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Tables of Contents for Tensor Calculus and Analytical Dynamics
Chapter/Section Title
Page #
Page Count
Preface
ix
4
Acknowledgments
xiii
10
Summary of Conventions, Notations, and Basic Formulae
xxiii
Part I: Tensor Calculus
3
166
Chapter 1 Introduction and Background
3
28
1.1 Some History
3
5
1.1.1 Aims of Tensor Calculus
3
1
1.1.2 Tensors and Geometry
4
1
1.1.3 Tensors and Physics
5
1
1.1.4 Tensors and Mechanics
6
1
1.1.5 Exterior Forms (Cartan Calculus)
7
1
1.2 Some Algebra
8
12
1.2.1 Indices and Their Order
8
1
1.2.2 Index Conventions
9
2
1.2.3 Symmetry and Antisymmetry
11
1
1.2.4 Special Symbols
11
1
1.2.4.1 The Kronecker Delta
11
1
1.2.4.2 The Levi-Civita Permutation Symbols
12
1
1.2.4.3 The Generalized Kronecker Delta
13
2
1.2.4.4 The Generalized Permutation Symbols
15
3
1.2.5 Linear Equations, Cramer's Rule
18
1
1.2.6 Functional Determinants (Jacobians)
18
1
1.2.7 Derivatives of Determinants
19
1
1.3 Some Geometry
20
11
1.3.1 Primitive Concepts
20
1
1.3.1.1 Set
20
1
1.3.1.2 Group (G)
20
1
1.3.1.3 Space
21
1
1.3.1.4 Algebraic Definition
21
1
1.3.1.5 Examples
22
1
1.3.2 Coordinate System(s) (CS)
22
1
1.3.3 Manifold
22
1
1.3.3.1 Examples
23
1
1.3.4 Coordinate Transformation(s) (CT)
23
1
1.3.4.1 Systems of Coordinates vs. Frames of Reference
24
1
1.3.5 Successive CT, Group Property
25
1
1.3.6 Admissible CT
25
1
1.3.7 Invariance
26
1
1.3.7.1 Entity, or Object, Invariance
26
1
1.3.7.2 Form Invariance
26
1
1.3.8 Manifold Orientation
27
1
1.3.9 Tensor Calculus (TC)
27
1
1.3.10 Subspaces in a Manifold; Curves, Surfaces, etc
28
1
1.3.10.1 Special Cases
29
2
Chapter 2 Tensor Algebra
31
62
2.1 Introduction: Affine and Euclidean, or Metric, Vector Spaces
31
3
2.1.1 Vectors
31
1
2.1.1.1 0.Sum
31
1
2.1.1.2 O'. Product with a Scalar
31
1
2.1.1.3 O". Scalar, or Dot, Product
32
1
2.1.2 Affine and Euclidean Point Spaces
32
1
2.1.3 Linear Independence, Dimension and Basis
33
1
2.1.3.1 Local Basis, or Frame
33
1
2.1.3.2 Vector Subspace of an X(n)
34
1
2.1.3.3 Complementary Vector Subspaces
34
1
2.2 Vector Algebra in a Euclidean Vector Space
34
3
2.2.1 Dot Product
34
1
2.2.2 Covariant and Contravariant Components of a Vector
35
1
2.2.3 Dual of (or to) a Basis
36
1
2.2.4 Change of Basis
36
1
2.3 Introduction to Coordinate Transformations -- Affine/Rectilinear Coordinates
37
3
2.4 General Curvilinear (Nonlinear) Coordinate Transformations (CT)
40
2
2.4.1 Curvilinear Coordinates and Their Differentials
40
1
2.4.2 Integrability, Holonomic Coordinates
41
1
2.5 Tensor Definitions
42
11
2.5.1 Tensors of Rank One (Vectors)
42
2
2.5.2 Tensors of Rank Two
44
1
2.5.3 General Tensors
45
1
2.5.4 Relative Tensors (RT)
46
1
2.5.4.1 Some Motivation for RTs
47
2
2.5.5 Geometrical Objects (GO)
49
1
2.5.6 Rectangular Cartesian Coordinates (RCC)
49
1
2.5.7 Polar vs. Axial Vectors
50
1
2.5.8 Tensors vs. Matrices
51
1
2.5.9 Numerical Tensors
51
