VI PREFACE Hamilton published his fundamental paper on geometrical optics, introducing his "characteristics", as a key tool in the study of optical instruments. It wasn't until substantially later that Hamilton realized that his method applied equally well to the study of mechanics. Hamilton's method was developed by Jacobi and has been a cornerstone of theoretical mechanics ever since. In Chapter IV we discuss the modern version of the Hamilton-Jacobi theory—the geometry of symplectic manifolds. It is interesting to note that although Hamilton was aware of the work of Fresnel, he chose to ignore it completely in his fundamental papers on geometrical optics. Nevertheless, he made a theoretical prediction in the wave theory—the possibility of "conical refraction"—which was experimentally veri- fied soon thereafter. Unfortunately, we shall have nothing to say in this book on the issue of conical refraction (the problem of multiple characteristics in hyperbolic differential equations) but hope that some of the methods that we develop will prove useful in this connection. In 1833, Airy published a paper dealing with the behavior of light near a caustic. (A caustic is a set where geometrical optics predicts an infinite intensity of light. An object placed at a caustic will become quite hot, and hence the name.) In this paper, he introduced the functions now known as Airy functions. In Chapter VII, we show how the theory of singularities of mappings can be applied to understand and extend Airy's results. In 1887, Kelvin introduced the method of stationary phase—a technique for the asymptotic evaluation of certain types of definite integrals—in order to explain the V shaped wake that trails behind a ship as it moves across the water. The method of stationary phase, in a form expounded by Hormander, will be our principal analytical tool. We now turn to a chapter by chapter description of the contents of the book. Chapter I. Introduction. The Method of Stationary Phase The solution of the reduced wave equation A/x 4- k2[i = 0 with initial data prescribed on a hypersurface S C R3 is given by an integral of the form / Jk\x-y\ In order to see how (*) behaves in the range of large frequencies we first consider more general integrals of the form / eik*{z)a(z)dz. The method of stationary phase, for evaluating such integrals for k large, is discussed in detail following Hormander's idea of making use of Morse's lemma. It is then applied to (*) to explain the phenomenon of shift in phase, when light rays pass through caustics. In the Appendix to this Chapter, we present a proof
Previous Page Next Page