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Science/Engineering Geometric Structures in Nonlinear Physics (Hermann, Robert//Interdisciplinary Mathematics): Robert Hermann

Posted on 2010-03-16




Name:Science/Engineering Geometric Structures in Nonlinear Physics (Hermann, Robert//Interdisciplinary Mathematics): Robert Hermann
ASIN/ISBN:0915692422
Language:English
File size:11.8 Mb
ISBN: 0915692422
Publish Date: January 1992
File Type: PDF (OCR)
Pages: 363 pages
File Size: 12362 KB
Other Info: Math Science Press
   Science/Engineering Geometric Structures in Nonlinear Physics (Hermann, Robert//Interdisciplinary Mathematics): Robert Hermann



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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

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