Tensor calculus and analytical dynamics : a classical introduction to holonomic and nonholonomic tensor calculus ; and its principal applications to the Lagrangean dynamics of constrained mechanical systems ; for engineers, physicists, and mathematicians

Tensor Calculus and Analytical Dynamics provides a concise, comprehensive, and readable introduction to classical tensor calculus - in both holonomic and nonholonomic coordinates - as well as to its principal applications to the Lagrangean dynamics of discrete systems under positional or velocity constraints. The thrust of the book focuses on formal structure and basic geometrical/physical ideas underlying most general equations of motion of mechanical systems under linear velocity constraints. Written for the theoretically minded engineer, Tensor Calculus and Analytical Dynamics contains uniquely accessbile treatments of such intricate topics as: - Tensor calculus in nonholonomic variables - Pfaffian nonholonomic constraints - Related integrability theory of Frobenius The book enables readers to move quickly and confidently in any particular geometry-based area of theoretical or applied mechanics in either classical or modern form. Cover Half Title Title Page Copyright Page Dedication Author Preface Acknowledgments Contents Summary of Conventions, Notations, and Basic Formulae Part I: Tensor Calculus Chapter 1 Introduction and Background 1.1 Some History 1.1.1 Aims of Tensor Calculus 1.1.2 Tensors and Geometry 1.1.3 Tensors and Physics 1.1.4 Tensors and Mechanics 1.1.5 Exterior Forms (Cartan Calculus) 1.2 Some Algebra 1.2.1 Indices and Their Order 1.2.2 Index Conventions 1.2.3 Symmetry and Antisymmetry 1.2.4 Special Symbols 1.2.4.1 The Kronecker Delta 1.2.4.2 The Levi-Civita Permutation Symbols 1.2.4.3 The Generalized Kronecker Delta 1.2.4.4 The Generalized Permutation Symbols 1.2.5 Linear Equations, Cramer's Rule 1.2.6 Functional Determinants (Jacobians) 1.2.7 Derivatives of Determinants 1.3 Some Geometry 1.3.1 Primitive Concepts 1.3.1.1 Set 1.3.1.2 Group (G) 1.3.1.3 Space 1.3.1.4 Algebraic Definition 1.3.1.5 Examples 1.3.2 Coordinate System(s) (CS) 1.3.3 Manifold 1.3.3.1 Examples 1.3.4 Coordinate Transformation(s) (CT) 1.3.4.1 Systems of Coordinates vs. Frames of Reference 1.3.5 Successive CT, Group Property 1.3.6 Admissible CT 1.3.7 Invariance 1.3.7.1 Entity, or Object, Invariance 1.3.7.2 Form Invariance 1.3.8 Manifold Orientation 1.3.9 Tensor Calculus (TC) 1.3.10 Subspaces in a Manifold; Curves, Surfaces, etc 1.3.10.1 Special Cases Chapter 2 Tensor Algebra 2.1 Introduction: Affine and Euclidean, or Metric, Vector Spaccs 2.1.1 Vectors 2.1.1.1 0. Sum 2.1.1.2 0'. Product with a Scalar 2.1.1.3 0" Scalar, or Dot, Product 2.1.2 Affine and Euclidean Point Spaces 2.1.3 Linear Independence, Dimension and Basis 2.1.3.1 Local Basis, or Frame 2.1.3.2 Vector Subspace of an Xn 2.1.3.3 Complementary Vector Subspaces 2.2 Vector Algebra in a Euclidean Vector Space 2.2.1 Dot Product 2.2.2 Covariant and Contravariant Components of a Vector 2.2.3 Dual of (or to) a Basis 2.2.4 Change of Basis 2.3 Introduction to Coordinate Transformations - Affine/Rectilinear Coordinates 2.4 General Curvilinear (Nonlinear) Coordinate Transformations (CT) 2.4.1 Curvilinear Coordinates and Their Differentials 2.4.2 Integrability, Holonomic Coordinates 2.5 Tensor Definitions 2.5.1 Tensors of Rank One (Vectors) 2.5.2 Tensors of Rank Two 2.5.3 General Tensors 2.5.4 Relative Tensors (RT) 2.5.4.1 Some Motivation for RTs 2.5.5 Geometrical Objects (GO) 2.5.6 Rectangular Cartesian Coordinates (RCC) 2.5.7 Polar vs. Axial Vectors 2.5.8 Tensors vs. Matrices 2.5.9 Numerical Tensors 2.6 Properties of Tensor Transformation 2.7 Algebraic Operations with Tensors 2.7.1 Symmetry