Geometric Mechanics Toward a Unification of Classical Physics

by Talman, Richard

• ISBN13: 9783527406838
• ISBN10: 3527406832
• Edition: 2nd
• Format: Hardcover
• Copyright: 2007-07-10
• Publisher: Wiley-VCH

Summary

For physicists, mechanics is quite obviously geometric, yet the classical approach typically emphasizes abstract, mathematical formalism. Setting out to make mechanics both accessible and interesting for non-mathematicians, Richard Talman uses geometric methods to reveal qualitative aspects of the theory. He introduces concepts from differential geometry, differential forms, and tensor analysis, then applies them to areas of classical mechanics as well as other areas of physics, including optics, crystal diffraction, electromagnetism, relativity, and quantum mechanics. For easy reference, the author treats Lagrangian, Hamiltonian, and Newtonian mechanics separately -- exploring their geometric structure through vector fields, symplectic geometry, and gauge invariance respectively. Practical perturbative methods of approximation are also developed. This second, fully revised edition has been expanded to include new chapters on electromagnetic theory, general relativity, and string theory. 'Geometric Mechanics' features illustrative examples and assumes only basic knowledge of Lagrangian mechanics.

Author Biography

Richard M. Talman is Professor of Physics at Cornell University, Ithaca, New York. He studied physics at the University of Western Ontario and received his Ph.D. at the California Institute of Technology in 1963. After accepting a full professorship for Physics at Cornell in 1971, he spent time as Visiting Scientist in Stanford, CERN, Berkeley, and the S.S.C. in Dallas and Saskatchewan. In addition, he has delivered lecture series at several institutions including Rice and Yale Universities. Professor Talman has been engaged in the design and construction of a series of accelerators, with special emphasis on x-rays.

Table of Contents

Preface
Introduction
Bibliography
Review of Classical Mechanics and String Field Theory
Preview and Rationale
Review of Lagrangians and Hamiltonians
Derivation of the Lagrange Equation from Hamilton's Principle
Linear, Multiparticle Systems
Effective Potential and the Kepler Problem
Multiparticle Systems
Longitudinal Oscillation of a Beaded String
Field Theoretical Treatment and Lagrangian Density
Hamiltonian Density for Transverse String Motion
String Motion Expressed as Propagating and ReflectingWaves
Problems
Bibliography
Geometry of Mechanics, I, Linear
Pairs of Planes as Covariant Vectors
Differential Forms
Algebraic Tensors
(Possibly Complex) Cartesian Vectors in Metric Geometry
Bibliography
Geometry of Mechanics, II, Curvilinear
(Real) Curvilinear Coordinates in n-Dimensions
Derivation of the Lagrange Equations from the Absolute Differential
Intrinsic Derivatives and the Bilinear Covariant
The Lie Derivative - Coordinate Approach
The Lie Derivative - Lie Algebraic Approach
Identification of Vector Fields with Differential Operators
Coordinate Congruences
Lie-Dragged Congruences and the Lie Derivative
Commutators of Quasi-Basis-Vectors
Bibliography
Geometry of Mechanics, III, Multilinear
Generalized Euclidean Rotations and Reflections
Multivectors
Curvilinear Coordinates in Euclidean Geometry (Continued)
Spinors in Three-Dimensional Space
Lagrange-Poincare Description of Mechanics
The Poincare Equation
Variational Derivation of the Poincare Equation
Restricting the Poincare Equation With Group Theory
Bibliography
Newtonian/Gauge Invariant Mechanics
Vector Mechanics
Single Particle Equations in Gauge Invariant Form
Gauge Invariant Description of Rigid Body Motion
The Foucault Pendulum
Tumblers and Divers
Bibliography
Hamiltonian Treatment of Geometric Optics
Analogy Between Mechanics and Geometric Optics
Variational Principles
Paraxial Optics, Gaussian Optics, Matrix Optics
Huygens' Principle
Bibliography
Hamilton-Jacobi Theory
Hamilton-Jacobi Theory Derived from Hamilton's Principle
Trajectory Determination Using the Hamilton-Jacobi Equation
The Kepler Problem
Analogies Between Optics and Quantum Mechanics
Bibliography
Relativistic Mechanics
Relativistic Kinematics
Relativistic Mechanics
Introduction of Electromagnetic Forces into Relativistic Mechanics
Bibliography
Conservation Laws and Symmetry
Conservation of Linear Momentum
Rate of Change of Angular Momentum: Poincare Approach
Conservation of Angular Momentum: Lagrangian Approach
Conservation of Energy
Cyclic Coordinates and Routhian Reduction
Noether's Theorem
Conservation Laws in Field Theory
Transition From Discrete to Continuous Representation
Angular Momentum of a System of Particles
Angular Momentum of a Field
Bibliography
The Electromagnetic Field Tensor
The Electromagnetic Field Equations
Bibliography
Relativistic Strings
Introduction
Area Representation in Terms of the Metric	p. 1
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