Application of Differential Equations by Li Qun (2nd Edition)

Author: P.J.Olver
Publisher:
Publish Date: 1999-11-01
Features: This book is devoted to explaining a wide range of applications of continuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can easily learn the basic computational techniques required for genuine physical problems. This book is in English. Excerpt: The first section gives a basic outline of the general concept of a manifold, the second doing the same for Lie groups, both local and global. In practice, Lie groups arise as groups of symmetries of some object, or, more precisely, as local groups of transformations acting on some manifold; the second section gives a brief look at these. The most important concept in the entire theory is that of a vector field, which acts as the "infinitesimal generator" of a one-parameter Lie group of transformations. This concept is fundamental for both the development of the theory of Lie groups and the applications to differential equations. It has the crucial effect of replacing complicated nonlinear conditions for the symmetry of some object under a group of transformations by easily verifiable linear conditions reflecting its infinitesimal symmetry under the corresponding vector fields. This technique will be explored in depth for systems of algebraic and differential equations in Chapter 2. The notion of a vector field then leads to the concept of a Lie algebra, which can be thought of as the infinitesimal generator of the Lie group itself, the theory of which is developed in Section 1.4. The final section of this chapter gives a brief introduction to differential forms and integration on manifolds. 1.1. Manifolds Throughout most of this book, we will be primarily interested in objects, such as differential equations, symmetry groups, and so on, which are defined on open subsets of Euclidean space \( \mathbb{R}^n \). The underlying geometrical features of these objects will be independent of any particular coordinate system on the open subset that might be used to define them, and it becomes of great importance to free ourselves from the dependence on particular local coordinates, so that our objects will be essentially "coordinate-free". More specifically, if \( U \subset \mathbb{R}^n \) is open and \( \phi: U \to V \), where \( V \subset \mathbb{R}^n \) is open, is any diffeomorphism, meaning that \( \phi \) is an infinitely differentiable map with a finitely differentiable inverse, then objects defined on \( U \) will have equivalent counterparts on \( V \). Although the precise formulas for the object on \( U \) and its counterpart on \( V \) will, in general, change, the essential underlying properties will remain the same. Once we have freed ourselves from this dependence on coordinates, it is a small step to the general definition of a smooth manifold. From this point of view, manifolds provide the natural setting for studying objects that do not depend on coordinates.