2
2.6 Properties of Tensor Transformation
53
1
2.7 Algebraic Operations with Tensors
54
8
2.7.1 Symmetry Properties
55
5
2.7.2 Outer (or Exterior, or Direct) Multiplication of Tensors
60
1
2.7.3 Contraction of Tensors
60
1
2.7.4 Inner Multiplication or Transvection of Tensors
60
1
2.7.5 Quotient Rule (QR)
61
1
2.7.5.1 QR for Relative, and General Tensors
62
1
2.7.6 Conjugate (or Associated) Tensors
62
1
2.8 The Metric Tensor
62
5
2.8.1 The Fundamental (Covariant) Metric Tensor
62
1
2.8.2 Definitions
63
1
2.8.3 Conjugate (Contravariant) Metric Tensor
64
1
2.8.4 Left-/Right-Handedness (Orientation) of a CS
65
1
2.8.5 Raising and Lowering of Indices
65
1
2.8.5.1 Mixed Metric Tensor
66
1
2.9 Tensorial Form of Ordinary Vector Algebra
67
13
2.9.1 Bases in Curvilinear Coordinates
67
2
2.9.1.1 Special Cases
69
1
2.9.2 Transformation of Basis Vectors
70
1
2.9.2.1 Length of Elementary Displacement Vector
71
1
2.9.2.2 Elementary Area
71
2
2.9.2.3 Elementary Volume
73
1
2.9.3 Permutation Tensors
73
1
2.9.4 Components of Vectors in Various Bases
74
3
2.9.4.1 Applications
77
3
2.10 Physical Components of Vectors and Tensors
80
1
2.11 On Direct, or Dyadic (Polyadic etc.), or Invariant, Representations of Tensors
81
3
2.11.1 General Tensors
81
1
2.11.2 Dyadics
82
2
2.12 Introduction to Riemannian Spaces
84
9
2.12.1 Riemannian Space, R(n)
84
1
2.12.2 Tangent (Point and Vector) Spaces
85
3
2.12.3 Flatness and Curvature
88
1
2.12.4 Linearly, or Affinely, Connected Manifolds, Integrability
89
1
2.12.5 Differences Between Affine (Nonmetric) and Metric Spaces
90
3
Chapter 3 Tensor Analysis
93
76
3.1 Introduction
93
1
3.2 Differentiation of Tensor Components
94
3
3.3 The Christoffel Symbols
97
4
3.3.1 Definitions
97
1
3.3.2 Properties
97
3
3.3.3 Successive Coordinate Transformations
100
1
3.3.4 Antisymmetric Part of the Christoffels
100
1
3.4 The Covariant Derivative (CD)
101
1
3.4.1 Definitions, Theorems
101
1
3.5 The Absolute, or Intrinsic, Derivative (AD)
102
3
3.5.1 Definitions
102
1
3.5.2 Some Properties/Theorems of CDs and ADs
103
2
3.6 Some Vector Analysis in Tensor Notation
105
3
3.7 Parallelism, Straight Lines
108
2
3.7.1 On Geodesics
109
1
3.7.2 (First) Proof of the Christoffel Transformation Equations (Equations 3.3.2e ff)
109
1
3.8 Geometrical Interpretation of Christoffels; Affine Manifolds; Torsion
110
13
3.8.1 Euclidean Manifolds
110
2
3.8.1.1 Relation of Equations 3.8.5a and b with the Earlier Christoffel Definitions (Equations 3.3.la and b)
112
1
3.8.1.2 Special Case: Moving Orthonormal Basis
112
1
3.8.1.3 Geometrical Interpretation of CDs and ADs
113
1
3.8.1.4 Geometrical Meaning of dV(k), d(*)V(k), and DV(k)
114
1
3.8.2 General Linearly, or Affinely, Connected and Metric-Equipped Manifolds
115
4
3.8.2.1 Fundamental Theorem of Riemannian Geometry
119
1
3.8.3 Asymmetric Affinities, Torsion
120
3
3.9 Geometrical Significance of Torsion of a Manifold
123
2
