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A Geometric Approach to Thermomechanics of Dissipating Continua

A Geometric Approach to Thermomechanics of Dissipating Continua
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Field name Details
Dewey Class 530.15
Title A Geometric Approach to Thermomechanics of Dissipating Continua (M) / by L. R. Rakotomanana
Author Rakotomanana, Lalaonirina R.
Publication Boston, MA : Birkhäuser , 2018
Physical Details XV, 265 pages : ill. ; 24 cm
Series Progress in Mathematical Physics ; 73
ISBN 9783319917818
Summary Note This book presents a Lagrangian approach model to formulate various fields of continuum physics, ranging from gradient continuum elasticity to relativistic gravito-electromagnetism. It extends the classical theories based on Riemann geometry to Riemann-Cartan geometry, and then describes non-homogeneous continuum and spacetime with torsion in Einstein-Cartan relativistic gravitation. It investigates two aspects of invariance of the Lagrangian: covariance of formulation following the method of Lovelock and Rund, and gauge invariance where the active diffeomorphism invariance is considered by using local Poincaré gauge theory according to the Utiyama method. Further, it develops various extensions of strain gradient continuum elasticity, relativistic gravitation and electromagnetism when the torsion field of the Riemann-Cartan continuum is not equal to zero. Lastly, it derives heterogeneous wave propagation equations within twisted and curved manifolds and proposes a relation between electromagnetic potential and torsion tensor.:
Contents note Intro; Preface; Contents; 1 General Introduction; 1.1 Classical Physics, Lagrangian, and Invariance; 1.2 General Covariance, Gauge Invariance; 1.3 Objectives and Planning; 2 Basic Concepts on Manifolds, Spacetimes, and Calculusof Variations; 2.1 Introduction; 2.2 Space-Time Background; 2.2.1 Basics on Flat Minkowski Spacetime; 2.2.2 Twisted and Curved Spacetimes; 2.3 Manifolds, Tensor Fields, and Connections; 2.3.1 Coordinate System, and Group of Transformations; 2.3.1.1 Manifolds, Tangent Space, Cotangent Space; 2.3.1.2 Change of Coordinate System 2.3.1.3 Examples of Group of Transformations2.3.1.4 Lorentz Invariance; 2.3.2 Elements on Spacetime and Invariance for Relativity; 2.3.2.1 Forms, Tensors and (Pseudo)-Riemannian Manifolds; 2.3.2.2 Hilbert's Causality Principle; 2.3.2.3 Euclidean Spacetime and Isometries; 2.3.2.4 Minkowski Spacetime and Lorentz Transformations; 2.3.2.5 Global Poincaré Transformations; 2.3.3 Volume-Form; 2.3.4 Affine Connection; 2.3.4.1 Affine Connection, Affinely Connected Manifold; 2.3.4.2 Example: Spherical Coordinate System; 2.3.4.3 Example: Elliptic-Hyperbolic Coordinate System 2.3.4.4 Practical Formula for Covariant Derivative2.3.4.5 Torsion and Curvature; 2.3.4.6 Newtonian Spacetime; 2.3.4.7 Levi-Civita Connection; 2.3.4.8 Normal Coordinate System and Inertial Frame; 2.3.5 Tetrads and Affine Connection: Continuum Transformations; 2.3.5.1 Transformation of a Continuum; 2.3.5.2 Holonomic Mapping; 2.3.5.3 Nonholonomic Mapping and Torsion e.g. (2000); 2.3.5.4 Nonholonomic Transformation and Curvature; 2.3.5.5 Torsion, Curvature, and Smoothness of Tensor Fields; 2.4 Invariance for Lagrangian and Euler-Lagrange Equations 2.4.1 Covariant Formulation of Classical Mechanics of a Particle2.4.2 Basic Elements for Calculus of Variations; 2.4.3 Extended Euler-Lagrange Equations; 2.5 Simple Examples in Continuum and Relativistic Mechanics; 2.5.1 Particles in a Minkowski Spacetime; 2.5.2 Some Continua Examples; 2.5.2.1 Energy-Momentum Tensor; 2.5.2.2 Dust in Relativistic Mechanics; 2.5.2.3 Perfect Fluids in Relativistic Mechanics; 2.5.2.4 Strain Gradient Continuum; 3 Covariance of Lagrangian Density Function; 3.1 Introduction; 3.2 Some Basic Theorems; 3.2.1 Theorem of Cartan; 3.2.2 Theorem of lovelockarma 3.2.3 Theorem of Quotient3.3 Invariance with Respect to the Metric; 3.3.1 Transformation Rules for the Metric and Its Derivatives; 3.3.2 Introduction of Dual Variables; 3.3.3 Theorem; 3.4 Invariance with Respect to the Connection; 3.4.1 Preliminary; 3.4.2 Application: Covariance of L; 3.4.3 Summary for Lagrangian Covariance; 3.4.4 Covariance of Nonlinear Elastic Continuum; 3.4.4.1 Covariance of Strain Energy Density; 3.4.4.2 Examples of Nonlinear Elastic Material Models; 4 Gauge Invariance for Gravitation and Gradient Continuum; 4.1 Introduction to Gauge Invariance
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