Lie Groups of Transformations and Infinitesimal Transformations

Abstract

In dimensional analysis the scaling transformations of the fundamental dimensions (1.86), the induced scaling transformations of the measurable quantities (1.88), the induced scaling transformations of all quantities (1.88), (1.89), and the induced scaling transformations preserving all constants (1.88), (1.91), are all examples of Lie groups of transformations. Scaling transformations are easily described in terms of their global properties as seen in Chapter 1. From the point of view of finding solutions to differential equations a general theory of Lie groups of transformations is unnecessary if transformations are restricted to scalings, translations, or rotations. However it turns out that much wider classes of transformations leave differential equations invariant including transformations composed of scalings, translations, and rotations. For the use and discovery of such transformations the notion of a Lie group of transformations is crucial—in particular the characterization of such transformations in terms of infinitesimal generators (which form a Lie algebra). This chapter introduces the basic ideas of Lie groups of transformations necessary in later chapters for the study of invariance properties of differential equations.