Properties 2.7.2 Outer (or Exterior, or Direct) Multiplication of Tensors 2.7.3 Contraction of Tensors 2.7.4 Inner Multiplication or Transvection of Tensors 2.7.5 Quotient Rule (QR) 2.7.5.1 QR for Relative, and General Tensors 2.7.6 Conjugate (or Associated) Tensors 2.8 The Metric Tensor 2.8.1 The Fundamental (Covariant) Metric Tensor 2.8.2 Definitions 2.8.3 Conjugate (Contravariant) Metric Tensor 2.8.4 Left-/Right-Handedness (Orientation) of a CS 2.8.5 Raising and Lowering of lndices 2.8.5.1 Mixed Metric Tensor 2.9 Tensorial Form of Ordinary Vector Algebra 2.9.1 Bases in Curvilinear Coordinates 2.9.1.1 Special Cases 2.9.2 Transformation of Basis Vectors 2.9.2.1 Length of Elementary Displacement Vector 2.9.2.2 Elementary Area 2.9.2.3 Elementary Volume 2.9.3 Permutation Tensors 2.9.4 Components of Vectors in Various Bases 2.9.4.1 Applications 2.10 Physical Components of Vectors and Tensors 2.11 On Direct, or Dyadic (Polyadic etc.), or Invariant, Representations of Tensors 2.11.1 General Tensors 2.11.2 Dyadics 2.12 Introduction to Riemannian Spaces 2.12.1 Riemannian Space, Rn 2.12.2 Tangent (Point and Vector) Spaces 2.12.3 Flatness and Curvature 2.12.4 Linearly, or Affinely, Connected Manifolds, Integrability 2.12.5 Differences Between Affine (Nonmetric) and Metric Spaces Chapter 3 Tensor Algebra 3.1 Introduction 3.2 Differentiation of Tensor Components 3.3 The Christoffel Symbols 3.3.1 Definitions 3.3.2 Properties 3.3.3 Successive Coordinate Transfonnations 3.3.4 Antisymmetric Part of the Christoffels 3.4 The Covariant Derivative (CD) 3.4.1 Definitions, Theorems 3.5 The Absolute, or Intrinsic, Derivative (AD) 3.5.1 Definitions 3.5.2 Some Properties/Theorems of CDs and ADs 3.6 Some Vector Analysis in Tensor Notation 3.7 Parallelism, Straight Lines 3.7.1 On Geodesics 3.7.2 (First) Proof of the Christoffel Transformation Equations (Equations 3.3.2e ff) 3.8 Geometrical Interpretation of Christoffels; Affine Manifolds; Torsion 3.8.1 Euclidean Manifolds 3.8.1.1 Relation of Equations 3.8.5a and b with the Earlier Christoffel Definitions (Equations 3.3.1a and b) 3.8.1.2 Special Case: Moving Orthonormal Basis 3.8.1.3 Geometrical Interpretation of CDs and ADs 3.8.1.4 Geometrical Meaning of dvt, d*vt, and Dvt 3.8.2 General Linearly, or Affinely, Connected and Metric-Equipped Manifolds 3.8.2.1 Fundamental Theorem of Riemannian Geometry 3.8.3 Asymmetric Affinities, Torsion 3.9 Geometrical Significance of Torsion of a Manifold 3.10 Curvature of a Manifold: Geometrical Aspects 3.10.1 Additional Derivations of the Riemann-Christoffel Tensor (R-C) and Path Dependence 3.10.1.1 Exactness (or Perfect Differential) Conditions 3.10.1.2 Via the Generalized Stokes' Theorem 3.11 Curvature of a Manifold: Algebraic Aspects 3.11.1 Symmetries-Antisymmetries of R-C 3.11.1.1 Number of Independent Components of R-C 3.11.2 Contraction(s) of R-C 3.11.3 Riemannian, or Sectional, Curvature 3.11.4 Curvature vs. Flatness (Riemannian vs. Euclidean Spaces) 3.11.5 Closing Remarks 3.12 Nonholonomic (NH) Tensor Algebra 3.12.1 Introduction 3.12.2 Nonholonomic Coordinates and Bases 3.12.3 On Notation 3.12.4 NH Metric Tensor 3.12.5 NH Vectors and Tensors 3.12.6 Mixed H-NH Transformations 3.13 NH Tensor Differentiation; Object of Anholonomicity 3.13.1 NH Differentiation 3.13.1.1 Definition 3.13.2 The Nonholonomicity (or Anholonomicity) Object 3.13.2.1 Remarks on AO 3.13.2.2 Additional Uses of the AO 3.13.2.3 Other Expressions for AO 3.13.2.4 The AO in Three-Dimensional Torsionless Space 3.14 NH Tensor Analysis: The Transitivity Equations 3.14.1 Basic Results 3.14.2 Transformation of the .Terms 3.14.3 Geometrical Interpretation of the Transitivity Equations 3.15 NH Tensor Analysis: NH Affinities and Christoffels 3.15.1 NH Basis Gradients