Diffeomorphism groups, hydrodynamics and relativity

Abstract

The goal of these lectures4 was to present some applications of global analysis to physical problems, specifically to hydrodynamics and general relativity. Parts I and II form a unit. Only a small amount of material from Part I is needed in Part III-an acquaintance with the rudiments of the diffeomorphism groups. The sort of global analysis used in hydrodynamics is developed in Part I. The machinery needed in relativity-infinite dimensional Hamiltonian systems-is developed as it is used. This organization should make it possible for one to read Part III separately if desired. Because of inevitable time restrictions, it was necessary to make some selection with regard to the topics and their depth of discussion. Tnere was an effort made to include topics and points of view that would be of interest to geometers and global analysts. Nevertheless, we hope the overall perspective presented gives a reasonably correct picture of some questions of interest to specialists in theoretical hydrodynamics and general relativity. As far as prerequisites go, we assume the following: A knowledge of the general facts about manifolds, differential forms and Riemannian geometry. For example Lang [1] and Bishop-Crittenden [1] contain more than enough background. No special knowledge of fluid mechanics is needed. In this regard we have included a certain amount of introductory material in § 2 below. One can supplement this by consulting standard texts; see for instance Feynman [1] and Landau-Lifschitz [1]. For general relativity we assume, in addition to the geometry above, some familiarity with the basic ideas of relativity; see for instance Taylor-Wheeler [1]. (This is mainly to motivate several points in the discussion.) Much of the material on hydrodynamics is taken from Ebin-Marsden [1]. However, our exposition here is more informal and gets at several points from a different direction. The exposition regarding turbulence is largely influenced by Ruelle-Takens [1]. The material on relativity is mostly taken from Fischer-Marsden [1].