Geometric Mechanics : Toward a Unification of Classical Physics


Richard. Talman
Bok Engelsk 2007 · Electronic books.
Utgitt
Hoboken : : Wiley, , 2007.
Omfang
1 online resource (607 p.)
Utgave
2nd ed.
Opplysninger
Description based upon print version of record.. - Geometric Mechanics; Contents; Preface; Introduction; 1 Review of Classical Mechanics and String Field Theory; 1.1 Preview and Rationale; 1.2 Review of Lagrangians and Hamiltonians; 1.2.1 Hamilton's Equations in Multiple Dimensions; 1.3 Derivation of the Lagrange Equation from Hamilton's Principle; 1.4 Linear, Multiparticle Systems; 1.4.1 The Laplace Transform Method; 1.4.2 Damped and Driven Simple Harmonic Motion; 1.4.3 Conservation of Momentum and Energy; 1.5 Effective Potential and the Kepler Problem; 1.6 Multiparticle Systems; 1.7 Longitudinal Oscillation of a Beaded String. - 1.7.1 Monofrequency Excitation1.7.2 The Continuum Limit; 1.8 Field Theoretical Treatment and Lagrangian Density; 1.9 Hamiltonian Density for Transverse String Motion; 1.10 String Motion Expressed as Propagating and Reflecting Waves; 1.11 Problems; Bibliography; 2 Geometry of Mechanics, I, Linear; 2.1 Pairs of Planes as Covariant Vectors; 2.2 Differential Forms; 2.2.1 Geometric Interpretation; 2.2.2 Calculus of Differential Forms; 2.2.3 Familiar Physics Equations Expressed Using Differential Forms; 2.3 Algebraic Tensors; 2.3.1 Vectors and Their Duals; 2.3.2 Transformation of Coordinates. - 2.3.3 Transformation of Distributions2.3.4 Multi-index Tensors and their Contraction; 2.3.5 Representation of a Vector as a Differential Operator; 2.4 (Possibly Complex) Cartesian Vectors in Metric Geometry; 2.4.1 Euclidean Vectors; 2.4.2 Skew Coordinate Frames; 2.4.3 Reduction of a Quadratic Form to a Sum or Difference of Squares; 2.4.4 Introduction of Covariant Components; 2.4.5 The Reciprocal Basis; Bibliography; 3 Geometry of Mechanics, II, Curvilinear; 3.1 (Real) Curvilinear Coordinates in n-Dimensions; 3.1.1 The Metric Tensor. - 3.1.2 Relating Coordinate Systems at Different Points in Space3.1.3 The Covariant (or Absolute) Differential; 3.2 Derivation of the Lagrange Equations from the Absolute Differential; 3.2.1 Practical Evaluation of the Christoffel Symbols; 3.3 Intrinsic Derivatives and the Bilinear Covariant; 3.4 The Lie Derivative - Coordinate Approach; 3.4.1 Lie-Dragged Coordinate Systems; 3.4.2 Lie Derivatives of Scalars and Vectors; 3.5 The Lie Derivative - Lie Algebraic Approach; 3.5.1 Exponential Representation of Parameterized Curves; 3.6 Identification of Vector Fields with Differential Operators. - 3.6.1 Loop Defect3.7 Coordinate Congruences; 3.8 Lie-Dragged Congruences and the Lie Derivative; 3.9 Commutators of Quasi-Basis-Vectors; Bibliography; 4 Geometry of Mechanics, III, Multilinear; 4.1 Generalized Euclidean Rotations and Reflections; 4.1.1 Reflections; 4.1.2 Expressing a Rotation as a Product of Reflections; 4.1.3 The Lie Group of Rotations; 4.2 Multivectors; 4.2.1 Volume Determined by 3- and by n-Vectors; 4.2.2 Bivectors; 4.2.3 Multivectors and Generalization to Higher Dimensionality; 4.2.4 Local Radius of Curvature of a Particle Orbit; 4.2.5 "Supplementary" Multivectors. - 4.2.6 Sums of p-Vectors. - 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
Emner
Sjanger
Dewey
516
ISBN
9783527406838

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