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Science/Engineering Statistical Mechanics, Third Edition
Science/Engineering Essentials of Toxic Chemical Risk: Science and Society
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Science/Engineering Testtheorie und Fragebogenkonstruktion (Springer-Lehrbuch)
Science/Engineering Centrifugal Pumps, 2nd Edition
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Science/Engineering Networks, Crowds, and Markets: Reasoning About a Highly Connected World {repost}
Science/Engineering Introduction to Biophotonics (repost)
Science/Engineering The Art and Science of Psychotherapy (repost)
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Science/Engineering "Emulsion Science: Basic Principles" (repost)
Science/Engineering Elementary Principles of Chemical Processes 3rd edition
Science/Engineering Boundary Element Analysis (repost)
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Science/Engineering Geometric Structures in Nonlinear Physics (Hermann, Robert//Interdisciplinary Mathematics): Robert Hermann
Posted on 2010-03-16
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More Geometric Structures in Nonlinear Physics (Hermann, Robert//Interdisciplinary Mathematics): Robert Hermann As a result of a Kuhnian revolution, geometry has transformed our ways of thinking about physics. In this book, I return to the foundational work I began in the 1960's in the books published by Benjamin, Academic Press, and Dekker. In these my emphasis was on the linear problems involving vector bundles, linear differential operators, etc. In contrast, this volume emphasizes the type of nonlinear physics exemplified by fluid mechanics, vortex dynamics, interacting quantum field theory and Einsteinian gravitation. As in the previous book in this series, "Geometric Computing Science: First Steps", Interdisciplinary Mathematics Vol. 2, much of this material is preliminary in nature. My own mathematical roots are grounded in the theory of geometric stroctures, as developed in Paris and Princeton in the 1950's. It is also striking how elementary particle physics tied in with these ideas in the period 1965 -1980, emphasizing Lie groups and fiber bundles. As a result of this revolution in the ways of thinking about fundamental physics, it is now conventional wisdom that a Lie group is at the foundation. For example, when elementary particle physicists talk about the quark model, they are positing such a choice as a finite dimensional compact Lie group. String theory is based on another type of structure, one of tbe infinite Lie pseudogroups studied by Lie, Cartan and Spencer. This volume suggests that a more complicated geometric structure, a deformation structure, is involved. Using deformation theory, I connect such topics as quantum field renormalization theory, Heisenberg-picture quantum field theory and the Colombeau-Rosinger theory of multiplication of Distributions. (In fact, the latter subject is related to nonstandard analysis, hence ultimately logic is at the bottom of it all). From approximately 1964 to 1974, I worked on particle physics. My 1966 book "Lie Groups for Physicisis" couulbuted to the particle physicisis' evolution from the viewpoint of 19th century mathematics to that of Lie groups and fiber bundles. In this volume, my aim is to further catalyze the development of new mathematical tools and ideas which can aid in the futher development of physics along nonlinear lines. Physicists now use the term 'nonlinear physics' as a shorthand for mathematically going beyond the theory of linear differential equations. It seems to me that the heart of nonlinear physics consists of the problems of multiplication of distributions and tbe singularity structure of generalized solutions of nonlinear partial differemial equations. Contents: PREFACE 1 quantum field theory, Riemannian metric, Pseudogroups The variational problem associated with LieSpencer 11 pseudogroupes, Lie Algebras, Quantum Field Theory A DEFORMATIONTHEORETIC STRUCTURE BASED ON 25 quantum field theory, Poisson Bracket, Dirac Delta Functions Computation of the simplest 1D field equations 36 Poisson Bracket, Canonical Commutation Relations, dual space The Euler field equations 46 vector bundle, quotient set, equivalence relation The Euler equations and variational principles for 53 pseudogroups, vector fields, Lie algebra SOME AIMS IN RESEARCH IN THE GEOMETRY OF FLUIDS 57 LIE ALGEBRA DEFORMATIONS AND QUATUM FIELD THEORY 59 Deformation Theory, Lie Algebras, Category Theory Some reserach areas 66 Lagrangian field theories, associative algebra, QUANTUM FIELD THEORY Bibliography 74 Pseudogroup, Cartan connections, Current algebra FRACTALS AND HEGEL 81 Pfaffian system, fiber bundle, direct sum RIEMANNIAN METRICS AFFINE CONNECTIONS 93 Moving Frame, tensor field, symmetric matrix Affine connections 100 affine connection, covariant derivative, moving frame THE VORTICITY GRADIENT CONVECTION AND DIVERGENCE 109 Lie Derivative, PSEUDOGROUP, linear differential operator The Spencer operators for the Lie algebra of 115 Lie derivative, local diffeomorphisms, Lie subalgebra The dual Spencer operators for the Lie algebra 121 differential operator, Vorticity, convection operator The Conservation of Vorticity in the Euler Perfect 127 Conservation Laws, RIEMANNIAN GEOMETRY, VARIATIONAL PRINCIPLE Integrability conditions for infinitesimal variations 133 Affine Connection, Variational Principle, Diffeomorphisms Variation of the Action generated by a differential 142 Affine Connection, adjoint representation, algebra structure DIFFERENTIAL FORM METHODS IN THE THEORY OF 163 calculus of variations, string theories, Grassmann algebra Extremals of constrained variational problems 41 variational problem, solder forms, orthonormal frame bundle Minimal submanifolds of Riemannian manifolds 56 Hodge dual, Jacobi identity, inner product The Cartan form in YangMills theory 72 Lagrangian Field Theories, Differential Forms, Vector Bundles DIFFERENTIAL GEOMETRY OF THE HILBERT VARIATIONAL 207 scalar curvature, curvature forms, metric tensor The vanishing of the Ricci curvature and the first 220 Ricci curvature, metric variation, Palatini Action field in General Relativity 225 diffeomorphisms, inverse matrix, tensor fields The Cartan Form for variational problems determined 238 metric tensor, Einstein equations, variational principle THE TENSOR ALGEBRA ON A MANIFOLD AND ITS 247 tensor algebra, AFFINE CONNECTION, covariant differentiation Total covariant derivatives 255 Tensor Fields, Moving Frame, Tensor Analysis The total covariant derivative operator acting on 265 Riemannian Metrics, Levi-Civita Connection, Transition Relations Tensorvalued differential forms 272 associative algebra, Tensor Algebra, Covariant Derivative CLASSICAL AND QUANTUM MECHANICS AND THEIR 283 Poisson bracket, quantum mechanics, Hilbert space To see my other books, click Download Link (Here).
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