and Affinities 3.15.2 Properties of the NH Affinities 3.15.3 NH Affinities in a Riemannian Space (i.e., Ln.Rn) 3.15.4 Transformation of the NH Affinities 3.15.5 The First-Kind NH Christoffel-Like Symbols and Their Properties 3.16 NH Tensor Analysis: NH Covariant Derivative 3.16.1 Nonholonomic Riemann-Christoffel Tensor Part II: Analytical Dynamics Chapter 4 Introduction to Analytical Dynamics 4.1 Fundamental Concepts 4.2 Configuration Space 4.2.1 Kinematics 4.2.2 Kinetics 4.3 Introduction to Constraints - Purpose of Analytical Mechanics (AM) 4.3.1 Whence the Need for AM Chapter 5 Particle on a Curve and on a Surface 5.1 Introduction 5.2 Particle in Ordinary Space: General Coordinates 5.3 Particle in Ordinary Space: Natural, or Intrinsic, Variables 5.4 Particle on a Curve 5.5 Particle on a Surface 5.5.1 Introduction to Surfaces, Velocity 5.5.2 Tensor Analysis on a Surface 5.5.3 Curve on a Surface 5.5.4 Acceleration 5.5.5 Forces, Equations of Motion 5.6 General n-Dimensional (Riemannian) Surfaces 5.6.1 Riemannian Space (Rn) Inside a Euclidean Space (EN: n< N < oo.) 5.6.2 Differences Between Euclidean and Riemannian Arc-Lengths 5.6.3 Problem of Embedding, or lmmersing 5.6.4 Problem of Equivalence, or Integrability 5.7 Perturbation of Trajectories in Configuration Space, and Their Stability 5.7.1 The Perturbation Equation 5.7.2 The Energy Integral 5.7.3 Alternative Forms 5.7.4 Normal (Nonisochronous) Perturbations Chapter 6 Lagrangean Mechanics: Kinematics 6.1 Introduction 6.2 Holonomic Constraints 6.2.1 Basic Definitions, System (or Generalized) Coordinates 6.2.2 Scleronomic vs. Rheonomic Constraints 6.2.3 Additional Holonomic Constraints 6.2.4 Geometrical Interpretation of Holonomic Constraints 6.2.4.1 Configuration Space 6.2.4.2 Extended Configuration Space 6.3 Velocity, Admissible and Virtual Displacements, and Acceleration in Particle and Holonomic System Variables 6.4 Nonholonomic Coordinates, Velocities, etc 6.4.1 Basic Definitions, Quasi-Coordinates 6.4.2 Properties of Pfaffian Transforrnations 6.4.3 Particle and System Kinematics in Quasi-Variables 6.5 The Transitivity Equations 6.5.1 Nonintegrability Conditions 6.5.2 Comprehensive Examples and Problems on Rigid-Body Kinematics 6.6 Additional Pfaffian Constraints 6.6.1 Holonomicity vs. Nonholonomicity 6.6.2 Scleronomicity vs. Rheonomicity 6.6.3 Catastaticity vs. Acatastaticity 6.6.4 Nonholonomic Constraints (Equations 6.6.1) 6.7 Theorem of Frobenius 6.8 Geometrical Interpretation of Pfaffian Constraints 6.8.1 Degrees of Freedom Revisited, Accessibility 6.9 Geometrical Interpretation of the Frobenius' Conditions (Equation 6.7.5e) 6.9.1 First lnterpretation 6.9.2 Second Interpretation Chapter 7 Lagrangean Mechanics: Kinetics 7.1 Introduction 7.2 The Fundamental Kinematico-Inertial Quantities 7.2.1 Kinetic Energy 7.2.1.1 Holonomic Variables 7.2.1.2 Nonholonomic Variables 7.2.2 Metric in Configuration/Event Space 7.2.3 Acceleration 7.2.4 Inertia Force: Holonomic Components and Holonomic Euler-Lagrange Operator 7.2.5 Inertia Force: Nonholonomic Components and Nonholonomic Euler-Lagrange Operator 7.2.5.1 First Derivation 7.2.5.2 Second Derivation 7.3 The Forces 7.4 The Physical Synthesis: Lagrange's Principle(s), Equations of Motion 7.4.1 Lagrange's Principle (LP) 7.4.2 Principle of Relaxation of Constraints (PRC) 7.4.3 Lagrangean Forms of the Equations of Motion 7.4.3.1 Holonomic Variables 7.4.3.2 Nonholonomic Variables 7.4.4 Appellian Forms of the Equations of Motion 7.4.5 Summary 7.5 The Central Equation 7.6 The Power, or Energy Rate, Equations 7.6.1 Holonomic Variables 7.6.2 Nonholonomic Variables 7.7 Comprehensive Examples and Problems on Lagrangean Dynamics Bibliography and References Index (Principal Authors and Subjects)