3.10 Curvature of a Manifold: Geometrical Aspects
125
4
3.10.1 Additional Derivations of the Riemann-Christoffel Tensor (R-C) and Path Dependence
127
1
3.10.1.1 Exactness (or Perfect Differential) Conditions
127
2
3.10.1.2 Via the Generalized Stokes' Theorem
129
1
3.11 Curvature of a Manifold: Algebraic Aspects
129
7
3.11.1 Symmetries-Antisymmetries of R-C
131
1
3.11.1.1 Number of Independent Components of R-C
131
2
3.11.2 Contraction(s) of R-C
133
1
3.11.3 Riemannian, or Sectional, Curvature
133
1
3.11.4 Curvature vs. Flatness (Riemannian vs. Euclidean Spaces)
134
1
3.11.5 Closing Remarks
135
1
3.12 Nonholonomic (NH) Tensor Algebra
136
6
3.12.1 Introduction
136
1
3.12.2 Nonholonomic Coordinates and Bases
136
2
3.12.3 On Notation
138
1
3.12.4 NH Metric Tensor
139
1
3.12.5 NH Vectors and Tensors
140
1
3.12.6 Mixed H-NH Transformations
141
1
3.13 NH Tensor Differentiation; Object of Anholonomicity
142
7
3.13.1 NH Differentiation
142
1
3.13.1.1 Definition
142
1
3.13.2 The Nonholonomicity (or Anholonomicity) Object
143
1
3.13.2.1 Remarks on AO
144
1
3.13.2.2 Additional Uses of the AO
145
1
3.13.2.3 Other Expressions for AO
145
1
3.13.2.4 The AO in Three-Dimensional Torsionless Space
146
3
3.14 NH Tensor Analysis: The Transitivity Equations
149
4
3.14.1 Basic Results
149
1
3.14.2 Transformation of the Gamma Terms
150
2
3.14.3 Geometrical Interpretation of the Transitivity Equations
152
1
3.15 NH Tensor Analysis: NH Affinities and Christoffels
153
7
3.15.1 NH Basis Gradients and Affinities
153
1
3.15.2 Properties of the NH Affinities
154
1
3.15.3 NH Affinities in a Riemannian Space (i.e., L(n) --> R(n))
154
2
3.15.4 Transformation of the NH Affinities
156
2
3.15.5 The First-Kind NH Christoffel-Like Symbols and Their Properties
158
2
3.16 NH Tensor Analysis: NH Covariant Derivative
160
9
3.16.1 Nonholonomic Riemann-Christoffel Tensor
162
7
Part II: Analytical Dynamics
169
202
Chapter 4 Introduction to Analytical Dynamics
169
12
4.1 Fundamental Concepts
169
4
4.2 Configuration Space
173
3
4.2.1 Kinematics
174
1
4.2.2 Kinetics
175
1
4.3 Introduction to Constraints -- Purpose of Analytical Mechanics (AM)
176
5
4.3.1 Whence the Need for AM
178
3
Chapter 5 Particle on a Curve and on a Surface
181
52
5.1 Introduction
181
1
5.2 Particle in Ordinary Space: General Coordinates
181
4
5.3 Particle in Ordinary Space: Natural, or Intrinsic, Variables
185
9
5.4 Particle on a Curve
194
3
5.5 Particle on a Surface
197
14
5.5.1 Introduction to Surfaces, Velocity
197
1
5.5.2 Tensor Analysis on a Surface
198
4
5.5.3 Curve on a Surface
202
4
5.5.4 Acceleration
206
2
5.5.5 Forces, Equations of Motion
208
3
5.6 General n-Dimensional (Riemannian) Surfaces
211
7
5.6.1 Riemannian Space (R(n)) Inside a Euclidean Space (E(N): n is less than N is less than XXX)
211
4
5.6.2 Differences Between Euclidean and Riemannian Arc-Lengths
215
1
5.6.3 Problem of Embedding, or Immersing
215
1
5.6.4 Problem of Equivalence, or Integrability
