forms of all transformations g∈G, carrying ξ(t) to η(t), are found. Similar results are given for curves. Moreover, analogous of the similarity groups in the three-dimensional space–time and in the four-dimensional space–time-mass are defined. Finally, applications to Newtonian mechanics of the above results are given.