Tensor Calculus and Analytical Dynamics provides a concise, comprehensive, and readable introduction to classical tensor calculus - in both holonomic and nonholonomic coordinates - as well as to its principal applications to the Lagrangean dynamics of discrete systems under positional or velocity constraints. The thrust of the book focuses on formal structure and basic geometrical/physical ideas underlying most general equations of motion of mechanical systems under linear velocity constraints.
Written for the theoretically minded engineer, Tensor Calculus and Analytical Dynamics contains uniquely accessbile treatments of such intricate topics as:

tensor calculus in nonholonomic variables

Pfaffian nonholonomic constraints

related integrability theory of Frobenius The book enables readers to move quickly and confidently in any particular geometry-based area of theoretical or applied mechanics in either classical or modern form.

Booknews

Introduces classical tensor calculus on both holonomic and nonholonomic coordinates, along with its principal applications to the Lagrangean analytical dynamics. Focuses on the basic geometrical/physical ideas and formal structure of the advanced dynamics of mechanical systems under general position and/or velocity (Pfaffian) constraints, via the kinetic principle of virtual work. Oriented toward the theoretically minded engineer, physicist, or mathematician; requires knowledge of conventional infinitesimal calculus and intermediate-level theoretical mechanics. Annotation c. by Book News, Inc., Portland, Or.