216
2
5.7 Perturbation of Trajectories in Configuration Space, and Their Stability
218
15
5.7.1 The Perturbation Equation
219
2
5.7.2 The Energy Integral
221
1
5.7.3 Alternative Forms
222
1
5.7.4 Normal (Nonisochronous) Perturbations
223
10
Chapter 6 Lagrangean Mechanics : Kinematics
233
52
6.1 Introduction
233
1
6.2 Holonomic Constraints
233
5
6.2.1 Basic Definitions, System (or Generalized) Coordinates
233
3
6.2.2 Scleronomic vs. Rheonomic Constraints
236
1
6.2.3 Additional Holonomic Constraints
236
1
6.2.4 Geometrical Interpretation of Holonomic Constraints
237
1
6.2.4.1 Configuration Space
237
1
6.2.4.2 Extended Configuration Space
237
1
6.3 Velocity, Admissible and Virtual Displacements, and Acceleration in Particle and Holonomic System Variables
238
3
6.4 Nonholonomic Coordinates, Velocities, etc
241
6
6.4.1 Basic Definitions, Quasi-Coordinates
241
2
6.4.2 Properties of Pfaffian Transformations
243
2
6.4.3 Particle and System Kinematics in Quasi-Variables
245
2
6.5 The Transitivity Equations
247
16
6.5.1 Nonintegrability Conditions
249
3
6.5.2 Comprehensive Examples and Problems on Rigid-Body Kinematics
252
11
6.6 Additional Pfaffian Constraints
263
8
6.6.1 Holonomicity vs. Nonholonomicity
264
1
6.6.2 Scleronomicity vs. Rheonomicity
265
1
6.6.3 Catastaticity vs. Acatastaticity
265
1
6.6.4 Nonholonomic Constraints (Equations 6.6.1)
265
6
6.7 Theorem of Frobenius
271
7
6.8 Geometrical Interpretation of Pfaffian Constraints
278
4
6.8.1 Degrees of Freedom Revisited, Accessibility
280
2
6.9 Geometrical Interpretation of the Frobenius Conditions (Equation 6.7.5e)
282
3
6.9.1 First Interpretation
282
1
6.9.2 Second Interpretation
282
3
Chapter 7 Lagrangean Mechanics: Kinetics
285
86
7.1 Introduction
285
1
7.2 The Fundamental Kinematico-Inertial Quantities
285
16
7.2.1 Kinetic Energy
285
1
7.2.1.1 Holonomic Variables
286
1
7.2.1.2 Nonholonomic Variables
287
1
7.2.2 Metric in Configuration/Event Space
288
2
7.2.3 Acceleration
290
2
7.2.4 Inertia Force: Holonomic Components and Holonomic Euler-Lagrange Operator
292
1
7.2.5 Inertia Force: Nonholonomic Components and Nonholonomic Euler-Lagrange Operator
293
1
7.2.5.1 First Derivation
293
2
7.2.5.2 Second Derivation
295
6
7.3 The Forces
301
2
7.4 The Physical Synthesis: Lagrange's Principle(s), Equations of Motion
303
22
7.4.1 Lagrange's Principle (LP)
303
4
7.4.2 Principle of Relaxation of Constraints (PRC)
307
1
7.4.3 Lagrangean Forms of the Equations of Motion
307
1
7.4.3.1 Holonomic Variables
307
3
7.4.3.2 Nonholonomic Variables
310
4
7.4.4 Appellian Forms of the Equations of Motion
314
1
7.4.5 Summary
314
11
7.5 The Central Equation
325
3
7.6 The Power, or Energy Rate, Equations
328
7
7.6.1 Holonomic Variables
328
2
7.6.2 Nonholonomic Variables
330
5
7.7 Comprehensive Examples and Problems on Lagrangean Dynamics
335
36
Bibliography and References
371
10
Index (Principal Authors and Subjects)
381