Marsden, Jerrold E. (1974) Applications of global analysis in mathematical physics. Mathematics Lecture Series. Vol.2. , Inc. , Berkeley, CA. ISBN 091409811X. https://resolver.caltech.edu/CaltechBOOK:1974.001

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Abstract
These notes are based on a series of ten lectures given at Carleton University, Ottawa, from June 21 through July 6, 1973. The notes follow the lectures fairly closely except for a few minor amplifications. The purpose of the lectures was to introduce some methods of global analysis which I have found useful in various problems of mathematical physics. Many of the results are based on work done with P. Chernoff, D. Ebin, A. Fischer and A. Weinstein. A more complete exposition of some of the points contained here may be found in ChernoffMarsden [1] and MarsdenEbinFischer [1] as well as in references cited later. "Global Analysis" is a vague term. It has, by and large, two more or less distinct subdivisions. On the one hand there are those who deal with dynamical systems emphasizing topological problems such as structural stability (see Smale [2]). On the other hand there are those who deal with problems of nonlinear functional analysis and partial differ equations using techniques combining geometry and analysis. It is to the second group that we belong. One of the first big successes of global analysis (in the second sense above) was Morse theory as developed by Palais [7] and Smale [3] and preceeded by the ideas of LeraySchauder, LusternikSchnirelman and Morse. The result is a beautiful geometrization and powerful extension of the classical calculus of variations. (See Graff [1] for more uptodate work.) It is in a similar spirit that we proceed here. Namely we want to make use of ideas from geometry to shed light on problems in analysis which arise in mathematical physics. Actually it comes as a pleasant surprise that this point of view is useful, rather than being a mere language convenience and an outlet for generalizations. As we hope to demonstrate in the lectures, methods of global analysis can be useful in attacking specific problems. The first three lectures contain background material. This is basic and more or less standard. Each of the next seven lectures discusses an application with only minor dependencies, except that lectures 4 and 5, and 9 and 10 form units. Lectures 4 and 5 deal with hydrodynamics and 9 and 10 with general relativity. Lecture 6 deals with miscellaneous applications, both mathematical and physical, of the concepts of symmetry groups and conserved quantities. Lecture 7 studies quantum mechanics as a hamiltonian system and discusses, e.g. the BargmannWigner theorem. Finally lecture 8 studies a general method for obtaining global (in time) solutions to certain evolution equations. It is a pleasure to thank Professors V. Dlab, D. Dawson and M. Grmela for their kind hospitality at Carleton.
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Subject Keywords:  Mathematics 
Series Name:  Mathematics Lecture Series 
Record Number:  CaltechBOOK:1974.001 
Persistent URL:  https://resolver.caltech.edu/CaltechBOOK:1974.001 
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ID Code:  25041 
Collection:  CaltechBOOK 
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Deposited On:  02 Oct 2007 
Last Modified:  03 Oct 2019 03:02 
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